Graded Identities and Central Polynomials for the Verbally Prime Algebras
Tóm tắt
Let F be a field of characteristic zero and let R be an algebra that admits a regular grading by an abelian group H. Moreover, we consider G a group and let A be an algebra with a grading by the group G × H, we define the R-hull of A as the G × H-graded algebra given by
$\mathfrak {R}(A)=\oplus _{(g,h)\in G\times H}A_{(g,h)}\otimes R_{h}$
. In this paper we provide a basis for the graded identities (resp. central polynomials) of the R-hull of A, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the G × H-graded algebra A is known. In particular, for any a,
$b\in \mathbb {N}$
, we find a basis for the graded identities and the graded central polynomials for the algebra Ma,b(E), graded by the group
$G\times \mathbb {Z}_{2}$
. Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural
$\mathbb {Z}_{2}$
-grading and the matrix algebra Ma+b(F) is equipped with an elementary grading by the group
$G\times \mathbb {Z}_{2}$
, so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on Ma,b(E) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product Ma,b(E) ⊗ Mr,s(E) is PI-equivalent to Mar+bs,as+br(E). Finally, when the grading group is
$\mathbb {Z}_{3}\times \mathbb {Z}_{2}$
(resp.
$\mathbb {Z}\times \mathbb {Z}_{2}$
), we present a complete description of a basis for the graded central polynomials for the algebra M2,1(E) (resp. Ma,b(E) in the case b = 1).
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