Z_2-graded cocharacters for superalgebras of triangular matrices

Let K be a field of characteristic zero, let A, B be K-algebras with polynomial identities and let M be a free (A; B)-bimodule. The algebra R of 2x2 upper triangular matrices, having the elements of the algebras A and B on the main diagonal and the elements of the free module M on the (1,2) position can be endowed with a natural Z2-grading. In this paper, we compute the graded cocharacter sequence, the graded codimension sequence and the superexponent of R. As a consequence of these results, we also study the above PI-invariants in the setting of upper triangular matrices over the field K. In particular, we completely classify these invariants for the algebra of 3 × 3 upper triangular matrices endowed with all possible Z2-gradings.