IC/83/188

/ f 3 5 6 / 8 ^ INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

QUATERNIONIC SUPERGROUPS

AND D = k EUCLIDEAN EXTENDED UPERSYMMETRIES

Jerzy Lukierski

and

Anatol Novicki

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGAN IZATION 1983 Ml RAMARE-TRIESTE

IC/83/188

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUATERNIONIC SUPERGROUPS AND D=4 EUCLIDEAN EXTENDED SUPERSYMMETRIES

Jerzy Lukierski ** and Anatol Howicki *•*

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

The classification of all standard classical super-

groups described by linear trangformatioris on real, complex

or quaternionic superspaces is given. The quaternionic groups

and supergroups are considered in more detail. Further?the real,

complex and quaternionic description of extended D=4 Euclidean

super-conformal algebra is given. We also construct the graded

extensions of Euclidean de-Sitter algebras 0(4,1)= 0(1,1:H) and

0(5)=U(2;H). The relation between D=4 Euclidean and D=4 Hinkowski

superalgebras via analytic continuation is outlined.

MIRAMARE - TRIESTE

October 1983

* To be submitted for publication.

** On leave of absence from Institute for Theoretical Physics, University

of Wroclaw, 50-205 Wroclaw, ul. Cybulskiego 36, Poland.

••* On leave of absence from Institute of Teacher's Training, 50-527 Wroclaw,

ul. Dawida la, Poland.

TW-

"•'•\r f l n a ^ i C r i l [ . ' i n o l g f t b r a a r e d i v : d i ' ; i M I ^ O f o u r C u r r a n f a m i l i e s

i "' • C , D , ¥i Mi The :-.-l.r. ud:.-r'i c o m p l e x for::^ o f theLr '-oramutat^r A I f^-bm^-•i r> yi • '

• '"-,". j j , 2J). With a given Car tan complex ?Jgebr;a one can assoc-.ate a'l

• ••-i'. Lie algebras which after complexif icat ion can be identified w;. .'•

.• :s--.'iding complex form. Lei. as recall further that

i '• Only one compact group ran be associated with a given standard

:.p' ex form, i.e.

A : SU(ni-l) compact B : S0(2n+l) compactn n

Sp(n)= USp(£n;C)compact D ; B0(2n) compact .n

C :n

Other associated Lie algebras describe nonoompact groups,

ii) All real forms of classical Lie groups can be represented by the

linear transformations or, real (R), complex (C) and quate-nionic (H) vector

spaces, with ristric and/or voliune-preserving conditions (see e.g.fsj'1.

During Lhe 1st decide it has been shown th.it sums Lie algebras G^ can

be extended by adding new generators G in such a way that they form the

Z ~ graded extension of Lie algebra structure called the Lie superalgebra

G= G .© G (see e.g.£3-8J). All complex finite-dimensional simple Lie

superalgebras were first • classified by Kac 13,^] and the classical super-

algebras are those in which the superalgebra G is reductive. The list of

classical Lie superalgebras can be divided into four parts (we use here the

Kac notation L3,1*)

a) standard classical Lie superalgebras A(n,m), B(n,m), C(n) and

D{n,m)

b) exceptional Lie superalgebras F M ) , G(3)

c) strange Lie superalgebras P(n), Q(n)

d) one-parameter family of deformations of D(2,l), denoted by

D(2,l;ot ).

The supersymntetric analogues of Cartan classical Lie algebras are

standard classical Lie superalgebras. In particular one can consider all

real forms of the complex superalgebras A(n,m), B(n,ml, C(n) and D(n,m),

and corresponding supergroups obtained by the exponentiation map

— 1 —

G -- Go€> G, —> , • .Br]

»her»- D £G , F, 6 G , ol arc commuting and i, anti cominu t i ng (Grassrnann)r i') « I r --^

sunergruup parameters [j~'j], "ft appfara that all these supergroups can be

realized as linear transformation on real, complex and quaternionic super-

space 3 .

The supergroups, similarly like Clifford modules describing the

fundamental spin representations [ 10-12] are equipped with real, complex or

quaternionic structure. The supergroups with real and complex structure,

which are represented by the linear transformations on real and complex

superspaces are well-known from physical applications. We have

i) Two real supergroups» i.e,

- ! inear real supergroups SL(n,m-,R), describing general linear

frames in fi"'"1 - {X . . ,x 6' . . . fiJ ) , with the superdeterminant equal toI n I m

one. For n-m this supergroup contains one-parameter Abelian center, which

should be factored out. The supergroup SL(n,m;R) is a real form of A(n,m).

- orthosymplectic real supergroup 0Sp(n,m;R) (even m = 2k), which

preserves the orthosymplectic scalar product ( r = l,...k)

where the canonical form of orthosymplectic metric is

(1.3a)

In general the metric G can be chosen as follows:

A& ~ \ 0 ; a w jand provides the family of orthosymplectic real supergroup OSp(n-s,s;m;R)

where s describes the signature in 0{n) sector. The supergroup

03p(n-s,9;iB;R) for n odd is a real form of B(n,m), for n = 2 - the real

fcrm of C (n) , and for n even, nj,4 - the real form of D(n,ni).

i.i J Three complex supergroups, i.e.

- I inear complex supergroup SL( n,m ; C ) describing general li near

frames in C ' - ( z z i ' "1 ), where z. - x. + iy. , = Q, + i 0, ,l n l m i i i w «

wl'. • i yup^rdeterminant equal to one. In order to obtain a simple supergroup»

'• >r r> = m the Abelian center should he factored out.

- graded unitary supergroup SU(n,m),

^cal&r product

preserving the graded unitary

(1.4a)

The general form of graded-unitary metric is

HAft 00

%n

and provides the family of supergroups SU(n-s,s; m-1,1), with s and 1

describing signatures in two unitary sectors. It should be mentioned that

if we put H = G (see (1.3)), one obtains the supergroups SU(n;k,k)/

- orthosymplectic complex supergroup 0Sp(n,2k;C), obtained by

considering the linear transformations of C ' preserving the complex-

analytic scalar product (1.2) in C ' .

In this paper we shall consider the supergroups with quaternionic

structure, i.e. which can be represented by linear endomorphisms of

quaternionic superspaces. In particular,we shall show how to describe

quaternionic superspace and quaternionic supergroups using complex super-

space coordinates.

Of interest for physical applications are the supersymmetric

extensions of groups with quaternionic fundamental spin representations.

From the list of fundamental representations of Clifford algebras (see [10-12])

it follows that many spin coverings of orthogonal and pseudoorthogonal

groups are endowed with quaternionic structure. In particular, because the

4points of four 'dimensional Euclidean space R can be parametrised by a real

quaternion x = x + e x ( r - 1,2,3) [l3.lt>} , we obtain the followingA r r

quaternionic 5p*n coverings of orthogonal £roupf-i do^erib ing thu physicrii

4imporUint geometric transformations i.n R :

i) Sp(l)®Sp(l) = 0(4) (rotations)

ii) Sp(2) = 0(5) (de-Sitter group)

iii) Sp(l,l) = 0(4,1) ( anti-de- Sitter group)

iv) S[.(2;H) = 0(5,1) (conformal group) .

In the second part of this paper we shall describe three superalgebras

corresponding to the supersymmetrization of spin groups ii)-iv).

In the final section we shall also briefly discuss the correspondence

in four dimensions between Euclidean and Minkowski supersymmetries.

It should be mentioned that another physically interesting spin

covering with quaternionic structure is 0(6,2), describing de-Sitter

D=7 symmetry or D = 6 conformal symmetry. The supersymmetrization of

these symmetries is beyond the scope of this paper but it is considered in

a separate publication

II. QUATERNIONIC GROUPS

There are three infinite families of Lie groups with quaternionic

structure which describe linear transformations of quaternionic vector

spaces:

- special linear quaternionic group SL(n;H) (4n-l real parameters),

describing nonsingular volume-preserving linear quaternionic frames in H ,

represented by n x n quaternionic matrix h. . with arbitrary quaternion-

valued entries, satisfying only the condition det h=l, where the quater-

nionic determinant is defined according to Dieudonne [l6,lT].-

- unitary quaternionic group U(n;tO = Sp(n) (n(2n+l) real para-

meters), describing the isometries of H with the following scalar product:

i - A

where q.=q.+ q.e , q,= q° _ii i r i i

~ ~ Ore

(2.1)

(2.2)

- 3 -

The quaternionic U(n;H) matrices are described by the subgroup of h sat-

+ + —isfying the condition h h - 1, where h = h .

ij Ji

- antiunitary quaternionic group U (n;H) = orthogonal quaternionic

group O(n,H) (n(2n-l) real parameters), describing the Isometries of H

with the invariant scalar product {see [lBj )

(2.3)

In general q. = q. + q.e 1'e9 + c'.e c a n ^e replaced by the involution

describing reflection with respect to any fixed plane in three-space spanned

by three imaginary units. The quaternionic U(n;H) = O(n;H) matrices are+ it

described by the condition h 6 h - e equivalent to h h = 1, where

h* = h .

The group U(n;H) is compact but has its noncompact version

U(n-s,s;H) which ia obtained if we generalize the canonical scalar product

(2.1) as fallows

In particular, i.f we choose h. . = e C. . me obtain the quaternionic group

Sp(2k;H) which satisfies the realtion

(a.

The group O(n;H), similarly like O(n;C)r does not have different non-

compact versions.

III. QUATEKNICNtC SUPERSPACES AND QUATERNIONIC SUPERGROUPS

nOne. can extern:] '.he 1 ineftr vector space H over the c;'Jii \.^ •

The

(3.1)

and the quaternionic superspace in a given linear frame is described by

n + ra coordinates (q,.©j )= Q € H ' (A= 1,... n-mi). All possible

linear frames are described by graded quaternionic matrices

•n -vnB £U - bosonic sector

(3.2)

\ y ft / F W " - Ferfliionic sector

where the elements K. . belong to the Z ^graded quaternion-valued Grassnmn

aLgebra G..T with B, t B in the even sector G . and F , F in the oddH 1 2 H 1 2

sector G,,, Because the superspace H ' is also Z -graded ( q. belong" , , 2 i

to even sector Q and to odd sector Q , we require that bath

gradings are consistent under the multiplication "J? ffi Q -> Q, i.e. H is

the left module over G with the propertyH

f L 1 ( I ) I |, v \ . . . •

The relation (4.3) implies that the Grassmannvariables in 'Pi and Q anti-

commute. Further,one can intiwli-e an infinitesimal change of the superspace

frame

(3.4)

where describe real infinitesimal parameters in G , and the parameters

represent resl infinitesimal Grassmann parameters in G ; the graded

transposition is given by the formula

rv. R 1 \(3.5)

the

quaternion-valued generators B , F form Z - garded superalgebra.

Because the generators are represented by the matrices over associative

field H, the Z -graded Jacobi identities are satisfied automatically.

It appears that all three quaternionic groups have their super-

symmetric extensions. We get

- special linear quaternionic superalgebra SL(n,m;H) by considering

the arbitrary quaternion-valued matrix generators f in odd sector

F © F . One obtains

(3.6)

Introducing the supertrace STr3t= TrB - TrB and quaternionic trace

Sp(e e ) = - S , one getsr s rs

= o (3.7)

We see that the bosonic generators B which can be expressed as a bilinear*

form in 5 satisfy the condition Sp(STrB) =0 . The exponentiation of such

a superalgebra gives the elements of special linear graded quaternionic

supergroup SL(n,m;H).

- one can extend supersymmetricaliy the scalar product (2.1) by

supplementing it with the scalar product (2.3) in the fermionic sector

where the canonical form of the super-metric is

(3.9)

The supergroup which leaves (3i3) invariant we shall call quaternionic unitary-

antiunitary supergroup and denote by UU^ (n;m;H). The superalgebra

L'U i (n;m;H) is gi^en by the quaternionic matrices Hatisfyini^ tho r-"l ai ion !

a-ririr

(3.10)

where Q, . - 0-<u denotes graded quaternionic conjugation of the first

kind, and the supergroup is defined by the condition

(3.11)

The supergroup UU^(n;m;H) has its noncompact versions UU^ (n-s,s ;rti;H)

which are obtained if we introduce in (3.8) the general Hermitean-anti-

Hermitean metric

.fAftH^ 0

0 RmjH-H (3.12)

In particular the supersymmetric extension of the orthogonal scalar product

(2.3)

(Q (3.13)

defining orthosymplectic quaternionic supergroup OSp(n,2k;H) can be

obtained by the choice

0

G.AP> 0

0(3.14)

Because UU^(n;m;H) = UjU(m;n;H), where

we obtain the following relation, consistent with (2.4) and (2.6);

(3.15)

(3.16)

•[•;-,,.,,,,..-.„.,. „, , •• , -h.it t h t r e »xii=ts '•>•< 'I :;n;'.;ue n e t r i c-pros(.>rving q u a t e r n i o n i c

:;uu';re.-f,.ip Ul; , ( n - a ,.:, ] m ; [I ! .

- 8 -

1

fJOMHLKX I-AKAMKTHi'/.ATrON OF QUATKilN J ON IC GROUPS AND OlMTFKNTONJ"

SUPERGROUPS'

a) Quaternionic groups.

Let us write the quaternion q as follows;

(4.1)

One can describe q* = (q ,...q )6H by complex vector (z[ , f ) 6 C , whereI n 12

(4.2)

In particular q.-+q. implies z -+z , z -»-z , and the quaternionic

scalar products can be described by pairs of complex scalar products.

We obtain

- quaternionic unitary scalar product (see (2.1))

9.3)

where j? -» (tt , u ). We see that

- quaternionic antiunitary scalar product (see (2.3))

(4.5)

We obtain

Other quaternionic scalar products, called orthogonal and symplectic, are

equivalent to the unitary and the antiunitary.

In order to describe the linear quaternionic group as the complex

- 9 -

.IMP we first '.h,\c-x vo thai thn grm:;- SL(n;H) car; be embedded in SL(2n;C).

Indeed, for asingLe quaternionic variable

(4.7)

where J = i 0"1 describes the 2 x 2 complex matrix realization ofr r

quaternionic algebra describing left multiplication , and

K ^ T X iC= -L5-, C ^ 3 = L C ^ (4.8)

where C denotes complex conjugation, is the 2 x 2 matrix realization of

quaternionic algebra describing right multiplication. We see that from the

right the multiplication by e and e is antilinear

o (4-9)

because left and right multiplicationicoromute.

We restrict the transformations A£EL(2n;C) to the ones representing

the left multiplication by GL (n;H) if we assume that fA, K r=0, where

K = I © K Because [A, K ! =0 for any A and K - K,K^t it is auf-r n r 1 3 1 2

ficient to assume that

(4.10)

2nwhere for the general choice of the quaternionic basis in C

(4.11)

The 2nx2n matrix algebra of SL(2n;C), restricted by the condition (4.10 )

is denoted by SU (2n) and has the form

- 10 -

The generators of SU (2n) are obtained from those of SU(2n) by

- -putting SU(2n) into a symmetric Riemsnnian pair (USp(2n ;C) (+)

w?n),'USpf2n;C) ) , corresponding in quaternionic notation to the splitting

:L(n;l-I) = U(n;H)© SL(n;H)/U(n;H)/

- multiplying the generators from the coset SU(2n)/USp(2n;C) =

:;-L(n;H)/U(n;H) by i.'

b) Quaternionic supergroups.

It was shown in Sec. Ill that there are only two quaternionic super-

groups:

i) SL(n,m;H) = SU*(2n,2m) .

This supergroup contains the bosonic sector SL(n;H)® GL(m;H) =

SU (2n) x U (2m). Because SU (2n) is not a metric-preserving group,

SL(n,m;H} cannot be described as the intersection of two complex super-

groups .

The superalgebra SU (2n,2nO is obtained from SU(2n,2m) in the

following two steps:

- one introduces Z -grading of SU(2n;2m) as follows [22]

L, L,(4.13)

where

triad

(4.14)

- one introduces the generators L of SU(2n;2m) by the following

rescalingr

(4.15)

- 11 -

•*-~v*t+ • » l « ^ ' l -•- -'.-•

i i ! UU^ (n;m;H).

n, mThe scalar- product (3.8) in Quaternionic superspace H can be

described equivalently by the following pair of complex scalar products in

2n,2mcomplex superspace C :

(4.16)

where n GrasRrnsnn- valued quaternions C7 , t are described by 2n

complex Grassrnann coordinates {& , @ 1, (? t

It is easy to see from the formula (4.16) that

complex Grassrnann coordinates {& , @ 1, (? t? ) via the formula (4.1].

UU0L('»i',w)H) - S

Taking possible signatures in U(n;H) sector one gets

and in particular

(4.18)

(4.19)

V. REAL PARAMETRIZATION OF QUATERNIONIC GROUPS AND QUATERNIONIC

SUPERGROUPS.

a) Quaternionic groups.

the unitary product of two quaternion-valued vectors q ,p € H

(5.1)

- 12 -

can be represented by four real scalar products in R

(5.2)

where the first one is Euclidean, and the remaining three symplectic. We

therefore see that one can describe U{n;H) as the intersection of four

real groups in R

Sp (^,R)n 5p(=),

(5.3)

with the aymplectic metrics £ = E @ l n where E is given by the

formulae:

Three real symplectic matrices satisfy the quaternion algebra

(5.5)

The generators UK

l,.,.n) of U(n;H) satisfying quater-

nionic anti hermicity condition can be written as follows:

(5.6)

where A - A and S = SJi U Ji

The quaternionic antiunitary scalar product is represented by the4n

Aillowing four real scalar products in R :

(5 .7 )

where the f i r s t one is syroplectic, and tiie three remaining are pseudoEuclidean with

the signature (2n,2n). Therefore one can write

The quaternionic generators ( U ) , , can be decomposed in terms of real ones

as follows:

(5.9)

(r) (r)where S = S.. and A . = - A.

ij Ji ij JiIn order to describe the linear quaternionic group SL(n;H) as the

dngroup of linear transformation in R let us observe that the multipli-

andcation of quaternions from the left/, right canbe represented as follows:

(5.10a)

where = -SyO vr

are given by (5.1*) and

0 -I 0 C

0 0 ij

0 O 1 0

C 0 0 1H Q <J O

o -'•• .- '-J

(5.10b)

(5.11)

satisfy the algebra of quaternionic imaginary units

and

(5.12a)

(5.12b)

We see that six 4 x 4 matrices describe the fundamental real representa-

tion of 0(4) algebra.

The transformations t/L t SL(4n;R) are restricted to those

representing left multiplication by SL(n;H ) if

r= 1,2,3

Due to the relation (5.12a) it is sufficient to assume that (5.13) is valid

only for two values of r. Therefore w e see that the qliaternionic structure

4n 2of R is described by two real conditions, selecting out of 16 n -1

parameters of SL(4n;R) the 4n - 1 parameters of SL(n;H).

b) Quaternionic supergroups.

i) SL(n,m;H)

In order to describe the quaternionic superalgebra SL(n,m,;H)

describing linear frames in K ' we should again use the condition (5,13)

with the replacement

(5.15)

Therefore we see that the quaternionic superalgebra UU^n.mjH) is the

4n ,4mintersection of the following four orthosymplectic supergroups in R

(5.16)

(5.14)

n,mwhere I is the unit matrix in Hn ,m

ii) UU^ (n;m;H)

The scalar product (3.8) in H ' can be represented as four real

4n,4mscalar products in R as follows:

- 15 -

VI. EUCLIDEAN D=4 EXTENDED SUPERCONtORMAL ALGEBRA

In lie saond part of this paper -we shall consider the supersymmeti-

zation of SL(2;H) 2S0(5,l) al"n4ra, describing D-A Euclidean conformal

symmetry. It fallows from Sees.Ill and 5/ that Euclidean D=4 extended super-

conformal algebra is SL(2,N;H) = SU (4,2N). Because for a given dimension

- 16 -

-• n»4>

L1"!! cor.for^al trans-format"!*"^ descr ibo?. the Lar£e:-;t ^lobai jjeonu-;tric symmetry,

;i 1 OLI-:H' geonu; UL Lc- :r-.v :[une t ri.cr - e.g. de Sitter, anti-de Sitter, fiat

i.u:: ! soran - csri bf; obtained by considering suitable subalgebras and/or

. ':T-L T-JC t i ons . We shall show that also the supersymmetrized de Sitter, anti-

:- -.• i'.-r and flat Euclidean symmetries can be obtained in a similar way Fr;-;

. :>:ryr; superconformal algebra.

The supergroup SL(2,N;H) can be considered also as describing D=5

1 Sitter supersymmetry or the supersymmetrization of D=6 Lorentz group.

;t jrf this last role of 31,(2.10 - as describing D=6 spinora - which n/Jas

;:-.! ..cursed i-ecently in the context of supersymmetry by Kngo and Townsend[2 ] j. The

case N=l his already bewi discussed in detail by the present authors [2'i].

the D-4 Euclidean anti-de Sitter supergroup for H-l was also O'-nsitfered i in

>Eef.21 . In th L;; paper we consider the case of general M, We shall

restrict our discussion only to the infinitesimal aspect,, i.e. wo shall

consider only the superalgebras.

a) Quaternionic parametrization.

We Introduce the fundamental quatermor.ic realization of SL(2,H;H)

superalgebra by means of the following (2+N) x (2+N) matrices:

SL(2,N;H):

£ \

.-. • c f

(6.1)

The bosonic sector describes

i) 0(5,1) Euclidean conformal algebra {c.. = £.= n = 0 ) ,

represented by the following 2 x 2 quaternionic matrices [13,lM

JJ-1 0 \ OL>,,--OJ,

'o 0'& 0,

0(4) rotations:

Four translations:

Conformal accelerations: o bc o

- 17 -

Dilatations: 1 G \

( S. 2 )

In quaternionic notation the 0(5,1) algebra can be written as follows:

]- - (ab+ba)D+(6.3)

where

-real quaternions describe four vectors, e.g. a = a, e.. (e=(e ,1)/* r* r

and P(e ) ^ describe the four momentum vector,

- imaginary quaternions describe selfdual (antiselfdual) parts of

0(4} antisymmetric tensors, e.g. H(u> ,0) = w M (M(O,tj) = 0> N } wherer r r r

(6.4)

or using covariant notation M(o; , o» ) = H (ui e , ui e ) = a M

whereuv uv

One can check that the realtions (6.3) are equivalent to the

standard form of 0(5,1) algebra (see 6.25).

ii) GL(N;H) = A©SL(N;H) internal symmetry algebra (a = b ;

_JW - flj =0) where A is the noncompact chiral generator and

EL(M;H) - SU (2H) is the nonabelian part, are represented by the following

N x N quaternionic matrices:

- 18 -

Nonrompact chiralgenerator

I nr a l

t;o-iorators:

A =•C \

1 (6.6)

(6.7)

(6.8)

The SL(N;H) algebra can be described in quaternionic language In the fol-

lowing compact way:

(6.3)

The fermionic sector is described by 4N quaterni.onic supercharges,

represented by the following matrices:

o ; o0 0 i

i ° : • • ° -\'o o

0.5-.U

0 • 6

0 • 0

0 11 O-

* o 0:

(6.1Ca)

- 19 -

I1I"«JB-B 2L;..

1'he ari+: i com;y;nt.y t. ion

follows:

lor the QLiat crnionic

The covariance relations of the superalgebra SL(2;N;H) are

We see that if N=2 the Abelian charge A commutes with all other generators,

i.e. it 13 a central charge.

b! Complex parametrization.

In order to obtain the complex description of SL(2;N;H)= SU (4,2N)

- 20 -

we use the correspondence

(6.13)

where £ = %^ + and The 4N quaternionic supercharges

Q.(£)> 11.(5) can be expressed by 16N complex supercharges S , UJ J ' A A

( * -1,2 A=l,,,4, j=l,,,N).

(6.14)

Introducing complex 0(5) Dirac matrices

o it (6.15)

where c^ = (- iff.,1), and complex 4x4 matrix representation of 0(5,1)

algebra ( r,s = ],..,5)

(6.16)

one can show that the supercharges S , U do satisfy the pseudo-Majorana

conditions [23] _

where C= f P satisfies the relation C I' = - P^C and C - -1.

In.- fundamental an ti commut at ion relations take the form

•'« ! f

(6.18)

where a,b- 0,1,.,.5 and

Therefore we see that we describe the bosonic sector in forms of real

generators,

f^ /*• (6.20)

The covariance relations for complex supercharges (6.14) take the form:

[A ,

The complex form of the superalgebra SU (4.2N) with the separation

of P., and K.i generators is obtained by the following choice of ther- f-

complex superchargesJ

- ; i ' : t i , - . ' • ( ) • : - J I i n ; i f • . n " ; > : n e n I r, i ; ) n ' u . o r r . T i t ' i t i o n r s l a t i o n s ?

- 2k' -

rermionic sector is described by 16N real quaternionic charges

and the relations (6.11) can be written as follows

( 6. 2 4 )

c) Real parametrization.

One can check that the relations(6.3) are equivalent to the following

standard form of 0(5,1) algebra for the generators {6. 3 )

(6.28)

(6.25)

where E = (E I ) and F = (F , I ) are two four-dimensional real reali-

zations (5.4) and (5.10b) of quaternionic algebra, and E = g E , whereA AB B

gAB = {~ ' -' -' + K

The covariance relations look as follows:

Introducing the real SL(N;H) generators via the formula (6.20) one

can write the real Lie algebra of the internal sector GL(n;H)=T,,© A in the

form

- U

n:

•iA

I- A

23 -

-. — f *•

(6.29)

VII. EUCLIDEAN D=4 EXTENDED DE- SITTER SUPERALGEBfiAS

It is known (see e.g.£25]) that the D=d conformal superalgebra

SU(2,2;N) in Minkowski space does contain as its sub-superalgebra the extended

de-Sitter superalgebra 0Sp(N;4). Similarly, if we define the following

quatet-nionic supercharges-.

we obtain the operator basis of two Euclidean D~4 extended de-Sitter

superalgebras

0Sp(N,2;H) = UI^d.lfNjH) (bosonic sector:

Sp(2;H) © O(H;H) ~ USp(2,2;C) © 0*(2N) »

0(4,1) © 0*(2N))(7.2)

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IJlj 'PjN;!!! (bosonic sec Lor:

U(2;H) S O(N;H) « USp(4;C)e

0(5) (J, 0 (2N))(7.3)

a) Quaternionic paranietrization

The arbitrary element of the superalgebras (7.2-3) generated by the

supercharges (7.1) looks as follows:

i^-D (7-4)

In quaternionic notation the fundamental anticommutation relations have the

form

(7.5)

where

The generators (7.6) describe the internal O(N;H) Lie algebra

(7.6)

w (7.7)the same in both superalgebras (7.2) and (7.3).

b) Complex and real parametrization.

The complex representation of the Euclidean de Sitter superalgebras

is obtained if we introduce the following 8N complex supercharges .'

B (7.8.)

.'hich, due t:o the reali ty of <; and C _P matrices ;;f: vl :•

oudo-Mojorana conditions:

•-> A ^ v •$ 'Aft ft

"'"he fundamental anticommutators look as follows

i"\. ihe rjuaLernion Lc

(7 .9 )

s

(7.10)

-AB

where t^ = t (ejx ) and

/ • * - / " - / * - (7.12)

In the relations (7.10-11) we describe the bosonic sector in terms of real

generators:

i ) geometric 0(4,1) (sign "+") or 0(5) (sign "-") algebra

t V >M

C I- C -

ii) internal sector 0(N;H)s0 (2N)

( 7 . 1 3 )

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fkJt

F

t l

(7.14)

(EA)£ ( , = ( F A ) s r and ^ ^ ( - , + , - , + } .

Finally the covariance relat ions do have the form

°A

V) e would l ike to mention ttiat by introducing 4N. real supercharges

(7.15)

{7.IB)

one can write the anticommutators (7.5) in real form.

VIII. FROM EUCLIDEAN TO HINKOWSKI SUPERSYMMETRIES VIA ANALYTIC <XWnKUA.TION

There exists the analytic continuation from N-extended D=4 Euclidean

conformal supersymmetry to 2N-extended D=4 Minkowski conformal supersym-

metry I2k\

and similarly

OSp (N • I \ M ) (.IN', k )(8.2)

We see that only even N-extended Minkowski conforrnal and de-Sitter super-

symmetries do have their Euclidean counterparts. In particular the Wess-

Zumino SU{2,2;1) auperalgebra cannot be continued to the Euclidean space.

The analytic continuation is obtained by the complex rescaling of

the part of generators only from the bosonic sector:

D=4 conformal superalgebras SU (4)©(2N)

D=4 anti-de Sitter

superalgebras USp{2,2;C)©0

We shall briefly describe four analytic continuations in the for-

mulae written above:

i) U (2N) .w^-v~> U(2N)

Due to the known decomposition of the generators of U(2N) and

u (2N)- [ l

(8.3)

the analytic continuation is straightforward.

ii) SU (4) -^*-~-> SU{2,2)

The analytic continuation consists of two steps:

SU (4) ~ w — . ~ » SU{4) ^v—•—•* SU(2,2} ,

The first step is described in i), and the second is obtained by muliplying

the elements of SU(4) matrix with one index 3 or 4 by i.

iii) 0 (2H) -w~> 0(2N)

In analogy to i)

0*(2.N) s U(N) ©tht analytic continuation is straightforward.

iv) USp(2,2;Cl~~~> S?>(4) -*, 0(4,1) ~~~-> 0(3,2)

lbs analytic continuation is obtained by muitipiyinr; thn ,y:ricm', or

M k l (k,1=0,1,2,3,4) of 0(4,1) by i if k or 1 is equal to 1.

The decompositions (8.3) and (8,4) can be also described in quater-

nionic notation. We have

-r-Si

U(N;H):

and the analytic continuation GL(N;H)~*U(2N) is deBcribed by

(B.5)

(8.6)

where q =iq + q e . Similarly, if we introduce the generators0 k k

we obtain

U(N):

and the analyitic continuation O(N;H)

(8.8)

0(211) is described Toy

We recall that in the procedure of analytic continuation (8.1) and (8.2) the

supercharges remain unchanged.

ACKNOWLEDGEMENT

Both authors would like to thank Professor Abdus Salam, the Inter-

national Atomic Energy Agency and UNESCO for warm hospitality at the

Internationrj I Centre for Theoretical Physios, Trieste .

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