Conference Non-Associative Algebras Related TopicsList of Participants 1 LIST OF PARTICIPANTS Marcia Aguiar (University of S~ao Paulo, Brazil) Helena Albuquerque (University of Coimbra,

Program and Abstracts

Conference

Non-Associative Algebrasand

Related Topics

25-29 July 2011Coimbra-Portugal

Foreword

We would like to welcome you to the NAART (Non-Associative Algebras andRelated Topics) and wish you a pleasant stay in our town.

The Theory of Non-Associative Algebras not only constitutes a well-developedand very active area of research but is also an intercurricular one, playing a cen-tral role in Mathematics and Physics. The aim of this conference is to bringtogether mathematicians and physicists interested in this field, with a central goalof increasing the quality of research, promoting interaction among researchers anddiscussing new directions for the future. The conference will last for five days and,besides a series of contributed sessions, there will also be five plenary talks andfour courses. Contributions from all researchers in the area are welcome, includingyoung researchers. The venue will be the historic University of Coimbra, foundedin 1290 - the first university in Portugal and one of the oldest in Europe. Thehost of the event is the Department of Mathematics of the University of Coimbra(DMUC), drawing on the support of several other Portuguese research institutions.

We are deeply indebted to the Centro de Matematica da Universidade de Coim-bra, to the Centro de Matematica da Universidade do Porto, to the Centro deMatematica da Universidade da Beira Interior, to the Fundacao para a Ciencia eTecnologia and also to the Banco Santander Totta for the material support.

Helena AlbuquerqueChair of the Organizing Committee

Contents

Contents i

List of Participants 1

Schedule 3

Schedule of Contributed Talks 5

Mini Courses 9

Plenary Sessions 13

Contributed Talks 18

Posters 61

Index 73

i


LIST OF PARTICIPANTS

Marcia Aguiar (University of Sao Paulo, Brazil)Helena Albuquerque (University of Coimbra, Portugal)Maribel Ramırez Alvarez (Universidad de Almerıa, Spain)Abu Zaid Ansari (AMU, Aligarh, India)Manuel Arenas (Universidad de Chile, Chile)Farkhad Arzikulov (I.M.I.T. , Andizhan State University, Uzbekistan)Raymond Aschheim (Polytopics, France)Imen Ayadi (Universite de Metz, ISGMP, France)Ignacio Bajo (Universidad de Vigo, Spain)Elisabete Barreiro (University of Coimbra, Portugal)Patrıcia Beites (University of Beira Interior, Portugal)Saıd Benayadi (Universite de Metz, ISGMP, France)Georgia Benkart (University of Wisconsin-Madison, USA)Ines Borges (ISCAC, Portugal)Said Boulmane (Faculty of Sciences Meknes, Morocco)Murray Bremner (University of Saskatchewan, Canada)Emmanuel Briand (University of Sevilla, Spain)Paula Carvalho (University of Porto, Portugal)Francesco Catino (University of Salento, Italia)Flavio U. Coelho (University of Sao Paulo, Brazil)Isabel Cunha (University of Beira Interior, Portugal)Erik Darpo (University of Oxford, UK)Jose Marıa Sanchez Delgado (Universidad de Cadiz, Spain)Suleyman Demir (Anadolu University, Turkey)Geoffrey Dixon (USA)Cristina Draper (Universidad de Malaga, Spain)Alberto Elduque (University of Zaragoza, Spain)Vyacheslav Futorny (University of Sao Paulo, Brazil)Maxim Goncharov (Inst. of Math. SB RAS, Novosibirsk, Russia)Edgar Goodaire (Memorial University, Canada)Valerio Guido (University of Salento, Italy)Malika Ait Ben Haddou (Faculty of Sciences Meknes, Morocco)Manuel Avelino Insua Hermo (Dpt. Matematica Aplicada I, University of Vigo, Spain)Radu Iordanescu (Institute of Mathematics of the Romanian Academy, Romania)Clara Jimenez-Gestal (Universidad de La Rioja, Spain)Santos Gonzalez Jimenez (University of Oviedo, Spain)Henrique Guzzo Jr (University of Sao Paulo, Brazil)Ji Hye Jung (Seoul National University, South Korea)Elamin Kaidi (Universidad de Almeria, Spain)Mustafa Emre Kansu (Dumlupinar University, Turkey)Iryna Kashuba (University of Sao Paulo, Brazil)Richard Kerner (University of Paris 6, France)Daehong Kim (Seoul National University, South Korea)Miran Kim (Seoul National University, South Korea)Myungho Kim (Seoul National University, South Korea)


Mikhail Kotchetov (Memorial University of Newfoundland, Canada)Jesus Laliena (Universidad de La Rioja, Spain)Christian Lomp (University of Porto, Portugal)Samuel Lopes (University of Porto, Portugal)Sara Madariaga (University of Logrono, Spain)Shahn Majid (University of London, UK)Consuelo Martınez (University of Oviedo, Spain)Vanesa Meinardi (Universidad Nacional de Cordoba, Argentina)Cesar Polcino Milies (University of Sao Paulo, Brazil)Jose Manuel Casas Miras (Dpt. Matematica Aplicada I, University of Vigo, Spain)Daniel Mondoc (Lund University, Sweden)Sophie Morier-Genoud (Universite Paris 6, France)Alejandro Nicolas (University of Cantabria, Spain)Valentin Ovsienko (CNRS, Universite Lyon 1, Lyon, France)Florin Panaite (Institute of Mathematics of the Romanian Academy, Romania)Sandra Pinto (University of Coimbra, Portugal)Rosemary Miguel Pires (Universidade Federal Fluminense, Brasil)Alexander Pozhidaev (Sobolev Inst. of Math., SB RAS, Novosibirsk, Russia)Natalia Maria Bessa Pacheco Rego (IPCA, Portugal)Roberto Rizzo (University of Salento, Italy)Rodrigo Rodrigues (University of Sao Paulo, Brazil)Hansol Ryu (Seoul National University, South Korea)Liudmila Sabinina (Facultad de Ciencias, UAEM, Mexico)Gil Salgado (UASLP, Mexico)Juana Sanchez-Ortega (University of Malaga, Spain)Ana Paula Santana (University of Coimbra, Portugal)Paulo Saraiva (University of Coimbra, Portugal)Ivan Shestakov (University of Sao Paulo, Brazil)Agata Smoktunowicz (University of Edinburgh, UK)Dragos Stefan (University of Bucharest)Ralph Stohr (University of Manchester, UK)Osamu Suzuki (Nihon University, Japan)Sergei Sverchkov (University of Sao Paulo, Brazil)Irina Sviridova (University of Brasilia, Brazil)Mohammad Reza Molaei Taherabadi (University of Kerman (Shahid Bahonar), Iran)Murat Tanıslı (Anadolu University, Turkey)Maribel Tocon (Universidad de Cordoba, Spain)Arkady Tsurkov (University of Sao Paulo, Brazil)Maria del Carmen Rodriguez Vallarte (UASLP, Mexico)Ivan Yudin (CMUC, University of Coimbra, Portugal)Efim Zelmanov (University of California, San Diego, USA)Pasha Zusmanovich (Tallinn University of Technology, Tallinn, Estonia)

Schedule 3

SCHEDULE

Monday, July 25th

9:00 - 9:50: Registration.9:50 - 10:00: Opening ceremony.

10:00 - 11:00: Opening plenary session: Georgia Benkart,Planar diagram algebras.

11:00 - 11:30: Coffee break.11:30 - 13:00: Mini course 1: Alberto Elduque,

Gradings on simple Lie algebras.

15:00 - 16:30: Contributed talks.16:30 - 17:00: Coffee break.17:00 - 18:00: Contributed talks.

Tuesday, July 26th

9:00 - 10:30: Mini course 2: Alexander Pozhidaev,Differentiably simple (super)algebras.

10:30 - 11:30: Mini course 1 (part two): Alberto Elduque,Gradings on simple Lie algebras.

11:30 - 12:00: Coffee break.12:00 - 13:00: Plenary session: Ivan Shestakov,

On speciality of Malcev algebras.

15:00 - 16:00: Contributed talks.16:00 - 16:30: Coffee break.16:30 - 18:00: Visit to the University of Coimbra.

Wednesday, July 27th

9:00 - 10:30: Mini course 3: Saıd Benayadi,Quadratic (resp. odd-quadratic) Lie superalgebras.

10:30 - 11:30: Mini course 2 (part two): Alexander Pozhidaev,Differentiably simple (super)algebras.

11:30 - 12:00: Coffee break.11:30 - 13:00: Plenary session: Shahn Majid,

Geometry of algebraic quasigroups.

14:30 - 20:00: Tour of Montemor-o-Velho and Figueira da Foz.20:00: Conference dinner.

Schedule 4

Thursday, July 28th

9:30 - 10:30: Mini course 4: Vyacheslav Futorny,Kac-Moody algebras, vertex operators and applications.

10:30 - 11:30 : Mini course 3 (part two): Saıd Benayadi,Quadratic (resp. odd-quadratic) Lie superalgebras.

11:30 - 12:00: Coffee break.12:00 - 13:00: Plenary session: Consuelo Martınez,

Cheng Kac superalgebras.

15:00 - 16:30: Contributed talks.16:30 - 17:00: Coffee break.17:00 - 17:30: Contributed talks.17:30 - 18:30: Poster Session.

Friday, July 29th

9:00 - 10:30: Mini course 4 (part two): Vyacheslav Futorny,Kac-Moody algebras, vertex operators and applications.

10:30 - 11:30: Contributed talks.11:30 - 12:00: Coffee break.12:00 - 13:00: Contributed talks.

15:00 - 16:00: Closing plenary session: Efim Zelmanov,Jordan superalgebras.

Schedule of contributed talks 5

SCHEDULE OF CONTRIBUTED TALKS

Monday, July 25th

Room 2.3 (Chair: Florin Panaite)

15:00 - 15:25: Murat Tanıslı,Maxwell equations for magnetic sources and massive photon in octonion algebra,

15:30 - 15:55: Mustafa Emre Kansu,Electromagnetism with complex octonions,

16:00 - 16:25: Richard Kerner,Space-time symmetry groups derived from cubic and ternary algebras,

17:00 - 17:25: Rosemary Miguel Pires,Code loops: automorphisms and representations,

17:30 - 17:55: Suleyman Demir,Hyperbolic octonionic massive gravitoelectromagnetism with monopoles.

Room 2.4 (Chair: Alberto Elduque)

15:00 - 15:25: Flavio U. Coelho,The representation dimension of Artin algebras,

15:30 - 15:55: Ignacio Bajo,Indefinite Kahler metrics on Lie algebras with abelian complex structure,

16:00 - 16:25: Mikhail Kotchetov,Weyl groups of fine gradings on Lie algebras,

17:00 - 17:25: Mohammad Reza Molaei Taherabadi,A generalization of left invariant vector fields and Lie algebras generated by them.

Room 2.5 (Chair: Henrique Guzzo Jr)

15:00 - 15:25: Murray Bremner,Malcev and Bol dialgebras,

15:30 - 15:55: Maxim Goncharov,Structures of Malcev Bialgebras on a simple non-Lie Malcev algebra,

16:00 - 16:25: Manuel Arenas,On speciality of binary-Lie algebras,

17:00 - 17:25: Liudmila Sabinina,Moufang loops and their automorphisms.


Tuesday, July 26th

Room 2.3 (Chair: Daniel Mondoc)

15:00 - 15:25: Irina Sviridova,Algebras with involution and their identities,

15:30 - 15:55: Juana Sanchez-Ortega,Leibniz triple systems.

Room 2.4 (Chair: Ignacio Bajo)

15:00 - 15:25: Ralph Stohr,On linear equations over free Lie algebras,

15:30 - 15:55: Vanesa Meinardi,Finite growth representations of infinite Lie conformal algebras.

Room 2.5 (Chair: Richard Kerner)

15:00 - 15:25: Iryna Kashuba,Representation type of Jordan algebras,

15:30 - 15:55: Jesus Laliena,The derived algebra of skew elements in a semiprime superalgebra with superinvolution.


Thursday, July 28th

Room 2.3 (Chair: Saıd Benayadi)

15:00 - 15:25: Elisabete Barreiro,Homogeneous symmetric antiassociative quasialgebras,

15:30 - 15:55: Gil Salgado,Heisenberg Lie superalgebras and its invariant superorthogonal and supersymplectic forms,

16:00 - 16:25: Manuel Avelino Insua Hermo,On (co)homology of Hom-Leibniz algebras.

Room 2.4 (Chair: Ivan Shestakov)

15:00 - 15:25: Erik Darpo,Four-dimensional power-commutative real division algebras,

15:30 - 15:55: Henrique Guzzo Jr,Multiplicative mappings of alternative rings,

16:00 - 16:25: Sara Madariaga,Abelian groups in the slice category of bialgebras,

17:00 - 17:25: Valentin Ovsienko,A series of algebras generalizing the octonions and Hurwitz-Radon identity.

Room 2.5 (Chair: Liudmila Sabinina)

15:00 - 15:25: Arkady Tsurkov,Strongly stable automorphisms of the category of free linear algebras,

15:30 - 15:55: Ana Paula Santana,Exact sequences in the Borel-Schur algebra,

16:00 - 16:25: Ivan Yudin,Normalized bar resolution and Schur algebra.


Friday, July 29th

Room 2.3 (Chair: Manuel Arenas)

10:30 - 10:55: Raymond Aschheim,Non associative quantum gravity,

11:00 - 11:25: Osamu Suzuki,Nonassociative algebra associated to genetics and Jordan algebra,

12:00 - 12:25: Sergei Sverchkov,Algebraic theory of DNA recombination,

12:30 - 12:55: Maribel Tocon,A capacity 2 theorem for graded Jordan systems.

Room 2.4 (Chair: Consuelo Martınez Lopez)

10:30 - 10:55: Abu Zaid Ansari,Lie ideals and generalized derivations in semiprime rings,

11:00 - 11:25: Geoffrey Dixon,Octonions, lattices, and preferred parentheses,

12:00 - 12:25: Daniel Mondoc,On generalized Jordan triple systems of second order and extended Dynkin diagrams,

12:30 - 12:55: Florin Panaite,Alternative twisted tensor products and Cayley algebras.

Room 2.5 (Chair: Jesus Laliena)

10:30 - 10:55: Cesar Polcino Milies,Alternative loop algebras over finite fields,

11:00 - 11:25: Pasha Zusmanovich,A commutative 2-cocycles approach to classification of simple Novikov algebras.

Mini Courses 9

Non-Associative Algebras and Related TopicsCoimbra, July 25–29, 2011

Quadratic (resp. odd-quadratic) Lie

superalgebras

Saıd Benayadi1

A quadratic (resp. odd-quadratic) Lie superalgebra is a Lie superalgebra with a non-degenerate supersymmetric, even (resp. odd), invariant bilinear form. Quadratic (resp.odd-quadratic) Lie superalgebras appear in particular in differential (super-)geometryand in physical models based on Lie superalgebras. The classification of quadratic (resp.odd-quadratic) classical simple Lie superalgebras had been obtained by V. Kac. Butmany solvable Lie superalgebras also belong to this class. In this mini course:

1. We will construct some non-trivial examples of quadratic (resp. odd-quadratic)Lie superalgbras.

2. Methods of double extension of quadratic (resp. odd-quadratic) Lie superalgebraswill be developed.

3. We will give some inductive descriptions of quadratic (resp. odd-quadratic) Liesuperalgebras.

4. We will introduce the quadratic dimension of a quadratic Lie superalgebra andwe will establish some relations between this invariant and others invariants ofquadratic Lie superalgebras.

5. At the end, we will discuss some open problems on quadratic (resp. odd-quadratic)Lie superalgebras.

1Laboratoire de Mathematiques et Applications de Metz, CNRS UMR 7122,Universite Paul Verlaine-Metz, Ile du Saulcy, F-57045 Metz cedex 1, [email protected]

Mini Courses 10


Gradings on simple Lie algebras

Alberto Elduque1

A survey of some recent classification results for gradings over abelian groups onfinite dimensional simple Lie algebras over algebraically closed fields will be given. Thiswill require the classification of gradings on matrix algebras, on the algebra of octonionsand on the Albert algebra (exceptional simple Jordan algebra).

Any grading of a finite dimensional algebra over an abelian group is equivalent toa representation of a diagonalizable affine group scheme on the affine group scheme ofautomorphisms of the algebra. This point of view, necessary to deal with the modularcase, will be stressed throughout.

1Departamento de Matematicas e Instituto Universitario de Matematicas y Apli-caciones, Universidad de Zaragoza, 50009 Zaragoza, [email protected]

Mini Courses 11


Kac-Moody algebras, vertex operators and

applications

Vyacheslav Futorny1

This mini-course will focus on vertex type constructions for certain classes of infinite-dimensional Lie algebras, including Affine Kac-Moody algebras and elliptic Affine alge-bras (the latter are particular cases of Krichever-Novikov algebras associated with ellipticcurves). During the first lecture we will discuss some aspects of the representation theo-ry of Kac-Moody algebras based on free field realizations of Affine Lie algebras andtheory of vertex algebras. Vertex algebras have origin in string theory, they provide amathematical foundation of 2-dimensional conformal field theory. Vertex algebras havenumerous applications in many areas of mathematics, in particular they are ubiquitousin representation theory of infinite-dimensional Lie algebras. Some generalizations ofvertex constructions for Affine Lie algebras will be considered. In the second lecturevertex realizations of Affine Lie algebras will be applied to the study of representationsof the Lie algebra of vector fields on N-dimensional torus (these are joint results with Y.Billig). Finally, we will discuss free field realizations of elliptic Lie algebras which wererecently obtained in a joint work with B. Cox and A. Bueno.

1Instituto de Matematica e Estatıstica, Universidade de Sao Paulo, Caixa Postal66281, Sao Paulo, CEP 05315-970, [email protected]

Mini Courses 12


Differentiably simple (super)algebras

Alexander Pozhidaev1

A superalgebra A is called differentiably simple if it lacks homogeneous ideals in-variant under Der(A). In this talk we discuss the theory of differentiably simple (su-per)algebras and some its applications.

1Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk,[email protected]

Plenary sessions 13


Planar diagram algebras

Georgia Benkart1

Planar diagram algebras arise in the representation theory of low-rank Lie algebras,Lie superalgebras and quantum groups. They arise in statistical mechanics and haveplayed a prominent role in Jones’ work on subfactors of von Neumann algebras and oninvariants of knots and links. This talk will focus on planar diagram algebras, someintroduced in our recent joint work with Halverson, and on their combinatorial connec-tions with various well-studied sequences of numbers such as the Catalan and Motzkinnumbers.

1University of Wisconsin-Madison, [email protected]

Plenary sessions 14


Geometry of algebraic quasigroups

Shahn Majid1

Building on recent work we use a Hopf-algebra like theory of “Hopf coquasigroups”to study quasigroups such as S7. As well as tangent and cotangent bundles we now lookat further aspects of the geometry such as metrics and other bundles, using algebraicmethods previously developed for quantum groups.

1University of London, [email protected]

Plenary sessions 15


Cheng Kac superalgebras

Consuelo Martınez1

Cheng-Kac superconformal algebra CK(6) was found by S. J. Cheng and V. Kac in1997 and independently by Grozman, Leites and Shchepochkina in 2001. Its existenceobliged to reformulated the previous conjecture by V. Kac and van de Leur about thestructure of superconformal algebras. In 2001 they appear in a more general context inthe classification of simple Jordan superalgebras in prime characteristic in the case ofnon-semisimple even part.

We will explain what is know about those superalgebras and related structures. Wewill pay also special attention to the case of prime characteristic, explaining some resultsobtained jointly with E. Barreiro and A. Elduque.

1Departamento de Matematicas, Universidad de Oviedo, Oviedo, [email protected]

Plenary sessions 16


On speciality of Malcev algebras

Ivan Shestakov1

A Malcev algebra is called special if it can be embedded into a commutator algebraof a certain alternative algebra. We will give some new results related with the problemof speciality of Malcev algebras.

1University of Sao Paulo, [email protected]

Plenary sessions 17


Jordan superalgebras

Efim Zelmanov1

I will try to give a broad survey of the theory of Jordan superalgebras and theirconnections.

1University of California, San Diego, [email protected]



Lie ideals and generalized derivations in

semiprime rings †

Nadeem Ur Rehman1, Abu Zaid Ansari 1

Let R be an associative ring with center Z(R). For each x, y ∈ R denote the commu-tator xy−yx by [x, y] and the anti-commutator xy+yx by x◦y. An additive subgroup Lof R is said to be Lie ideal of R if [L,R] ⊆ L. An additive mapping F : R→ R is calledgeneralized inner derivation if F (x) = ax + xb for fixed a, b ∈ R. For such a mappingF , it is easy to see that F (xy) = F (x)y+ xIb(y) for all x, y ∈ R. This observation leadsto the following definition given in [Communication Algebra 26(1998), 1149-1166]. Anadditive mapping F : R→ R is called generalized derivation with associated derivationd if F (xy) = F (x)y + xd(y) for all x, y ∈ R. In the present paper we shall show thatL ⊆ Z(R) such that R is semiprime ring satisfying several conditions.

Keywords: Derivations, generalized derivations, prime rings, semiprime rings, Lieideals

Mathematics Subject Classification 2010: 16W25, 16N60, 16U80

References

[1] Ashraf M., Ali A. and Ali S., Some commutativity theorems for rings with generalizedderivations, Southeast Asian Bull. Math. 32(2) (2007),415-421.

[2] Ashraf M. Khan A., Lie ideals and generalized derivations of semiprime rings, (to appear).

[3] Ashraf M., Rehman N., On derivations and commutativity in prime rings, East west J.Math. 3(1) (2001) 87-91.

[4] Ashraf M., Rehman N., On commutativity of rings with derivations, Results Math. 42(2002),3-8.

[5] Argac, N., On prime and semiprime rings with deivations, Algebra Colloq. 13(3) (2006),371-380.

[6] Awtar R., Lie and Jordan structures in prime rings with derivations, Proc. Amer. Math.Soc. 41 (1973) 67-74.

[7] Awtar R., Lie structure in prime rings with derivations, Publ. Math. (Debrecen) 31 (1984)209-215.

[8] Bell, H.E. and Daif, M.N. On derivations and commutativity in prime rings, Acta Math.Hungar. 66(4)(1995), 337-343.

[9] Bell, H.E. and Rehman, N., Generalized derivations with commutativity and anti- commu-tativity conditions, Math. J. Okayama Univ. 49(2007), 139-147.

†This research is supported by UGC, India, Grant No. 36-8/2008(SR).


[10] Bergen, J., Herstein, I.N. and Kerr, J.W., Lie ideals and derivations of prime rings, J.Algebra 71(1981), 259-267.

[11] Bresar, M., On the distance of the compositions of two derivations to the generalized deriva-tions, Glasgow Math. J.33(1)(1991), 89-93.

[12] Daif, M.N. and Bell, H.E. Remarks on derivations on semiprime rings, Internat. J. Math.Math. Sci. 15 (1) (1991) 204-206.

[13] Deng, Q. and Ashraf, M., On strong commutativity preserving mappings, Results in Math.30(1996), 259-263.

[14] Herstein, I.N., On the Lie structure of associative rings, J. Algebra 14(4) (1970) 561-571.

[15] Herstein, I.N., Topics in ring theory, Univ. Chicago Press, Chicago (1969).

[16] Herstein, I.N., Rings with invilotion, Univ. Chicago Press, Chicago and Londan (1976).

[17] Hongan, M., Rehman, N. and Al Omary, R., Lie ideals and Jordan triple derivations ofrings, (to appear)

[18] Posner, E.C., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.

[19] Quadri, M.A. , Khan, M.S. and Rehman, N., Generalized derivations and commutativity ofprime rings, Indian J. Pure Appl. Math. 34(9)(2003), 1393-1396.

[20] Rehman, N., On commutativity of rings with generalized derivations, Math. J. Okayama

Univ. 44(2002) 43-49.

1Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, [email protected], [email protected]



On speciality of binary-Lie algebras

Manuel Arenas ‡1

In the present talk, I will present some results obtained in a joint work with ProfessorIvan Shestakov. We proved that, for every assocyclic algebra A, the algebra A− is binary-Lie. We found a simple not Malcev binary-Lie superalgebra T that can not be embeddedin A−s for any assocyclic superalgebra A. We used the Grassman envelope of T to provethe corresponding result for algebras. This solve negatively a problem by V.T. Filippovfrom [1]. Finally, we proved that the superalgebra T is isomorphic to the commutatorsuperalgebra A−s for a simple binary (-1,1) superalgebra A.

Keywords: Assocyclic algebra, binary-Lie algebra, speciality problem, superalgebra,(-1,1)-algebra.

Mathematics Subject Classification 2010: 17D02

References

[1] The Dniestr notebook. Unsolved problems in the theory of rings and modules. (Dnes-trovskaya tetrad’. Nereshennye problemy teorii kolets i modulej). ed. by V. T. Filippov,V. K. Kharchenko, I. P. Shestakov, 4th ed. (Russian), Novosibirsk: Institut Matematiki SORAN, 74 p. (1993).English transl. in Non-associative algebra and its applications, edited by L. V. Sabinin,L. Sbitneva, I. Shestakov, Proceedings of the 5th international conference, Oaxtep, Mexico,July 27 - August 2, 2003. Lecture Notes in Pure and Applied Mathematics 246, (2006).

[2] A. T. Gainov, Identical relations for binary-Lie rings, (Russian) Uspehi Mat. Nauk (N.S.)12, 3 (75), 141-146, (1957).

[3] E. Kleinfeld and L. Widmer, Rings satisfying (x, y, z) = (y, z, x), Comm. in algebra, 17(11),2683-2687, (1989).

[4] E. N. Kuzmin, Binary Lie Algebras of Small Dimentions, Algebra and Logic, vol. 37 (1998),No 3, 181-186.

[5] I. P. Shestakov, Superalgebras and counter-examples, Sibirskii Matem. Zh., 32, N 6 (1991),187–196; English transl.: Siberian Math.J. 32, N 6 (1991), 1052–1060.

[6] I. P. Shestakov, Simple (-1,1) superalgebras, (Russian) Algebra i Logika, 37, N 6 (1998),

721–739; English transl.: Algebra and Logic 37 (1998), no.6, 411–422.

1Departamento de Matematicas, Facultad de Ciencias, Universidad de Chile, LasPalmeras 3425, Nunoa, Santiago, [email protected], [email protected]

‡supported by FONDECYT grant 11100092



Non associative quantum gravity

Raymond Aschheim1

Non associative algebras have an emergent and fundamental place in quantum gra-vity. I will first present the infinion algebra, which is the limit for large n of algebras buildby Cayley-Dickson process, and give explicit table for its restriction to 1024 dimensionsand general construction; then review three possible applications in quantum gravity.Their imaginary part can replace su(2) in Regge Calculus for embedded spinfoams [1],and in Loop Quantum Gravity for abstract spinfoams [2], so that a non associative be-havior emerging in dimensions 5 to 8 extend naturally the spacetime to internal encodingof all fermions, with e8 roots as quantum numbers. This concept of matter as an effectof spacetime nonassociativity is also shown in the new framework of relative locality [3],where we propose to use logarithms of imaginary infinions as momentum space basis.

Keywords: division algebra, quantum gravity

Mathematics Subject Classification 2010: 17A35, 17B81

References

[1] E. Maglioro and C. Perini, Regge gravity from spinfoams, in Proceedings of LOOPS11International Conference on Quantum Gravity, J. of Physics: Conference Series to ap-pears in (2011), available in arxiv:1105.0216v6 [gr-qc]. ,

[2] R. Aschheim, Spin Foam with topologically encoded tetrad on trivalent spin networks,in Proceedings of LOOPS11 International Conference on Quantum Gravity, J. of Physics:Conference Series to appears in (2011). ,

[3] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, The

principle of relative locality, in Proceedings of LOOPS11 International Conference on

Quantum Gravity, J. of Physics: Conference Series to appears in (2011), available in

arxiv:1101.0931 [hep-th].

1POLYTOPICS, 8 villa Haussmann, 92130 Issy, [email protected]



Indefinite Kahler metrics on Lie algebras with

abelian complex structure

Ignacio Bajo1

We study indefinite Kahler metrics on Lie algebras endowed with abelian complexstructures. It is seen that each of those Lie algebras is completely determined by anassociative and commutative complex algebra admitting a special hermitian form. Wepropose several constructions and provide an inductive description of such Lie algebras.As a geometrical application, the curvatures of the pseudo-Kahler metric are computedand sufficient conditions to ensure flatness or Ricci-flatness are given.

1Departamento de Matematica Aplicada II, E.T.S.I. Telecomunicacion, Univer-sidad de Vigo, 36280 Vigo, [email protected]



Homogeneous symmetric antiassociative

quasialgebras

Helena Albuquerque1, Elisabete Barreiro1, Saıd Benayadi2

This talk will focus on homogeneous symmetric antiassociative quasialgebras, mea-ning antiassociative quasialgebras equipped with a non-degenerate, supersymmetric,homogeneous (even or odd), associative bilinear form. We recall that any Z2-gradedquasialgebra is either an associative superalgebra or an antiassociative quasialgebra.Symmetric homogeneous associative superalgebras were studied by I. Ayadi on her doc-toral work under S. Benayadi’s supervision [3, 4]. We show that any odd-symmetricantiassociative quasialgebra is, in particular, an associative superalgebra. On even-symmetric antiassociative quasialgebras we present some notions of generalized doubleextensions in order to give inductive descriptions of this class of superalgebras. Ourgoal is to complete the study of Z2-graded quasialgebras provided with homogeneoussymmetric structures.

Keywords: antiassociative quasialgebras, homogeneous symmetric structures, genera-lized double extension, inductive description

References

[1] H. Albuquerque, A. Elduque, and J.M. Perez-Izquierdo, Z2-quasialgebras, Comm. Algebra30 (2002), 2161–2174.

[2] H. Albuquerque and Shahn Majid, Quasialgebra structure of the octonions, J. Algebra 220(1999), 188–224.

[3] I. Ayadi, “Super-algebres non Associatives avec des Structures Homogenes”, Doctoral The-sis, Universite Paul Verlaine-Metz, 2011.

[4] I. Ayadi and S. Benayadi, Symmetric Novikov superalgebras J. Math. Phys. 51 (2010), no.2, 023501, 15 pp.

1CMUC, Departamento de Matematica, Universidade de Coimbra, 3001-454Coimbra, [email protected], [email protected]

2Laboratoire LMAM, CNRS UMR 7122, Universite Paul Verlaine-Metz, Ile duSaulcy, F-57012 Metz cedex 01, [email protected]



Malcev and Bol dialgebras

Murray R. Bremner1

We apply the algorithm of Kolesnikov to the defining identities for Malcev algebrasto determine the defining identities for Malcev dialgebras. We then use computer algebrato show that these identities are equivalent to the identities of degree ≤ 4 satisfied bythe dicommutator in every alternative dialgebra. We generalize Kolesnikov’s algorithmto the case of general multioperator algebras, and apply this to the defining identitiesfor Bol algebras to determine the defining identities for Bol dialgebras. We then usecomputer algebra to show that these identities are equivalent to the identities of degree≤ 5 satisfied by the dicommutator and the Jordan diassociator in every right alternativedialgebra. This is joint work with Alexander Pozhidaev, Luiz Peresi, and Juana Sanchez-Ortega.

1Department of Mathematics and Statistics, University of Saskatchewan 106Wiggins Road (McLean Hall) Saskatoon, SK, S7N 5E6 [email protected]



The representation dimension of Artin algebras

Flavio Ulhoa Coelho1

The representation dimension of an algebra was introduced by Maurice Auslanderin the 70’s of the last century with the aim of measuring how far an algebra is to berepresentation-finite [1]. Recall that an algebra A is representation finite provided thereare only finitely many non-isomorphic indecomposable finitely generated A-modules.However, more than 25 years passed before it was considered again in the mainframeof the Representation Theory of Algebras. We shall give a very quick survey on thedevelopment of such notion and end our talk with new results on two classes of algebras:(i) Iterated tilted algebras, jointly with D. Happel and L. Unger [3]; and (ii) Stronglysimply connected algebras of polynomial growth, jointly with I. Assem and S. Trepode[2].

Keywords: representation dimension, Artin algebras

Mathematics Subject Classification 2010: 16G70, 16G20, 16E10

References

[1] M. Auslander, Representation dimension of artin algebras, Math. Notes, Queen Mary Col-lege, London (1971).

[2] I. Assem, F. U. Coelho and S. Trepode, The representation dimension of a class of tamealgebras, Preprint, 2010.

[3] F. U. Coelho, D. Happel and L. Unger, Auslander generators for iterated tilted algebras,Proc. Amer. Math. Soc. 138 (2010), 1587-1593.

1Departamento de Matematica - IME, Universidade de Sao Paulo, Rua do Matao,1010, Cidade Universitaria, Sao Paulo, 05508-090, SP, [email protected]



Four-dimensional power-commutative real

division algebras

Erik Darpo1

An algebra is said to be power-commutative if any subalgebra generated by a sin-gle element is commutative. While this is always true for associative algebras, thereare many examples of non-associative algebras failing to satisfy the power-commutativecondition, as well as power-commutative algebras that are not associative. However,most types of algebras that have been studied hitherto are power-commutative: al-ternative algebras, Lie algebras, non-commutative Jordan algebras, flexible algebras,power-associative algebras etc.

In the talk, I shall present a classification of all four-dimensional power-commutativereal division algebras, which naturally extends existing classifications in the power-associative and flexible cases.

1Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, [email protected]



Hyperbolic octonionic massive

gravitoelectromagnetism with monopoles

Suleyman Demir1, Murat Tanıslı1

The non-associative hyperbolic octonions can be expressed in the sub-algebra of conicsedenions [1]. This type of octonions differs from classical octonions because of havingboth imaginary and hyperbolic basis. The gravitoelectromagnetism term states thetheoretical analogy between the field equations in gravity and Maxwell’s equations inelectromagnetism. In relevant literature, the possible relations between gravitation andelectromagnetism have been formulated by using many hypercomplex number systemsincluding biquaternions [2, 3], octonions [4], Clifford numbers [5] and sedenions [6].In this study, after presenting the hyperbolic octonion formalism, a new formulationis proposed for the generalization of massive gravitoelectromagnetism with monopoleterms. The gravitoelectromagnetic wave equation including Proca-type generalizationand monopole terms is obtained in compact and elegant manner. Similarly, by usinghyperbolic octonion formalism, the most generalized form of Klein-Gordon equation ingravity has been developed for the particle carrying gravitational masses.

Keywords: Hyperbolic octonion, gravitoelectromagnetism, Proca equation

Mathematics Subject Classification 2010: 78A25, 83C22, 83C50, 83E99

References

[1] K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions-further re-sults, Appl. Math. Comput. 84 (1997), 27–47.

[2] V. Majernik, Some astrophysical consequences of the extented Maxwell-like gravitationalfield equations, Astrophys. Space. Sci. 84 (1982), 191–204.

[3] S. Demir and M. Tanıslı, Biquaternionic Proca-type generalization of gravity, Eur. Phys.J. Plus, 126 (2011), no. 5, 51–58.

[4] P. S. Bisht, S. Dangwal and O. P. S. Negi, Unified split octonion formulation ofdyons, Int. J. Theor. Phys., 47 (2008) 2297–2313.

[5] S. Ulrych, Gravitoelectromagnetism in a complex Clifford algebra, Phys. Lett. B, 633,(2006) 631–635.

[6] J. Koplinger, Gravity and electromagnetism on conic sedenions, Appl. Math. Comput.

188 (2007), no. 1, 948–953.

1Anadolu University, Science Faculty, Department of Physics, 26470, Eskisehir,[email protected], [email protected]



Octonions, lattices, and preferred parentheses

Geoffrey Dixon1

The octonions are the poster child for algebraic non-associativity, but although ingeneral, x(yz) 6= (xy)z, and there seems to be no reason to prefer one placement of paren-theses over the other, a preference does arise when providing E8 lattices, representedover O, algebraic structures.

[email protected]



Structures of Malcev bialgebras on a simple

non-Lie Malcev algebra§

Maxim Goncharov1

Lie bialgebras were introduced by Drinfeld [1] while studying the solutions to theclassical Yang-Baxter equation. In [3], the definition of a bialgebra in the sense ofDrinfeld (D-bialgebra) related with any variety of algebras was given.

Malcev algebras were introduced by Malcev as tangent algebras for local analyticMoufang loops. The class of Malcev algebras generalizes the class of Lie algebras.

In [2] some properties of Malcev bialgebras were studied. In particular, there werefound conditions for a Malcev algebra with a comultiplication to be a Malcev bialgebra.In the present work we describe all Malcev bialgebra structures on a simple non-LieMalcev algebra.

Keywords: Lie bialgebra, Malcev bialgebra, classical Yang-Baxter equation, nonas-sociative coalgebra, simple non-Lie Malcev algebra

Mathematics Subject Classification 2010: 17D10, 17B62,16T25

References

[1] V.G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometricmeaning of the classical Yang-Baxter equation, Sov, Math, Dokl. 27, (1983), 68–71.

[2] V.V. Vershinin, On Poisson-Malcev Structures, Acta Applicandae Mathematicae 75 (2003),281–292.

[3] V.N. Zhelyabin, Jordan bialgebras and their relation to Lie bialgebras, Algebra and logic

36 (1997), 1-15.

1Sobolev Institute of Mathematics, Novosibirsk, [email protected]

§The author was supported by Lavrent’ev Young Scientists Competition (No 43 on04.02.2010), by the Program ”Development of scientific potential of the higher school” (project2.1.1.10726), Russian Foundation for basic Research (Grant 09-01-00157-a, 11-01-00938-), theState Support Programs for the Leading Scientific Schools and the Young Scientists of theRussian Federation (Grands Nsh-3669.2010.1), and the Federal Targeting Programs (contracts02.740.11.0429, 02.740.11.5191, 14.740.11.0346), the Integration Grant of the Siberian Divisionof the Russian Academy of Sciences (No. 97).



Multiplicative mappings of alternative rings

H. Guzzo Jr1, J. C. M. Ferreira2

Let R and S be arbitrary alternative rings. A mapping ϕ of R onto S is calleda multiplicative isomorphism if ϕ is bijective and satisfies ϕ(xy) = ϕ(x)ϕ(y) for allx, y ∈ R. In this work we prove the following result, let R′

be an alternative ringcontaining a family Λ of idempotents. Suppose that R′

has a nonzero subring R whichsatisfies:

(i) For each e ∈ Λ, eR ⊆ R and Re ⊆ R;

(ii) If x ∈ R is such that xR = 0, then x = 0;

(iii) If x ∈ R is such that (eR)x = 0 (or e(Rx) = 0) for all e ∈ Λ, then x = 0 (andhence Rx = 0 implies x = 0);

(iv) For each e ∈ Λ and x ∈ R, if (exe)(R(1− e)

)= 0 then exe = 0.

Then any multiplicative isomorphism ϕ of R onto an arbitrary alternative ring is addi-tive.

As an application we generalize the famous Martindale’s result.

Keywords: Alternative rings, multiplicative mappings

Mathematics Subject Classification 2010: 17D05, 17A36

References

[1] F. Y. Lu and J. H. Xie, Multiplicative Mappings of Rings, Acta MathematicaSinica 22 (2006), 1017–1020.

[2] W. S. Martindale III, When are multiplicative mappings additive?, Proc. Amer.Math. Soc. 21 (1969), 695–698.

[3] M. Slater, Prime alternative rings, I, Journal of Algebra 15 (1970), 229–243.

1Universidade de Sao Paulo, Departamento de Matematica, Rua do Matao, 1010,05508-090 - Sao Paulo, [email protected]

2Universidade Federal do ABC, Centro de Matematica, Computacao e Cognicao,Rua Santa Adelia, 166, 09210-170 - Santo Andre, [email protected]



On (co)homology of Hom-Leibniz algebras

Jose Manuel Casas1 ¶, Manuel Avelino Insua1, Natalia Pacheco2

The Hom-Lie algebra structure was initially introduced in [2] motivated by examplesof deformed Lie algebras coming from twisted discretizations of vector fields. Hom-Liealgebras are K-vector spaces endowed with a bilinear skew-symmetric bracket satisfyinga Jacobi identity twisted by a map. When this map is the identity map, then thedefinition of Lie algebra is recovered.

Following the generalization in [4] from Lie to Leibniz algebras, it is natural todescribe same generalization in the framework of Hom-Lie algebras. In this way, thenotion of (multiplicative) Hom-Leibniz algebra was firstly introduced in [3] as K-vectorspaces L together with a linear map α : L → L, endowed with a bracket operation[−,−] : L⊗ L→ L which satisfies (α[x, y] = [α(x), α(y)]) the Hom-Leibniz identity

[α(x), [y, z]] = [[x, y], α(z)]− [[x, z], α(y)], for all x, y, z ∈ L (1)

Our goal in the present talk is to present the construction of (co)homology (co)chaincomplexes which allow us the computation of (co)homologies of multiplicative Hom-Leibniz algebras, its application to classify a particular type of abelian extensions anddevelop a theory of universal central extensions.

We introduce the notions of representation and co-representation, which are the ade-quate coefficients for defining the co-chain and chain complexes from which we computethe cohomology and homology of a Hom-Leibniz algebra with coefficients. We establishthe equivalence between the categories of representations and co-representations, we in-terpret low dimensional (co)homology vector spaces, in particular the second cohomologyis the set of isomorphism classes of abelian α-extensions (the structure of representationsinduced by the extension doesn’t coincide with the previous action, but it coincides overthe image of α), and we establish the relationships with Hom-Lie algebra (co)homology.We use the homology complex to construct universal central extensions of perfect Hom-Leibniz algebras. Our second homology appears as the kernel of the universal centralextension.

When we write the results obtained for the particular case α = Id, then we recoverthe corresponding results for Leibniz algebras in [4, 5].

Keywords: Hom-Leibniz algebra, (co)-representation, (co)homology, extensions

Mathematics Subject Classification 2010: 17A32, 16E40

References

[1] J. M. Casas, M. A. Insua and N. Pacheco, (Co)homology of Hom-Leibniz algebras,submitted to Linear Algebra and its Applications (2011).

¶(1) supported by MICINN, grant MTM2009-14464-C02-02 (European FEDER support in-cluded) and by Xunta de Galicia, grant Incite09 207 215 PR


[2] J. T. Hartwing, D. Larson and S. D. Silvestrov, Deformations of Lie algebras usingσ-derivations, J. Algebra 295 (2006), 314–361.

[3] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2(2008), no. 2, 51–64.

[4] J.-L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz,L′Enseignement Mathematique 39 (1993), 269–292.

[5] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and

(co)homology, Math. Ann. 296 (1993), 139–158.

1Dpto. Matematica Aplicada I, Univ. de Vigo, E. I. Forestal, 36005 Pontevedra,[email protected], [email protected]

2IPCA, Dpto. de Ciencias, Campus do IPCA, Lugar do Aldao, 4750-810 VilaFrescainha, S. Martinho, Barcelos, [email protected]



Electromagnetism with complex octonions

Mustafa Emre Kansu1, Murat Tanıslı2, Suleyman Demir2

Octonions, which have both non-commutative and non-associative algebraic pro-perties, generate alternative division algebra [1]. In this study, the general informationabout complex octonions are introduced by using Cayley-Dickson multiplication rules.Maxwell’s equations, which are very important for electromagnetism, are formed withoutelectric and magnetic charge densities. Local energy conservation equation, which havepreviously produced in biquaternionic form [2], is rearranged by using complex octonions.By defining complex octonionic differential operator and field equation, local energyconservation equation is rewritten. So, the electromagnetic energy density and flow areobtained [3].

Keywords: Octonion, Maxwell’s equations, electromagnetic energy

Mathematics Subject Classification 2010: 08Axx, 35Q61

References

[1] S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics, Cam-bridge University Press, Cambridge, UK (1995); J. C. Baez, The Octonions, Bull. Amer.Math. Soc. 39 (2002), 145–205.

[2] M. Tanıslı, Gauge transformation and electromagnetism with biquaternions, EurophysicsLetters 74 (2006), issue 4, 569–573.

[3] J. D. Jackson, Classical Electrodynamics (Third Edition), John Wiley - Sons Inc., New

York, U.S.A. (1999).

1Dumlupınar University, Faculty of Art and Science, Department of Physics,43100, Kutahya, [email protected]

2Anadolu University, Science Faculty, Department of Physics, 26470, Eskisehir,[email protected], [email protected]



Representation type of Jordan algebras

Iryna Kashuba1

This talk is devoted to the problem of the classification of indecomposable Jordanbimodules over finite dimensional Jordan algebras when squared radical is zero. Recall,that to each Jordan algebra corresponds the Lie algebra TKK(J) called Tits-Kantor-Koecher construction. Moreover the category of one-sided Jordan (bi)modules over J isequivalent to some subcategory S 1

2of Lie modules over TKK(J). We describe the quivers

of S 12

and consequently give a characterization of Jordan algebras of finite and tame type

with respect to their one-sided bimodules. This is joint results with V. Serganova.

1University of Sao Paulo, [email protected]



Space-time symmetry groups derived from cubic

and ternary algebras

Richard Kerner1

We investigate certain Z3-graded associative algebras with cubic Z3 invariant cons-titutive relations, introduced by one of us some time ago. The invariant forms on finitealgebras of this type are given in the lowest-dimensional cases with two and three gene-rators. Various types of non-associative ternary algebras of such forms are investigated.We show how the Lorentz symmetry represented by the SL(2, C) group can be intro-duced without any notion of metric, just as the symmetry of Z3-graded cubic algebraand its constitutive relations. Its representation is found in terms of the Pauli matrices.The relationship of such algebraic constructions with quark states is also considered.

1Laboratoire de Physique Theorique de la Matiere Condensee, Universite Pierre-et-Marie-Curie - CNRS UMR 7600 Tour 23, 5-eme etage, Boıte 121, 4, PlaceJussieu, 75005 Paris, [email protected]



Weyl groups of fine gradings

Mikhail Kotchetov1

Given a grading Γ : U =⊕

g∈G Ug on a nonassociative algebra U by an abelian groupG, we have two subgroups of Aut(U): the automorphisms that stabilize each componentUg (as a subspace) and the automorphisms that permute the components. By the Weylgroup of Γ we mean the quotient of the latter subgroup by the former. In the case ofa Cartan decomposition of a semisimple complex Lie algebra, this is the automorphismgroup of the root system, i.e., the classical extended Weyl group. A grading is calledfine if it cannot be refined.

We compute the Weyl groups of all fine gradings on matrix algebras, octonions, theAlbert algebra, and the simple Lie algebras of types A, B, C, D (except D4), F4 andG2 over an algebraically closed field. The characteristic is assumed to be different from2 in the case of the Albert algebra and the simple Lie algebras (also different from 3 fortype G2). This is joint work with A. Elduque.

1Memorial University of Newfoundland, [email protected]



The derived algebra of skew elements in a

semiprime superalgebra with superinvolutionJesus Laliena1

The study of the relationship between the structure of an associative algebra A andthat of the Lie algebra A− was started by I. N. Herstein and W. E. Baxter in the fifties.Regarding superalgebras, this line of research was motivated by the classification of the fi-nite dimensional simple Lie superalgebras given by V. Kac in 1977, particularly the typesgiven from simple associative superalgebras and from simple associative superalgebraswith superinvolution. The Lie structure of prime superalgebras and simple superalge-bras without superinvolution was studied by F. Montaner [4] and S. Montgomery [5],and later C. Gomez-Ambrosi and I. Shestakov [1] investigated the Lie structure of theLie superalgebra of skew elements, K, and also of its derived algebra, [K,K], when wehave a simple associative superalgebra with superinvolution over a field of characteristicnot 2. These results were extended to prime associative superalgebras with superinvo-lution [2]. In the case of semiprime associative superalgebras, there have been describedin [3] the Lie ideals of the superalgebra K. In this talk we will deal with the case of thederived superalgebra [K,K].

Keywords: semiprime associative superalgebras, superinvolutions, skewsymmetricelements, Lie structure, derived superalgebra

Mathematics Subject Classification 2010: MSC 17B60, 16W50

References

[1] C. Gomez-Ambrosi and I. P. Shestakov, On the Lie structure of the skew elements ofa simple superalgebras with superinvolution, J. Algebra 208 (1998), 43–71.

[2] C. Gomez-Ambrosi, J. Laliena and I. P. Shestakov, On the Lie structure of theSkew Elements of a Prime Superalgbra sith Superinvolution, Comm. Algebra 28 (7) (2000),3277–3291.

[3] J. Laliena and S. Sacristan, Lie structure in semiprime superalgebras with superinvo-lution, J. Algebra (2007), .

[4] F. Montaner, On the Lie structure of associative superalgebras, Comm. Algebra 26 (7)(1998), 2337–2349.

[5] S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras,

J. Algebra 195 (1997), 558–579.

1C/ Luis de Ulloa s/n, Edificio Vives, Departamento de Matematicas y Com-putacion, 26004 Logrono, [email protected]



Abelian groups in the slice category of bialgebras

Sara Madariaga1

A representation of a loop Q in a variety of loops V is an abelian group in the slicecategory V/Q [1]. In this talk we discuss the slice category of bialgebras in connectionwith the representation theory of formal loops and Sabinin algebras.

References

[1] A. Dharwadker and J.D.H. Smith, Split extensions and representations of Moufangloops, Communications in Algebra 23 (1995), 4245–4255.

[2] J. Mostovoy and J. M. Perez-Izquierdo, Formal multiplications, bialgebras of distri-

butions and nonassociative Lie theory. Transform. Groups 15 (2010), 625–653.

1Departamento de Matematicas y Computacion, Universidad de La Rioja,Logrono, [email protected]



Finite growth representations of infinite Lie

conformal algebras

Carina Boyallian1, Vanesa Meinardi1

In the present paper we classify all finite growth representations of all infinite rankconformal subalgebras of gcN that contain a Virasoro subalgebra. This problem reducesto the study of finite growth representations on the corresponding extended annihilationalgebras, which are certain subalgebras of DN (see Ref. [8]). The main tools used hereare the recent results [2] on the classification of quasifinite highest weight modules overthe central extension of DN and some of its important subalgebras (Refs.[4, 6]).

Keywords: representations, weight

Mathematics Subject Classification 2010: 06B15, 17B10

References

[1] C. Boyallian, V.G. Kac and J.I. Liberatti, On the classification of subalgebras of CendN andgcN , Journal of Algebra 260 (2003) 32-63.

[2] C. Boyallian, V.G. Kac, J. Liberati and C. Yan, Quasifinite highest weight modules over theLie algebra of matrix differential operators on the circle, Journal of Math. Phys. 39 (1998),2910-2928.

[3] V. G. Kac, Infinite-dimensional Lie Algebras, 3rd edition, Cambridge University Press, Cam-bridge, 1990.

[4] C. Boyallian and V.B. Meinardi, QHWM of the orthogonal type Lie subalgebra of the Liealgebra of matrix differential operators on the circle. Journal of Mahtematical Physics. 51online (2010).

[5] C. Boyallian and V.B. Meinardi, Quasifinite highest weight modules over WN∞ . Journal

Physics A., Math. Theor. 44 235201 (2011).

[6] C. Boyallian and V.B. Meinardi, Representations of a sympletic type subalgebra of WN∞ .

Journal of Mahtematical Physics. 54 online (14 de junio del 2011.)

[7] V. G. Kac and A. Radul, Quasifinite highest weight modules over the Lie algebra of differentialoperators on the circle, Comm. Math. Phys. 157 (1993), 429-457.

[8] V.G. Kac, Vertex Algebras for beginners, University Lecture Series, 10 (American Matema-

tical Society, Providence, RI, 1996), Second edition 1998.

1FaMAF, Facultad de Matematica Astronomıa y Fısica, 5000 Cordoba, [email protected], [email protected].



Alternative loop algebras over finite fields

Cesar Polcino Milies1

We determine the structure of the semisimple group algebra of certain groups overthose finite fields that produce the fewest number of simple components. We use thisinformation to obtain similar information about the loop algebras of all indecomposableRA loops and observe that the isomorphism problem has a negative answer for alternativeloop algebras over finite fields. This is joint work with R. Ferraz and E.G. Goodaire.

1Instituto de Matematica e Estatıstica, Universidade de Sao Paulo, Caixa Postal66.281, CEP 05314-970, Sao Paulo SP, [email protected]



On generalized Jordan triple systems of second

order and extended Dynkin diagrams

Daniel Mondoc1

In this talk we give the correspondence between simple complex generalized Jor-dan triple systems of second order and extended Dynkin diagrams (joint work with N.Kamiya).

1Centre for Mathematical Sciences, Lund University, [email protected]



A series of algebras generalizing the octonions

and Hurwitz-Radon identity

Sophie Morier-Genoud1, Valentin Ovsienko2

We study non-associative twisted group algebras with cubic twisting functions. Weconstruct a series of algebras that extend the classical algebra of octonions in the sameway as the Clifford algebras extend the algebra of quaternions. We study their proper-ties, give several equivalent definitions and prove their uniqueness within some naturalassumptions. We then prove a simplicity criterion. We present an application of theconstructed algebras to the celebrated Hurwitz problem on square identities.

Keywords: Graded commutative algebras, non-associative algebras, Clifford alge-bras, octonions, square identities, Hurwitz-Radon function, code loop, Parker loop

References

[1] H. Albuquerque and S. Majid, Quasialgebra structure of the octonions, J. Algebra 220(1999), no. 1, 188–224.

[2] A. Elduque, Gradings on octonions, J. Algebra 207 (1998), no. 1, 342–354.

[3] S. Morier-Genoud and V. Ovsienko, Simple graded commutative algebras, J. Algebra

323 (2010), no. 6, 1649–1664.

1Universite Paris Diderot Paris 7, UFR de mathematiques case 7012, 75205 ParisCedex 13, [email protected]

2CNRS, Institut Camille Jordan, Universite Claude Bernard Lyon 1, 43 boule-vard du 11 novembre 1918, 69622 Villeurbanne cedex, [email protected]



Alternative twisted tensor products and Cayley

algebras

Helena Albuquerque1, Florin Panaite2

We present a construction from [1] called alternative twisted tensor products fornot necessarily associative algebras, which arose as a common generalization of severaldifferent constructions: the Cayley-Dickson process, the Clifford process and the twistedtensor product of two associative algebras, one of them being commutative. Some verybasic facts concerning the Cayley-Dickson process (the equivalence between the twodifferent formulations of it and the lifting of the involution) are particular cases of generalresults about alternative twisted tensor products of algebras. As a class of examples ofalternative twisted tensor products, we present a tripling process for an algebra endowedwith a strong involution, containing the Cayley-Dickson doubling as a subalgebra andsharing some of its basic properties.

Keywords: Cayley algebras, twisted tensor products

Mathematics Subject Classification 2010: 17A01

References

[1] H. Albuquerque and F. Panaite, Alternative twisted tensor products and Cayley al-

gebras, Comm. Algebra 39 (2011), no. 2, 686–700.

1CMUC, Departamento de Matematica, Universidade de Coimbra, 3001-454Coimbra, [email protected]

2Institute of Mathematics of the Romanian Academy, P.O.Box 1-764, RO-014700, Bucharest, [email protected]



Code loops: automorphisms and representations

Rosemary Miguel Pires1, Alexandre Grichkov2

We present a introduction about code loops based on the paper of Robert L. Griess,[2]. We characterized code loops of rank n as homomorphic image of certain free Moufangloops with n generators and we introduce the concept of characteristic vectors associatedwith code loops. With the results of the theory studied, we classify all the code loopsof rank 3, we find all the groups of outer automorphisms of these loops and finally, wedetermine all their basic representations.

This work was obtained as part of the PhD thesis under the supervisor AlexandreGrichkov.

Keywords: Code Loops, even codes, characteristic vectors, outer automorphisms,representations.

Mathematics Subject Classification 2010: MSC20-02, MSC20N05.

References

[1] R. M. Pires, Loops de codigo: automorfismos e representacoes, IME-USP (2011) - Tese

de Doutorado.

[2] R. L. Griess Jr., Code loops, J.Algebra 100 (1986), 224–234.

[3] R. H. Bruck, A survey of binary systems, Springer-Verlag (1958).

[4] O. Chein and E. G. Goodaire, Moufang Loops with a Unique Nonidentity Commutator

(Associator, Square), J. Algebra 130 (1990), 369–384.

[5] H. O. Pflugfelder, Quasigroups and Loops: An Introduction, Berlin:Heldermann

(1990).

[6] O. Chein and H. O. Pflugfelder, Quasigroups and Loops: Theory and Applications,

Berlin:Heldermann (1990).

[7] E. G. Goodaire, E. Jaspers and C. Polcino, Alternative Loop Rings, North Holland

Math 184 (1996).

1Instituto de Ciencias Exatas (ICEx), Departamento de Matematica, Universi-dade Federal Fluminense (UFF), Volta Redonda-RJ, [email protected], [email protected]

2Instituto de Matematica e Estatıstica (IME), Universidade de Sao Paulo (USP),Sao Paulo-SP, [email protected], [email protected]



Moufang loops and their automorphisms

Liudmila Sabinina1

I would like to speak about Moufang loops in which inner mappings are automor-phisms. Main result of the talk is a theorem proved jointly with A. Grishkov and P.Plaumann. Let Fn be the free group with a basis X = {x1, . . . , xn} and let Cn be the freecommutative Moufang loop with a basis Y = {y1, . . . , yn}. Put zi = (xi, yi) ∈ Fn × Cnfor 1 ≤ i ≤ n and denote by An the subloop of Fn × Cn generated by Z = {z1, . . . , zn}.Theorem. The loop An is a free automorphic Moufang loop of rank n.

Some corollaries of the theorem will be given.

Keywords: Automorphic Moufang loops

Mathematics Subject Classification 2010: 20N05

1Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, AvenidaUniversidad 1001, 62209 Cuernavaca, Morelos, [email protected]



Heisenberg Lie superalgebras and its invariant

superorthogonal and supersymplectic forms

M.C. Rodrıguez-Vallarte1, G. Salgado1, O.A. Sanchez-Valenzuela2

Finite-dimensional complex Lie superalgebras of Heisenberg type obtained from agiven Z2-homogeneous supersymplectic form defined on a vector superspace, are classi-fied up to isomorphism. Those arising from even supersymplectic forms, have an ordinaryHeisenberg Lie algebra as its underlying even subspace, whereas those arising from oddsupersymplectic forms get based on an abelian Lie algebras. The question of whetherthis sort of Heisenberg Lie superalgebras do or not do support a given invariant super-geometric structure is addressed, and it is found that none of them do. It is proved,however, that 1-dimensional extensions by appropriate Z2-homogeneous derivations do.Such “appropriate” derivations are characterized, and their invariant supergeometricstructures are fully described [1].

Keywords: Heisenberg Lie superalgebras, ad-invariant supersymmetric bilinear forms

Mathematics Subject Classification 2010: 17B30, 17B70, 81R05

References

[1] M.C. Rofrıguez-Vallarte, G. Salgado and O.A. Sanchez-Valenzuela, Heisen-

berg Lie superalgebras and its invariant superorthogonal and supersymplectic forms, Jour-

nal Algebra 332 (2011), no. 1, 71–86.

1Facultad de Ciencias, UASLP, Av. Salvador Nava s/n, Zona Universitaria, C.P.78290, SLP, [email protected], [email protected],[email protected]

2CIMAT, Apdo. Postal 402, C.P. 36000, Guanjuato, Gto., [email protected], [email protected]



Leibniz triple systems

Murray R. Bremner1, Juana Sanchez-Ortega2

We introduce a new variety of triple systems, which are related to Leibniz algebras inthe same way that Lie algebras are connected with Lie triple systems. More precisely, weshow that the defining identities for these structures are satisfied by the iterated bracketin a Leibniz algebra and the (permuted) associator in a Jordan dialgebra. Thereby, wecall them “Leibniz triple systems”, and we construct their universal Leibniz envelopes.

1Department of Mathematics and Statistics, University of Saskatchewan 106Wiggins Road (McLean Hall) Saskatoon, SK, S7N 5E6 [email protected]

2Department of Algebra, Geometry and Topology, University of Malaga, [email protected]



Exact sequences in the Borel-Schur algebra

Ana Paula Santana1

Schur algebras are fundamental tools in the representation theory of the general lineargroup and of the symmetric group. In this talk I explain how we can use the inductionfunctor from the Borel-Schur algebra to the Schur algebra, over a commutative ring, toconstruct projective resolutions of Weyl modules. This is joint work with I. Yudin.

Keywords: Schur algebras, projective resolutions

Mathematics Subject Classification 2010: 20G43, 16E05

References

[1] A. P. Santana and I. Yudin, Characteristic free resolutions for Weyl and Specht modules,

http://arxiv.org/abs/1104.1959




On linear equations over free Lie algebras

Ralph Stohr1

We investigate equations of the form

[x1, u1] + [x2, u2] + · · ·+ [xk, uk] = 0

over a free Lie algebra L. In the case where the coefficients u1, u2, . . . , uk are free gene-rators of L, we generalize a number of earlier results on equations with two variablesobtained in [1] to equations with an arbitrary number of indeterminates. Our mainresult is a complete description of the multilinear fine homogeneous component of thesolution space. This is joint work with Alaa Altassan.

References

[1] V.N. Remeslennikov and Ralph Stohr, The equation [x, u] + [y, v] = 0 in free Lie

algebras, Internat. J. Algebra Comput. 17 (2007), no. 5/6, 1165–1187.

1University of Manchester, [email protected],web:http://www.maths.manchester.ac.uk/ rs/



Non associative algebra associated to genetics

and Jordan algebra

Osamu Suzuki1

A non associative algebra associated to genetics is introduced and it is proved thatit is a Jordan algebra. By this fact we see that Jordan algebra is important not onlyin physics but in biology. Following Prof. A. Michali, we introduce a gametic algebraand Mendel algebra as a special case. The Mendel algebra is defined on a n-dimensionalvector space R[S1, S2, ..., Sn] by the following product:

Si ∗ Sj =1

2(Si + Sj)

Then we obtain a commutative and non associative algebra. We may understand theproduct is an algebraic description of separation law in Mendel’s law. Then we can provethe following theorem:

Theorem A Mendel algebra is a Jordan algebra.

Proof is done by a direct calculation. From the fact that specialization of Mendel algebraand the tensor product is Mendel algebra, we see that the family of Jordan algebras aboveobtained are closed under these operations. We notice that the specialization and tensorproduct are mathematical description of mating process and independent law in genetics.

Remark(1)It is proved that a Mendel algebra is at the same time flexible algebra;(2)We can obtain a family of non associative algebras from Mendel algebras and we candiscuss a hierarchy structure of these algebras.

Keywords: Mendelian genetics, Mendel algebra, Jordan algebra

References

[1] A. Micali and Ph. Revoy, Sur la algebres gametiques , Proc. Edinburgh Math. Soc. 43

(1986), no. 1,2, 187–197.

1Department of Computer Sciences and System Analysis, Setagaya-ku, Sakura-jjousui 3-25-40,Tokyo, [email protected]



Algebraic theory of DNA recombination

Sergei R. Sverchkov1

In this presentation we give a brief introduction of the algebraic theory for DNArecombination within of intersection of two areas: theoretical genetics and theory of non-associative algebras. We investigate the structure and representation of n-ary algebrasarising from DNA recombination, where n is a number of DNA segments participatingin recombination. For the algebraic formalization of DNA recombination we will use thebasic idea of the formalization of algebra of observables in quantum mechanics, which wasfirst realized by Pascual Jordan. We proved that every identity satisfied by n-ary DNArecombination, with no restriction on the degree, is consequence of n-ary commutativityand a single n-ary identity of the degree 3n-2. It solves the well- known open problemin the theory of n-ary intermolecular recombination.

Keywords: Jordan algebras, DNA recombination, theoretical genetics, theory of n-ary algebras.

1Novosibirsk State University, [email protected]



Algebras with involution and their identities‖

Irina Sviridova1

We will consider associative non-commutative algebras over a field F of zero characte-ristic.

Anti-automorphism of order 2 of an algebra is called involution of this algebra.If we will fix an involution ∗ on an associative F -algebra A then it can be considered

as a new unary operation on A. We can study identities of the algebra A in this newsignature. These identities are called identities with involution of A (∗-identities). Itis well-known (see for example [1]) that all ∗-identities of an F -algebra with involutionform two-side verbal ideal of the free associative algebra with involution closed underthe action of involution. If this ideal is non-zero then the algebra A is also PI-algebra(it satisfies the non-trivial ordinary polynomial identity without involution).

It is well-known also that any algebra A with involution ∗ as a vector space overF is the direct sum A = S + T of the subspace S = {a ∈ A|a∗ = a} of symmetricelements (satisfying equality a∗ = a) and of the subspace T = {a ∈ A|a∗ = −a} of skew-symmetric elements (satisfying a∗ = −a). Then the vector subspace S with operationx◦y = xy+yx forms Jordan algebra, and the subspace T with operation [x, y] = xy−yxforms Lie algebra.

The next relation between finitely generated associative algebras with involution andfinite dimensional algebras with involution over a field of zero characteristic is obtained.

Theorem. For any finitely generated associative algebra A with involution overa field F of zero characteristic there exists a finite dimensional over F algebra C withinvolution which has exactly the same ∗-identities as the algebra A.

Keywords: Algebras with involution, identities with involution

Mathematics Subject Classification 2010: 16R50, 16W10

References

[1] A. Giambruno and M. Zaicev, Polynomial identities and asymptotic methods. Mathe-

matical Surveys and Monographs 122 American Mathematical Society, Providence, RI, 2005.

1Departamento de Matematica, Universidade de Brasılia, 70910-900 Brasılia,DF, [email protected]

‖Supported by CNPq, and by CNPq-FAPDF PRONEX grant 2009/00091-0(193.000.580/2009)



A generalization of left invariant vector fields

and Lie algebras generated by them

M.R. Molaei Taherabadi1

In this talk we consider top spaces [1] as a class of completely regular semigroups withthe C∞ structures which are compatible with their product and their inverse mappings.We will present a non-associative algebra by using of a top space with a finite numberof identities [2].

Keywords: Top space, left invariant vector field

Mathematics Subject Classification 2010: 17B45, 17B66

References

[1] M.R. Molaei, Top spaces, Journal of Interdisciplinary Mathematics 7 (2004), no. 2, 173–181.

[2] M.R. Molaei and M.R. Farhangdoost, Lie algebras of a class of top spaces, Balkan

Journal of Geometry and Its Applications 14 (2009), no. 1, 46–51.

1Mahani Mathematical Research Center, and Department of Mathematics, Uni-versity of Kerman (Shahid Bahonar), Kerman, [email protected]



Maxwell equations for magnetic sources and

massive photon in octonion algebra

Murat Tanıslı1, Bernard Jancewicz2

In this study, the octonion algebra and its general properties are defined by theCayley-Dickson’s multiplication rules for octonion units. The field equations, potentialand Maxwell equations for electromagnetism are investigated with the octonionic equa-tions and these equations can be compared with their vectorial representations. Also,the potential and wave equations for fields with sources are given. Then, using Maxwellequations, a Lorenz like condition is suggested for electromagnetism. Because the exis-ting equations give the well known Lorenz condition for magnetic monopole and source.These equations include the photon mass.

Keywords: complex octonion, Lorenz condition, electromagnetism, magnetic monopole,photon mass, Proca-Maxwell equations

Mathematics Subject Classification 2010: 35Q61, 08Axx

References

[1] V. Majernık, Gen. Rel. Grav. 35, 2003, 1833.

[2] V. Majernık and M. Nagy, Lett. Nuovo Cimento 16, 1976, 265.

[3] V. Majernık, Adv. Appl. Clifford Algebras 9, 1999, 119.

[4] A. Gamba, Il Nuovo Cimento 111A, 1998, 293.

[5] M. Tanıslı and G. Ozgur, Acta Phys. Slovaca, 53,2003,243.

[6] M. Tanıslı, Europhysics Letters, 74(4), 2006, 569.

[7] T. Tolan, K. Ozdas and M. Tanıslı, Il Nuovo Cimento B, 121(1), 2006, 43-55.

[8] N. Candemir, M. Tanıslı, K. Ozdas and S. Demir, Zeitschrift fur Naturforschung A, 63(a),2008, 15-18.

[9] F. Toppan, hep-th/0503210v1

[10] B.A. Bernevig, J.P. Hu, N. Toumbas and S.C. Zhang, Phys. Rev. Lett. 91, 2003, 23.

[11] K.J. Vlaenderen and A. Waser, Electrodynamics with the Scalar Field,http://www.info.global-scaling-verein.de/Documents/Electrodynamics-WithTheScalarField03.pdf

[12] O.P.S. Negi, S. Bisht and P.S. Bisht, Nuovo Cimento B, 113,1998,1449.

[13] B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics, (World Scientific),1988.


[14] S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, (Oxford UniversityPress, New York), 1995.

[15] J. Daboul and R. Delbourgo, J. Math. Phys. 40, 1999,4134.

[16] Y. Tian, Adv. Appl. Clifford Algebras 10, 2000, 61.

[17] Okubo S., Introduction to Octonion and Other Non-Associative Algebras in Physics, (Cam-bridge University Press, Cambridge), 1995.

[18] H.L. Carrion, M. Rojas and F. Toppan, hep-th/0302113v1.

[19] J.D. Jackson, Classical Electrodynamics, (John Wiley and Sons, New York), 1999.

1Science Faculty, Department of Physics, Anadolu University, 26470 Eskisehir,[email protected]

2Intitute of Theoretical Physics, University of Wroc law, Wroc law, [email protected]



A capacity 2 theorem for graded Jordan systems

Maribel Tocon1, Erhard Neher2

The Osborn Capacity 2 Theorem for Jordan algebras tackles the hardest case in theclassification of Jordan algebras with finite capacity, namely that of capacity two, i.e., theidentity element is a sum of two orthogonal division idempotents. Removing the divisioncondition on the diagonal Peirce spaces leads to the definition of triangulated Jordanalgebras: The Jordan algebra has two complementary orthogonal idempotents which areconnected by an invertible element in the off-diagonal Peirce space. The extension ofthis notion to triangulated Jordan systems (Jordan triple systems and Jordan pairs) isimmediate. A Jordan system is called graded-simple-triangulated if it is triangulatedand graded-simple with respect to a Λ-grading compatible with its triangularization, forΛ an abelian group. These are the Jordan systems which, via the Tits-Kantor-Koecherconstruction, correspond to B2-graded Lie algebras that are graded-simple with respectto a second compatible grading by Λ, a case of interest to the theory of extended affineLie algebras.

In this talk, we present a classification of graded-simple-triangulated Jordan systemswith division-graded diagonal Peirce spaces, based on the classification of graded-simple-triangulated Jordan systems ([NT]), with the restriction that Λ be torsion-free. As aconsequence we extend the Osborn Capacity 2 Theorem to the graded setting.

Keywords: Jordan structures covered by a triangle, coordinatization

Mathematics Subject Classification 2010: 17C10, 17C27

References

[1] E. Neher and M. Tocon, Lie tori of type B2 and graded-simple Jordan

structures covered by a triangle, preprint available at http://homepage.uibk.ac.at./

∼c70202/jordan/index.html.

1Departamento de Estadıstica e Investigacion Operativa, Universidad deCordoba, Puerta Nueva s/n, Cordoba, 14071, [email protected]

2Department of Mathematics and Statistics, University of Ottawa, Ottawa, On-tario K1N 6N5, [email protected]



Strongly stable automorphisms of the category

of free linear algebras

Arkadt Tsurkov1

This research is motivated by universal algebraic geometry. We consider in universalalgebraic geometry the some variety of universal algebras Θ and algebrasH ∈ Θ from thisvariety. We take a “set of equations” T ⊂ F × F in some finitely generated free algebraF of the arbitrary variety of universal algebras Θ and we “resolve” these equations inHom (F,H), where H ∈ Θ. The set Hom (F,H) serves as an “affine space over thealgebra H”. Denote by T ′H the set {µ ∈ Hom (F,H) | T ⊂ kerµ}. This is the set ofall solutions of the set of equations T . For every set of “points” R of the affine spaceHom (F,H) we consider a congruence of equations defined by this set: R′H =

⋂µ∈R

kerµ.

For every set of equations T we consider its algebraic closure T ′′H in respect to the algebraH. A set T ⊂ F × F is called H-closed if T = T ′′H .

One of the central questions of the theory is the following: When do two algebrashave the same geometry? What does it mean that the two algebras have the samegeometry? The notion of geometric equivalence of algebras gives a sort of answer tothis question. Algebras H1 and H2 are called geometrically equivalent if and only if theH1-closed sets coincide with the H2-closed sets. The notion of automorphic equivalenceis a generalization of the first notion. Algebras H1 and H2 are called automorphicalyequivalent if and only if the H1-closed sets coincide with the H2-closed sets after some“changing of coordinates”.

We can detect the difference between geometric and automorphic equivalence ofalgebras of the variety Θ by researching of the automorphisms of the category Θ0 of thefinitely generated free algebras of the variety Θ. By [?] the automorphic equivalence ofalgebras provided by inner automorphism degenerated to the geometric equivalence. Sothe various differences between geometric and automorphic equivalence of algebras canbe found in the variety Θ if the factor group A/Y is big. Hear A is the group of allautomorphisms of the category Θ0, Y is a normal subgroup of all inner automorphismsof the category Θ0. We have [1, Theorem 2] this decomposition

A = YS.

Hear S is a group of strongly stable automorphisms of the category Θ0. So A/Y ∼= S/S ∩Y.In [1] the variety of all Lie algebras and the variety of all associative algebras over

the field k were studied. The group A/Y in both these case is small, in particular, ifthe field k has not nontrivial automorphisms. We consider in this paper the variety ofall linear algebras over the infinite field k which has not nontrivial automorphisms. Thegroup A/Y ∼= S/S ∩Y is isomorphic to the group G/k∗I2, where G is the group of the

regular 2× 2 matrices over field k, which have a form

(a bb a

)where a, b ∈ k. So the

group A/Y is enough rich in this case.

Keywords: Linear algebras, automorphisms of categories, universal algebraic geome-try


Mathematics Subject Classification 2010: 17A50, 18A22

References

[1] B. Plotkin and G. Zhitomirski, On automorphisms of categories of free algebras of

some varieties, Journal of Algebra 306 (2006), no. 2, 344 – 367.

1Institute of Mathematics and Statistics, University of Sao Paulo, Rua do Matao,1010, 05508-090, [email protected]



Normalized bar resolution and Schur algebra

Ivan Yudin1

One of the basic results in Representation Theory of Schur algebras and of symmetricgroups are Weyl character formula and Frobenius determinant formula, respectively. Inthis talk I will explain how normalized bar resolution for Borel-Schur algebra can be usedto obtain resolutions of Weyl and Specht modules that “realize” the above mentionedformulae. This is joint work with A. P. Santana.

Keywords: Schur algebras, projective resolutions

Mathematics Subject Classification 2010: 20G43, 16E05

References

[1] A. P. Santana, I. Yudin, Characteristic free resolutions for Weyl and Specht modules,

http://arxiv.org/abs/1104.1959




A commutative 2-cocycles approach to

classification of simple Novikov algebras

Pasha Zusmanovich1

Finite-dimensional simple Novikov algebras were classified by Zelmanov in charac-teristic 0 and by Osborn and Xu in characteristic p. We will outline an alternativeapproach to these classifications, based on the notion of commutative 2-cocycles of Liealgebras – bilinear maps satisfying the usual cocycle equation, only being symmetricinstead of skew-symmetric.

1Depart. of Mathematics, Tallinn University of Technology, Tallinn, [email protected]

Posters 61


On finite-dimensional absolute-valued algebras

satisfying (xp, xq, xr) = 0

M. I. Ramırez Alvarez1

By means of principal isotopes lH(a, b) of the algebra lH [Ra 99] we give an exhaustiveand not repetitive description of all 4-dimensional absolute-valued algebras satisfying(xp, xq, xr) = 0 for fixed integers p, q, r ∈ {1, 2}. For such an algebras the numberN(p, q, r) of isomorphism classes is either 2 or 3, or is infinite. Concretely

1. N(1, 1, 1) = N(1, 1, 2) = N(1, 2, 1) = N(2, 1, 1) = 2,

2. N(1, 2, 2) = N(2, 2, 1) = 3,

3. N(2, 1, 2) = N(2, 2, 2) =∞.

Besides, each one of the above algebras contains 2-dimensional subalgebras. However,the problem in dimension 8 is far from being completely solved. In fact, there are8-dimensional absolute-valued algebras, containing no 4-dimensional subalgebras, satis-fying (x2, x, x2) = (x2, x2, x2) = 0.

1Universidad de Almerıa, [email protected]

Posters 62


On purely real algebras Lie on a Hilbert space

Farkhad Arzikulov1

In the given work we investigated a real von Neumann algebra R on a Hilbert spacesuch that R ∩ iR = {0} and an ultraweakly closed real algebra Lie L of sky-adjointoperators on a Hilbert space such that R(L) ∩ iR(L) = {0} for the real von Neumannalgebra R(L), generated by L in B(H). These algebras were called a purely real vonNeumann algebra and a ultraweakly closed purely real algebra Lie. It is proved that everymaximal purely real algebra Lie on a Hilbert space H is isomorphic to B(HR)k⊕B(HH)kfor some Hilbert spaces HR and HH on R and H (quaternions) correspondingly such thatB(H) = B(HF ) + iB(HF ), where F = R, H, B(HF )k = {x ∈ B(HF ) : x∗ = −x}, andfor such Hilbert subspaces HR and HH, that HR = HR ⊕ H⊥R , HH = HH ⊕ H⊥H , theidentity elements of the algebras B(HR) and B(HH) are mutually orthogonal and theirsum is equal to the identity element of B(H).

Also it was proved that any ultraweakly closed purely real abelian algebra Lie L ofsky-adjoint operators on a Hilbert space is isomorphic to the algebra Lie of all continuous

functions f : X →(

0 1−1 0

)R on a hyperstonean compact X.

Keywords: real von Neumann algebra, Algebra Li

Mathematics Subject Classification 2010: 17B10, 46L10

1Institute of Mathematics and Information technologies, Tashkent, [email protected], [email protected]

Posters 63


On homogeneous symmetric associative

superalgebras

Imen Ayadi1, Saıd Benayadi1

The main purpose of this work is to study associative superalgebras endowed withsymmetric, even or odd, non-degenerate associative bilinear forms, i.e the so calledsymmetric homogeneous structures. First we show that any associative superalgebrawith a non null product cannot admit simultaneously even symmetric and odd symmetricstructures. Next, we prove that, unlike the Lie case, all simple associative superalgebrasadmit either even symmetric or odd symmetric structures by giving explicitly in everycase the homogeneous symmetric structures. Second, we introduce some notions ofgeneralized double extensions in order to give inductive descriptions of even symmetricassociative superalgebras and odd symmetric associative superalgebras. Finally, we givean other interesting description of odd symmetric associative superalgebras whose evenparts are semi-simple bimodules with no use of doubles extensions.

Keywords: Simple associative superalgebras, associative superalgebras, homogeneoussymmetric structures, generalized double extension, inductive description

1Laboratoire de Mathematiques et Applications de Metz, CNRS UMR 7122,Universite Paul Verlaine-Metz, Ile du Saulcy, F-57045 Metz cedex 1, [email protected], [email protected]

Posters 64


On some Hurwitz reduced algebras of a ternary

quaternion algebra

Patrıcia D. Beites1, Alejandro P. Nicolas2

The definition of n-Lie algebra (n ≥ 2) was introduced in [2], being the most rele-vant and natural known generalization of the Lie algebra concept. Following the actualtendency, we use the term n-ary Filippov algebra instead of n-Lie algebra.

Let F be a field such that ch(F ) 6= 2. Consider the 4-dimensional ternary Filippovalgebra A4 over F equipped with a bilinear, symmetric and non-degenerate form (· , ·),and the canonical basis {e1, e2, e3, e4}. The multiplication table of A4 is the subsequentone

[e1, . . . , ei, . . . , e4] = (−1)iei, i ∈ {1, . . . , 4}, (1)

where ei means that ei is omitted. The remaining products are zero or obtained from(1) by anticommutativity.

We define a new multiplication on the underlying vector space of A4,

{x, y, z} := −(y, z)x+ (x, z)y − (x, y)z + [x, y, z],

and denote the obtained ternary quaternion algebra by A, as in [1]. In this reference,simplicity, identities of levels 1 and 2, and derivations of A were studied. Furthermore,envelopes for ternary Filippov algebras were obtained.

In the present work we consider some reduced algebras of A and its identities of levelless or equal to 2. We focus on those that are Hurwitz algebras, namely (A, •2,1) wherex •2,1 y = {x, e1, y}. Moreover, being a, b 6= 0, conditions for the existence of non-zerosolutions of the linear equation

a •2,1 x = x •2,1 b

are established. Under these, using matrix representations inspired by some results in[3], the general solution is determined.

Keywords: Filippov algebra, ternary quaternion algebra, reduced algebra, composi-tion algebra, matrix representation, linear equation


References

[1] P. D. Beites, A. P. Nicolas, A. P. Pozhidaev and P. Saraiva, On identities of aternary quaternion algebra, Comm. Algebra 39 (2011), no. 3, 830–842.

[2] V. T. Filippov, n-Lie Algebras, Siberian Math. J. 26 (1985), no. 6, 879–891.

Posters 65

[3] Y. Tian, Matrix representations of octonions and their applications, Adv. Appl. Clifford

Algebr. 10 (2000), no. 1, 61–90.

1Centro de Matematica and Departamento de Matematica, Universidade daBeira Interior, Rua Marques d’Avila e Bolama, 6201-001 Covilha, [email protected]

2Departamento de Matematica Aplicada, Universidad de Valladolid, CampusDuques de Soria s/n, 42004 Soria, [email protected]

Posters 66


On split Lie superalgebras

Antonio J. Calderon Martın1, Jose M. Sanchez Delgado1

We study the structure of arbitrary split Lie superalgebras. We show that any of suchsuperalgebras L is of the form L = U +

∑jIj with U a subspace of the abelian (graded)

subalgebra H and any Ij a well described (graded) ideal of L satisfying [Ij , Ik] = 0 ifj 6= k. Under certain conditions, the simplicity of L is characterized and it is shown thatL is the direct sum of the family of its minimal (graded) ideals, each one being a simplesplit Lie superalgebra.

Keywords: Infinite dimensional Lie superalgebras, structure theory

Mathematics Subject Classification 2010: 17B65, 17B05, 17B20

References

[1] C. Boyallian and V. Meinardi, Quasifinite representations of the Lie superalgebra ofquantum pseudodifferential operators. J. Math. Phys. 49 (2008), no. 2, 023505, 13 pp.

[2] A.J. Calderon, On split Lie algebras with symmetric root systems. Proc. Indian. Acad.Sci, Math. Sci. 118 (2008), 351-356.

[3] A.J. Calderon, On simple split Lie triple systems. Algebr. Represent. Theory 12 (2009),401-415.

[4] K. Iohara and Y. Koga, Note on spin modules associated to Z-graded Lie superalgebras.J. Math. Phys. 50 (2009), no. 10, 103508, 9 pp.

[5] E. Poletaeva, Embedding of the Lie superalgebra D(2, 1;α) into the Lie superalgebra ofpseudodifferential symbols on S1|2. J. Math. Phys. 48 (2007), no. 10, 103504, 17 pp.

[6] N. Stumme, The structure of Locally Finite Split Lie Algebras, J. Algebra. 220 (1999),

664-693.

1Departamento de Matematicas, Universidad de Cadiz, 11510 Puerto Real,Cadiz, [email protected], [email protected]

Posters 67


Gradings on e6

Cristina Draper1

A model of each fine grading on e6 is given. There are 14 of such fine gradings, 5of them are inner. Not every outer fine grading comes from extending to e6 a gradingeither on c4 or on f4, there is one fine grading produced by a quasitorus with outerautomorphisms but without order two outer automorphisms.

1Universidad de Malaga, [email protected]

Posters 68


Crystal bases for the quantum queer

superalgebra and semistandard decomposition

tableaux

Dimitar Grantcharov1, Ji Hye Jung2, Seok-Jin Kang2, Masaki Kashiwara3,Myungho Kim2

We give an explicit combinatorial realization of the crystal B(λ) for an irreduciblehighest weight Uq(q(n))-module V (λ) in terms of semistandard decomposition tableaux.We present an insertion scheme for semistandard decomposition tableaux and give algo-rithms of decomposing the tensor product of Uq(q(n))-crystals. Consequently, we obtainexplicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.

Keywords: Quantum queer superalgebras, crystal bases, odd Kashiwara operators,semistandard decomposition tableaux, shifted Littlewood-Richardson coefficients

Mathematics Subject Classification 2010: 17B37, 81R50

1Department of Mathematics University of Texas at Arlington, Arlington, TX76021, [email protected]

2Department of Mathematical Sciences, Seoul National University, Seoul 151-747, [email protected], [email protected], [email protected]

3Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan& Department of Mathematical Sciences, Seoul National University, Seoul 151-747, [email protected]

Posters 69


Crystal bases for the quantum queer

superalgebra

Dimitar Grantcharov1, Ji Hye Jung2, Seok-Jin Kang2, Masaki Kashiwara3,Myungho Kim2

We develop the crystal basis theory for the quantum queer superalgebra Uq(q(n)).We define the notion of crystal bases and prove the tensor product rule for Uq(q(n))-

modules in the category O≥0int. Our main theorem shows that every Uq(q(n))-module in

the category O≥0int has a unique crystal basis.

Keywords: quantum queer superalgebras, crystal bases, odd Kashiwara operators.

Mathematics Subject Classification 2010: 17B37, 81R50

1Department of Mathematics University of Texas at Arlington, Arlington, TX76021, [email protected]

2Department of Mathematical Sciences, Seoul National University, Seoul 151-747, [email protected], [email protected], [email protected]

3Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan& Department of Mathematical Sciences, Seoul National University, Seoul 151-747, [email protected]

Posters 70


Universal central extensions of Hom-Leibniz

algebras

Jose Manuel Casas1 ∗∗, Manuel Avelino Insua1, Natalia Pacheco2

The Hom-Lie algebra structure was initially introduced in [2] motivated by examplesof deformed Lie algebras coming from twisted discretizations of vector fields.

In [3] is defined a (multiplicative) Hom-Leibniz algebra as a triple (L, [−,−], α) con-sisting of a K-vector space L, a bilinear map [−,−] : L × L → L and a K-linear mapα : L→ L (which preserves the bracket) satisfying:

[α(x), [y, z]] = [[x, y], α(z)]− [[x, z], α(y)] (Hom− Leibniz identity) (2)

for all x, y, z ∈ L.If we consider α = id, then the Hom-Leibniz identity becomes to Leibniz identity, so

Leibniz algebras are a particular instance of (multiplicative) Hom-Leibniz algebras.Our goal in the present note is the study of universal central extensions of multiplica-

tive Hom-Leibniz algebras. A new notion of centrality for an extension π : (K,αK) �(L,αL) is necessary. An extension π is called α-central if [α(Ker (π)),K] = 0 =[K,α(Ker (π))], and is called central if [Ker (π),K] = 0 = [K,Ker (π)]. Clearly centralextension implies α-central extension and both notions coincide in case α = Id.

Also two notions of universality are needed: a central extension π : (K,αk) � (L,αL)is universal if for every central extension τ : (A,αA) � (L,αL), there exists a uniquehomomorphism ϕ : (K,αk)→ (A,αA) such that τϕ = π. If this property holds only forα-central extensions τ : (A,αA) � (L,αL), then π is called universal α-central. Clearlyuniversal α-central implies universal central and both notions coincide when α = Id.

After that, we can generalize some classical properties on universal central extensionsof Leibniz algebras to the multiplicative Hom-Leibniz algebras setting, in particular,we show that a multiplicative Hom-Leibniz algebra (L,αL) admits universal centralextension if and only if (L,αL) is perfect; in this case, the kernel of the universal centralextension is canonically isomorphic to HLα2 (L) [1], the second homology with trivialcoefficients of the multiplicative Hom-Leibniz algebra L. Nevertheless other results areweaker than the classical ones. For instance, the composition of two central extensions

is not a central extension, but it is an α-central extension; if 0→ (M,αM )i→ (K,αK)

π→(L,αL) → 0 is a universal α-central extension, then HLα1 (K) = HLα2 (K) = 0, but theconverse doesn’t hold.

Keywords: Hom-Leibniz algebra, universal α-central extension


∗∗(1) supported by MICINN, grant MTM2009-14464-C02-02 (European FEDER support in-cluded) and by Xunta de Galicia, grant Incite09 207 215 PR

Posters 71

References

[1] J. M. Casas, M. A. Insua and N. Pacheco, (Co)homology of Hom-Leibniz algebras,submitted to Linear Algebra and its Applications (2011).

[2] J. T. Hartwing, D. Larson and S. D. Silvestrov, Deformations of Lie algebras usingσ-derivations, J. Algebra 295 (2006), 314–361.

[3] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2

(2008), no. 2, 51–64.

1Dpto. Matematica Aplicada I, Univ. de Vigo, E. I. Forestal, 36005 Pontevedra,[email protected], [email protected]

2IPCA, Dpto. de Ciencias, Campus do IPCA, Lugar do Aldao, 4750-810 VilaFrescainha, S. Martinho, Barcelos, [email protected]

Posters 72


The nil-radical and the bar-radical for

alternative baric algebras

Henrique Guzzo Junior1, Rodrigo Lucas Rodrigues2

Let F be a field and let A be an algebra over F , not necessarily associative, neithercommutative nor finite dimensional. If ω : A→ F is a nonzero homomorphism, then theordered pair (A,ω) is called a baric algebra over F and ω is its weight function. Thekernel of ω is denoted by bar(A), and an ideal I is baric if I ⊆ bar(A). An alternativealgebra A over F is an algebra satisfying the equalities x2y = x(xy) and yx2 = (yx)x,for all x, y ∈ A.

We assume during this work that A is a finite dimensional algebra and F is a fieldof characteristic distinct of two, and we study two radicals. We denote by R(A) thenil-radical, which is the maximal nilpotent ideal of A, and by Rb(A) the bar-radical ofA, defined as the intersection of maximal baric subalgebras.

It is known that a baric alternative algebra possess an idempotent of weight 1, andin this case the bar-radical of A is the intersection of all maximal baric ideals.

Our objective is investigate relations between these two radicals. We discovery thatthese radicals are not necessarily equals and we can prove Rb(A) = RA ∩ (bar(A))2.

Keywords: bar-radical, baric algebra, nil-radical.

Mathematics Subject Classification 2010: MSC17A65, MSC17D05, MSC17D92.

References

[1] H. Guzzo Jr. and M. A. Couto, The radical in alternative baric algebras, Arch. Math.(Basel) 75 (2000), no. 3, 178–187.

[2] H. Guzzo Jr, The bar-radical of baric algebras, Arch. Math. (Basel) 67 (1996), no. 2,106–118.

[3] H. Guzzo Jr., Some properties of bar-radical of baric algebras, Comm. Algebra 30 (2010),no. 10, 4827–4835.

[4] H. Guzzo Jr., The structure of baric algebras, Groups, rings, and Group rings, Lec. Notes

Pure Appl. Math. Chapman and Hall/CRC, Boca Raton, FL (2006), 233–242.

1Instituto de Matematica e Estatıstica (IME), Departamento de Matematica,Universidade de Sao Paulo, Sao Paulo (USP), [email protected], [email protected]

2Instituto de Matematica e Estatıstica (IME), Departamento de Matematica,Universidade de Sao Paulo, Sao Paulo (USP), [email protected], [email protected]

Index

Alvarez, M. I. Ramırez, 61Ansari, Abu Zaid, 18Arenas, Manuel, 20Arzikulov, Farkhad, 62Aschheim, Raymond, 21Ayadi, Imen, 63

Bajo, Ignacio, 22Barreiro, Elisabete, 23Beites, Patrıcia D., 64Benayadi, Saıd, 9Benkart, Georgia, 13Bremner, Murray R., 24

Casas, Jose Manuel, 70Coelho, Flavio Ulhoa, 25

Darpo, Erik, 26Delgado, Jose M. Sanchez, 66Demir, Suleyman, 27Dixon, Geoffrey, 28Draper, Cristina, 67

Elduque, Alberto, 10

Futorny, Vyacheslav, 11

Goncharov, Maxim, 29Guzzo Jr, H., 30

Insua, Manuel Avelino, 31

Jung, Ji Hye, 68

Kansu, Mustafa Emre, 33Kashuba, Iryna, 34Kerner, Richard, 35Kim, Myungho, 69Kotchetov, Mikhail, 36

Laliena, Jesus, 37

Madariaga, Sara, 38Majid, Shahn, 14Martınez, Consuelo, 15Meinardi, Vanesa, 39Milies, Cesar Polcino, 40Mondoc, Daniel, 41

Nicolas, Alejandro P., 64

Ovsienko, Valentin, 42

Pacheco, Natalia, 70Panaite, Florin, 43Pires, Rosemary Miguel, 44Pozhidaev, Alexander, 12

Rodrigues, Rodrigo Lucas, 72

Sabinina, Liudmila, 45Salgado, Gil, 46Sanchez-Ortega, Juana, 47Santana, Ana Paula, 48Shestakov, Ivan, 16Stohr, Ralph, 49Suzuki, Osamu, 50Sverchkov, Sergei R., 51Sviridova, Irina, 52

Taherabadi, M.R. Molaei, 53Tanıslı, Murat, 54Tocon, Maribel, 56Tsurkov, Arkadt, 57

Yudin, Ivan, 59

Zelmanov, Efim, 17Zusmanovich, Pasha, 60

73