Group Gradings on Associative Algebras

Abstract

Let R s [ R be a G-graded ring. We describe all types of gradings on R ifG is torsion free and R is Artinian semisimple. If R is a matrix algebra over an algebraically closed field F, then we give a description of all G-gradings on R provided that G is an abelian group. In the case of an abelian group G we also classify all finite-dimensional graded simple algebras and finite-dimensional graded division algebras over an algebraically closed field of characteristic zero. ᮊ 2001Academic Press

Identities of graded algebras and codimension growth

“…All abelian gradings on matrix algebras have also been described in [9]. Let G be an abelian group and S, T two subgroups in G. First, we consider an S-graded algebra A = ⊕ s∈S A s and a T -graded algebra B = ⊕ t∈T B t .…”

“…In the proof of Theorem 6 in [9] it was shown that R = BC, where B M p (F ) and C M q (F ) are commuting G-graded subalgebras of R with elementary and "fine" gradings, respectively. Moreover, R e = B e , where e is the identity of G. Set T = Supp B, H = Supp C and suppose that T ∩ H contains a non-identity element g. Then C g −1 = 0 by Lemma 2.4 and there exist b ∈ B g , c ∈ C g −1 with 0 = bc ∈ R e \ B.…”

“…Since A is semisimple, all factors A α can be taken semisimple. From the description of all possible gradings on the full matrix algebra (see [9], Theorems 5 and 6) and graded simple algebras ( [9], Theorem 7) it follows that over an algebraically closed field of characteristic zero there exist only finitely many pairwise non-isomorphic G-graded semisimple algebras of fixed dimension. Hence one can take a finite set of G-graded homomorphic images A 1 , .…”

“…In the case of abelian groups the G-graded simple algebras of finite dimensions have been described in [9].…”

Counting Fine Gradings on Matrix Algebras and on Classical Simple Lie Algebras

“…[EK13] and references therein). In particular, a classification of fine gradings up to equivalence is known for matrix algebras over an algebraically closed field F of arbitrary characteristic [HPP98a,BSZ01,BZ03] and for classical simple Lie algebras except D 4 in characteristic different from 2 [Eld10,EK12c]. Type D 4 is different from all other members of Series D due to the phenomenon of triality.…”