Group gradings on upper triangular matrices. (Q2642252)

Graded algebras and their graded identities play an important role in the theory of algebras with polynomial identities. The paper under review studies the possible gradings on the algebra \(UT_n(F)\) of the \(n\times n\) upper triangular matrices over a field. The authors obtain a complete description of all possible gradings. Namely they prove the following theorem. Let \(F\) be an arbitrary field and \(G\) an arbitrary group. Suppose \(A=UT_n(F)\) is \(G\)-graded. Then \(A\) is isomorphic as a graded algebra to \(UT_n(F)\) equipped with some elementary \(G\)-grading. We recall that a grading on a subalgebra of \(M_n(F)\) is elementary whenever all matrix units \(E_{ij}\) are homogeneous in the given grading. Furthermore the authors state the interesting problem of describing the gradings on the algebras of block triangular matrices.