Identities and isomorphisms of graded simple algebras.
The authors study graded polynomial identities of finite dimensional \(G\)-graded simple algebras, where \(G\) is an Abelian group, the base field \(F\) is algebraically closed and such that the orders of all finite subgroups of \(G\) are invertible in \(F\). The main result is that two finite dimensional \(G\)-graded simple algebras are isomorphic as graded algebras if and only if they satisfy the same \(G\)-graded polynomial identities. The proof gives also the explicit form of the algebras and the graded polynomial identities which distinguish them. The algebras are of the form \(R=A\otimes_FB\), where \(A=M_n(F)\) is the \(n\times n\) matrix algebra with an elementary grading and \(B\) is a graded division algebra (i.e., the non-zero homogeneous elements of \(B\) are invertible). The identities used to distinguish the algebras are modifications of standard identities. The algebra \(B\) is a twisted group algebra \(F^\sigma[H]\) for a finite subgroup \(H\) of \(G\). Hence the classification the finite dimensional \(G\)-graded simple algebras is up to equivalence of the 2-cocycles \(\sigma\colon H\times H\to F^*\) defining the twisted group structure of \(B\).
scientific article; zbMATH DE number 5706972