Gradings on the algebra of upper triangular matrices and their graded identities. (Q1883027)

Let \(UT_n(K)\) be the algebra of \(n\times n\) upper triangular matrices over an infinite field \(K\). The authors describe the group gradings of \(UT_n(K)\) which are elementary, i.e. all the matrix units in \(UT_n(K)\) are homogeneous elements. The generators of the ideals of graded polynomial identities of \(UT_n(K)\) are described for the elementary gradings, and linear bases of the corresponding relatively free graded algebra are found. It is shown that non-isomorphic gradings produce different graded identities. Several examples with explicit computations are given for gradings by cyclic groups.