Graded polynomial identities of matrices
Gradings by an Abelian group \(G\) on the \(p\times p\) matrix algebra \(M_p(K)\) over an algebraically closed field \(K\) were classified by \textit{Yu. A. Bahturin, S. K. Sehgal} and \textit{M. V. Zaicev} [J. Algebra 241, No. 2, 677-698 (2001; Zbl 0988.16033)]. The so called fine gradings and elementary gradings are the building blocks of this classification. The present paper studies \(G\)-graded identities of \(M_p(K)\), where \(K\) is a field of characteristic zero, and \(G\) is an arbitrary group. The authors find relations between the \(G\)-graded identities of \(M_p(K)\), and the \(G\times H\)-graded identities of the tensor product of \(M_p(K)\) and \(M_q(K)\), where the latter is endowed with a fine \(H\)-grading. This allows to reduce the study of \(G\)-graded identities (\(G\) Abelian) of matrices to the case of elementary gradings. Generalizing a result of \textit{S. Yu. Vasilovsky} [Proc. Am. Math. Soc. 127, No. 12, 3517-3524 (1999; Zbl 0935.16012)], the authors give a basis of the \(G\)-graded identities of \(M_p(K)\) with an elementary grading such that the identity component coincides with the diagonal of \(M_p(K)\).
scientific article; zbMATH DE number 1864766
scientific article