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

Let K be an infinite field and let UTn(K) denote the algebra of n x n upper triangular matricesover K. We describe all elementary gradings on this algebra. Further we describe the generators of theideals of graded polynomial identities of UTn(K) and we produce linear bases of the correspondingrelatively free graded algebras. We prove that one can distinguish the elementary gradings by theirgraded identities. We describe bases of the graded polynomial identities in several “typical” cases.Although in these cases we consider elementary gradings by cyclic groups, the same methods servefor elementary gradings by any finite group.