On ℤn-graded identities of block-triangular matrices

The algebra of (Formula presented.) matrices over a field (Formula presented.) has a natural (Formula presented.) -grading. Its graded identities have been described by Vasilovsky who extended a previous work of Di Vincenzo for the algebra of (Formula presented.) matrices. In this paper, we study the graded identities of block-triangular matrices with the grading inherited by the grading of (Formula presented.). We show that its graded identities follow from the graded identities of (Formula presented.) and from its monomial identities of degree up to (Formula presented.). In the case of blocks of sizes (Formula presented.) and 1, we give a complete description of its monomial identities and exhibit a minimal basis for its (Formula presented.) -ideal.The algebra of n × n matrices over a field F has a natural Zn-grading. Its graded identities have been described by Vasilovsky who extended a previous work of Di Vincenzo for the algebra of 2 × 2 matrices. In this paper, we study the graded identities of632302313FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2012/16838-0Kemer, A.R., Varieties and ℤ2-graded algebras (Russian) (1984) Izv. Akad. Nauk SSSR, Ser. Mat, 48, pp. 1042-1059. , Translation: Math. USSR, Izv. 25 (1985), 359–374Sviridova, I., Identities of pi-algebras graded by a finite abelian group (2011) Comm. Algebra, 39, pp. 3462-3490Aljadeff, E., Kanel-Belov, A., Representability and Specht’s problem for G-graded algebras (2010) Adv. Math, 225, pp. 2391-2428Giambruno, A., Zaicev, M.V., Exponential codimension growth of P.I. algebras: an exact estimate (1999) Adv. Math, 142, pp. 221-243Giambruno, A., Zaicev, M.V., On codimension growth of finitely generated associative algbras (1998) Adv. Math, 140, pp. 145-155Giambruno, A., Zaicev, M.V., Codimension growth and minimal superalgebras (2003) Trans. Amer. Math. Soc, 355, pp. 5091-5117Lewin, J., A matrix representation for associative algebras I (1974) Trans. Amer. Math. Soc, 188, pp. 293-308Giambruno, A., Zaicev, M., Polynomial identities and asymptotic methods (2005) Mathematical surveys and monographs, 122. , Providence (RI): American Mathematical SocietyCentrone, L., de Mello, T.C., A model for the relatively-free graded algebra of block-triangular matrices with entries from a graded PI-algebra (2013) J. Pure. Appl. AlgebraDi Vincenzo, O.M., La Scala, R., Block-triangular matrix algebras and factorable ideals of graded polynomial identities (2004) J. Algebra, 279, pp. 260-279Di Vincenzo, O.M., On the graded identities of M1,1 (1992) Israel J. Math, 80, pp. 323-335Vasilovsky, S.Y., ℤn-graded polynomial identities of the full matrix algebra of order n (1999) Proc. Amer. Math. Soc, 127, pp. 3517-3524Valenti, A., Zaicev, M., Abelian gradings on upper block triangular matrices (2012) Canad. Math. Bull, 55, pp. 208-213Di Vincenzo, O.M., Koshlukov, P., Valenti, A., Gradings on the algebra of upper triangular matrices and their graded identities (2004) J. Algebra, 275, pp. 550-566Di Vincenzo, O.M., Koshlukov, P., Valenti, A., Gradings and graded identities for the upper triangular matrices over an infinite field, groups, rings, and group rings (2006) Lect. Notes Pure Appl. Math, 248, pp. 91-106Koshlukov, P., Valenti, A., Graded identities for the algebra of n×n upper triangular matrices over an infinite field (2003) Internat. J. Algebra Comput, 13, pp. 517-52