Lucio Centrone

The graded Gelfand--Kirillov dimension of verbally prime algebras

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M n (F), M n (E) and M a,b (E) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand--Kirillov dimensions.

ORDINARY AND Z2-GRADED COCHARACTERS OF UT2(E)

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic 0. In this article we consider the algebra R of 2 × 2 matrices with entries in E and its subalgebra G, which is one of the minimal algebras of polynominal identity (PI) exponent 2. We compute firstly the Hilbert series of G and, as a consequence, we compute its cocharacter sequence. Then we find the Hilbert series of R, using the tool of proper Hilbert series, and we compute its cocharacter sequence. Finally we describe explicitely the ℤ2-graded cocharacters of R.

On ℤn-graded identities of block-triangular matrices

The algebra of n × n matrices over a field F has a natural Zngrading. 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 of block-triangular matrices with the grading inherited by the grading of Mn(F ). We show that its graded identities follow from the graded identities of Mn(F ) and from its monomial identities of degree up to 2n − 2. In the case of blocks of sizes n − 1 and 1, we give a complete description of its monomial identities, and exhibit a minimal basis for its TZn ideal.The algebra of matrices over a field has a natural -grading. Its graded identities have been described by Vasilovsky who extended a previous work of Di Vincenzo for the algebra of matrices. In this paper, we study the graded identities of block-triangular matrices with the grading inherited by the grading of . We show that its graded identities follow from the graded identities of and from its monomial identities of degree up to . In the case of blocks of sizes and 1, we give a complete description of its monomial identities and exhibit a minimal basis for its -ideal.

On the growth of graded polynomial identities of

Let be a field of characteristic 0 and L be a G-graded Lie PI-algebra where the support of L is a finite subset of G. We define the G-graded Gelfand–Kirillov dimension (GK) of L in k-variables as the GK dimension of its G-graded relatively free algebra having k homogeneous variables for each element of the support of L. We compute the G-graded GK dimension of , where G is any abelian group. Then, we compute the exact value for the -graded GK dimension of endowed with the -grading of Vasilovsky.

A note on cocharacter sequence of Jordan upper triangular matrix algebra

ABSTRACT Let UJn(F) be the Jordan algebra of n × n upper triangular matrices over a field F of characteristic zero. This paper is devoted to the study of polynomial identities satisfied by UJ2(F) and UJ3(F). In particular, the goal is twofold. On one hand, we complete the description of G-graded polynomial identities of UJ2(F), where G is a finite abelian group. On the other hand, we compute the Gelfand–Kirillov dimension of the relatively free algebra of UJ2(F) and we give a bound for the Gelfand–Kirillov dimension of the relatively free algebra of UJ3(F).

A note on graded polynomial identities for tensor products by the Grassmann algebra in positive characteristic

Let G be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the TG-ideal of G-graded identities of a G-graded algebra in characteris...

Action of Pontryagin Dual of Semilattices on Graded Algebras

Let A be a ℂ-algebra and G be a finite abelian group. Then a G-graded algebra is merely a G-algebra and viceversa because of the fact that G and its group of characters are isomorphic. This fact is no longer true if we substitute G with infinite or non-abelian groups. In this paper we try to obtain similar results for a special class of abelian monoids, i.e., the bounded semilattices. Moreover, if S is such a monoid, we are going to investigate the role of S and its Pontryagin dual over the algebra A, in the case A is S-graded.