Contemporary Mathematics

Volume 420, 2006

Group Gradings on Associative Superalgebras.

Y. A. Bahturin and I. P. Shestakov

ABSTRACT. In this paper we describe all group gradings by a finite abelian

group

G

of any simple associative superalgebra over an algebraically closed

field F. Some restrictions on the characteristic of F apply.

1. Introduction

Let

T

be an abelian group,

F

a field. An associative algebra

A

is called a

T-

superalgebra if

A

is equipped with a grading by T, that is,

A=

ffitET

At

where each

At

is a vector subspace of

A

and

At As CAts.

A subspace (subalgebra, ideal)

B

of

A

is called graded if

B

=

ffitET(B

nAt).

A superalgebra

A

is called simple if

A

has

no proper nonzero graded ideals. Using other terminology, a simple superalgebra

is a graded simple algebra.

Before we start our discussion we introduce two types of gradings by groups on

the matrix algebras

[4].

If

A= Mn(F)

the any n-tuple

(g1, ...

,gn)

of elements of

G defines an elementary G-grading of A if we define A

9

= Span{Eij

I gi

1g1 = g}.

Here

Eij

is usual matrix unit. Any grading obtained from this by an automorphism

of

A

is also called elementary.

A grading of

A= Mn(F)

by

G

~

Zn

x

Zn

is called an c-grading, where cis a

primitive nth root of 1, if

A

9

=Span {X9

},

for any

g E

G.

If

a,

bare the generators

of

G

and

g

=

aib1

then

X

9

= X~Xl. Finally,

(1) Xo

~

[

T

0

: ] , x.

~

0

1

0 0

n

cn-2

0 0 1 0

0

0 0 0 0

1 0 0 0

The mapping a::

G

x

G---- F*

given by

a(aib1,akb1)

=

c1k

is a multiplicative

bicharacter on

G

and

X

9

Xh

=

a(g, h)Xgh·

The ratio

f3(g,

h) =

a(g, h)ja(h,

g) is

2000

Mathematics Subject Classification.

Primary 16W20, 16W22, 16W50, 16W55, 17A70,

17B70, 17C70.

Key words and phrases.

Graded algebra, simple associative superalgebra, matrix algebra.

The research of the first author was partially supported by NSERC grant 227060-04 and

FAPESP, grant 02-01-00219.

The research of the second author was partially supported by CNPq, grant 304633/03-8 and

FAPESP, grant 05/54063-7.

@2006 American Mathematical Society

http://dx.doi.org/10.1090/conm/420/07964