Gradings on the algebra of upper triangular matrices and their graded identities

Graded polynomial identities play an important role in the structure theory of PI algebras.Many properties of the ideals of identities are described in the language of graded identities and graded algebras. In this paper we study the elementary gradings on the algebra UT_n(K) of n × n upper triangular matrices over an infinite field. We describe these gradings by means of the graded identities that they satisfy. Namely we prove that there exist |G|^{n−1} nonisomorphic elementary gradings on UT_n(K) by the finite group G, and show that nonisomorphic gradings produce different graded identities. Furthermore we describe generators for the ideals of graded identities for a given (but arbitrary) elementary grading on UT_n(K), and produce linear bases of the corresponding relatively free graded algebra.