Classifying Quadratic Quantum Planes using Graded Skew Clifford

Classifying Quadratic Quantum Planes using

Graded Skew Clifford Algebras

Michaela Vancliff(supported in part by NSF grant DMS-0900239)

University of Texas at Arlington, USA

Journal of Algebra 346 (2011),152-164with Manizheh Nafari & Jun Zhang

M. Vancliff ([email protected]) J. Algebra 346 (2011), 152-164 uta.edu/math/vancliff 1 / 11

Motivation

2010: Cassidy & Vancliff → graded skew Clifford algebras (GSCAs)

geometry determines when GSCA is regular etc.

How useful are GSCAs in classifying (quadratic) regular algebras?

Regular algebras of gldim 2 (resp, 1) are GSCAs. Gldim 3?

The case of quadratic AS-regular algebras of gldim 3 (i.e., quadratic

quantum planes) is the goal of this talk & is joint work with

Manizheh Nafari and Jun Zhang.

Henceforth, k = algebraically closed field.


Motivation










Motivation










Motivation










Motivation










µ-symmetric Matrices

Definition

Let µ = (µij) ∈ M(n, k) be such that µijµji = 1 for all i , j such that i 6= j .

A matrix M ∈ M(n, k) is called µ-symmetric if Mij = µijMji for alli , j = 1, . . . , n.

Clearly,µij = 1 for all i , j ⇒ µ-symmetric = symmetricµij = −1 for all i , j ⇒ µ-symmetric = skew-symmetric (if char(k) 6= 2).

Example

n = 3:

a b cµ21b d eµ31c µ32e f

is µ-symmetric.

Assumption

For the rest of the talk, assume µii = 1 for all i .



Definition




Example

n = 3:


is µ-symmetric.

Assumption




Definition



Clearly,µij = 1 for all i , j ⇒ µ-symmetric = symmetric

µij = −1 for all i , j ⇒ µ-symmetric = skew-symmetric (if char(k) 6= 2).

Example

n = 3:


is µ-symmetric.

Assumption




Definition




Example

n = 3:


is µ-symmetric.

Assumption




Definition




Example

n = 3:


is µ-symmetric.

Assumption




Definition




Example

n = 3:


is µ-symmetric.

Assumption




Definition ([ Van den Bergh, Le Bruyn ] char(k) 6= 2 )

With µ as above,

Let M1, . . . ,Mn ∈ M(n, k) denote

µ-

symmetric

matrices.

A graded

skew

Clifford algebra, associated to

µ,

M1, . . . , Mn, is

a graded k-algebra A on degree-1 generators x1, . . . , xn and on degree-2

generators y1, . . . , yn with defining relations given by:

(i) xixj +

µij

xjxi =n∑

k=1

(Mk)ijyk for all i , j = 1, . . . , n, and

(ii) the existence of a normalizing sequence {y ′1, . . . , y

′n} ⊂ A2 that spans

ky1 + · · ·+ kyn.

Example

Skew polynomial rings on generators x1, . . . , xn with relationsxixj = −µijxjxi , for all i 6= j , are GSCAs.




With µ as above,


µ-

symmetric

matrices. A graded

skew


µ,

M1, . . . , Mn,

is



(i) xixj +

µij

xjxi =n∑

k=1




ky1 + · · ·+ kyn.

Example





With µ as above,


µ-

symmetric

matrices. A graded

skew


µ,

M1, . . . , Mn, is



(i) xixj +

µij

xjxi =n∑

k=1




ky1 + · · ·+ kyn.

Example





With µ as above,


µ-

symmetric

matrices. A graded

skew


µ,

M1, . . . , Mn, is



(i) xixj +

µij

xjxi =n∑

k=1




ky1 + · · ·+ kyn.

Example





With µ as above,


µ-

symmetric

matrices. A graded

skew


µ,

M1, . . . , Mn, is



(i) xixj +

µij

xjxi =n∑

k=1


(ii) yk is central for all k = 1, . . . , n.

the existence of a normalizing

sequence {y ′1, . . . , y

′n} ⊂ A2 that spans ky1 + · · ·+ kyn.

Example




Definition ([ Cassidy & Vancliff ] char(k) 6= 2 )

With µ as above, let M1, . . . ,Mn ∈ M(n, k) denote µ-symmetric

matrices. A graded

skew


µ,

M1, . . . , Mn, is



(i) xixj +

µij

xjxi =n∑

k=1






Example






matrices. A graded skew Clifford algebra, associated to

µ,

M1, . . . , Mn, is



(i) xixj +

µij

xjxi =n∑

k=1






Example






matrices. A graded skew Clifford algebra, associated to µ, M1, . . . , Mn, is



(i) xixj +

µij

xjxi =n∑

k=1






Example









(i) xixj + µijxjxi =n∑

k=1






Example










k=1




ky1 + · · ·+ kyn.

Example










k=1




ky1 + · · ·+ kyn.

Example



Following [CV], to the data µ & M1, . . . ,Mn in the definition of GSCA,

we associate

1. the skew polynomial ring S on generators z1, . . ., zn with defining

relations: zjzi = µijzizj , for all i 6= j , and

2. the elements qk = zTMkz ∈ S2 where z = [z1 . . . zn]T .

Definition ([Cassidy, Vancliff])

We call any (nonzero) element of S2 a quadratic form, and define the

quadric, V(q), determined by any quadratic form q to be the set of

points in P(S∗1 )× P(S∗

1 ) on which q and the defining relations of S vanish.

If Q1, . . . ,Qm ∈ S2, we call their span a quadric system. A quadric

system Q is said to be basepoint free (BPF) if⋂

q∈Q V(q) is empty; Q is

said to be normalizing if it is given by a normalizing sequence of S .



we associate















we associate















we associate





We call any (nonzero) element of S2 a quadratic form,

and define the










we associate















we associate









If Q1, . . . ,Qm ∈ S2, we call their span a quadric system.

A quadric






we associate











q∈Q V(q) is empty;

Q is




we associate














Theorem ([Cassidy, Vancliff])

A GSCA A = A(µ,M1, . . . ,Mn) is a quadratic, Auslander-regular algebra

of global dimension n that satisfies the Cohen-Macaulay property with

Hilbert series 1/(1− t)n iff

the quadric system associated to M1, . . . ,Mn

is normalizing & BPF; in this case, A is a noetherian AS-regular domain

and is unique up to isomorphism.





Hilbert series 1/(1− t)n iff the quadric system associated to M1, . . . ,Mn

is normalizing & BPF;

in this case, A is a noetherian AS-regular domain






Hilbert series 1/(1− t)n iff the quadric system associated to M1, . . . ,Mn

is normalizing & BPF; in this case, A is a noetherian AS-regular domain



Quadratic Quantum Planes

Returning to the classification of quadratic regular algebras D of global

dimension 3....

The classification depends on the point scheme X of D:

either X ⊆ P2 contains a line or it does not. The latter case, splits into 3

subcases, so in total we have 4 cases:

X contains a line

X is a nodal cubic curve in P2

X is a cuspidal cubic curve in P2

X is an elliptic curve in P2.

Note: our work attempts to classify all quadratic regular algebras D of

global dimension 3; not only the generic ones.




dimension 3.... The classification depends on the point scheme X of D:



X contains a line










either X ⊆ P2 contains a line or it does not.

The latter case, splits into 3


X contains a line












X contains a line












X contains a line












X contains a line












X contains a line












X contains a line












X contains a line







Theorem ( char(k) 6= 2 )

If X contains a line, then either D is a twist, by an automorphism, of a

GSCA,

or D is a twist, by a twisting system, of an Ore extension of a

regular GSCA of gldim 2.

Theorem

If X is a nodal cubic curve, then D = k[x1, x2, x3] with defining relations:

λx1x2 = x2x1, λx2x3 = x3x2 − x21 , λx3x1 = x1x3 − x2

2 ,

where λ ∈ k and λ(λ3 − 1) 6= 0. Moreover, for any such λ, any quadratic

algebra with these defining relations is regular & its point scheme X is a

nodal cubic curve in P2.

Suppose char(k) 6= 2.

If λ3 /∈ {0, 1}, then D is an Ore extn of a regular GSCA of gldim 2;

if λ3 = −1, then D is a GSCA.




GSCA, or D is a twist, by a twisting system, of an Ore extension of a


Theorem



2 ,












Theorem



2 ,

where λ ∈ k and λ(λ3 − 1) 6= 0.

Moreover, for any such λ, any quadratic











Theorem



2 ,












Theorem



2 ,












Theorem



2 ,








Theorem

X = cuspidal cubic curve in P2 iff char(k) 6= 3 & D = k[x1, x2, x3] with defrels:

x1x2 = x2x1 + x21 , x3x1 = x1x3 + x2

1 + 3x22 , x3x2 = x2x3 − 3x2

2 − 2x1x3 − 2x1x2.

(Moreover, any such algebra is regular, even if char(k) = 3.)

If char(k) 6= 2 & X = cuspidal cubic curve, then D is an Ore extn of a


It remains to consider X = elliptic curve in P2.

In [AS, ATV1], such algebras are classified into types A, B, E, H,

where some members of each type might not have an elliptic curve as their

point scheme, but a generic member does.


Theorem


x1x2 = x2x1 + x21 , x3x1 = x1x3 + x2

1 + 3x22 , x3x2 = x2x3 − 3x2

2 − 2x1x3 − 2x1x2.









Theorem


x1x2 = x2x1 + x21 , x3x1 = x1x3 + x2

1 + 3x22 , x3x2 = x2x3 − 3x2

2 − 2x1x3 − 2x1x2.









Theorem


x1x2 = x2x1 + x21 , x3x1 = x1x3 + x2

1 + 3x22 , x3x2 = x2x3 − 3x2

2 − 2x1x3 − 2x1x2.










Suppose X is an elliptic curve.

(i) Regular algebras of type H are GSCAs.

(ii) Regular algebras of type B are GSCAs.

(iii) As in [AS, ATV1], regular algebras D of type A are given by

D = k[x , y , z ] with def rels:

axy +byx + cz2 = 0, ayz +bzy + cx2 = 0, azx +bxz + cy2 = 0,

where a, b, c ∈ k, abc 6= 0, (3abc)3 6= (a3 + b3 + c3)3, char(k) 6= 3

& either a3 6= b3, or a3 6= c3, or b3 6= c3.

• If a3 = b3 6= c3, then D is a GSCA.

• If a3 6= b3 = c3 or if a3 = c3 6= b3, then D is a twist, by an

automorphism, of a GSCA.

In (iii), a3 6= b3 6= c3 6= a3 is still open.



















































where a, b, c ∈ k,

abc 6= 0, (3abc)3 6= (a3 + b3 + c3)3, char(k) 6= 3















&

either a3 6= b3, or a3 6= c3, or b3 6= c3.






























































Remarks & Questions

Up to isomorphism & anti-isomorphism, type E consists of at most 1

algebra;

it is still open whether or not this type is directly related to a

GSCA.

If D is a regular algebra of type A or E, then its Koszul dual is the

quotient of a regular GSCA; so, in this sense, such algebras are weakly

related to GSCAs.

Can cubic regular algebras of gldim 3 be classified using GSCAs?

Can quadratic regular algebras of gldim 4 be classified using GSCAs?


Remarks & Questions


algebra; it is still open whether or not this type is directly related to a

GSCA.



related to GSCAs.




Remarks & Questions



GSCA.



related to GSCAs.




Remarks & Questions



GSCA.



related to GSCAs.




Remarks & Questions



GSCA.



related to GSCAs.




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Classifying Quadratic Quantum Planes using Graded Skew Clifford