Graded Identities and Central Polynomials for the Verbally Prime Algebras

Autor(es)

Fidelis, Claudemir ^{[1, 2]} ; Diniz, Diogo ^[2] ; Bernardo, Leomaques ^[2] ; Koshlukov, Plamen ^[3]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Inst Matemat & Estat, BR-05508090 Sao Paulo, SP - Brazil

^[2] Univ Fed Campina Grande, Unidade Acad Matemat, BR-58429970 Campina Grande, PB - Brazil

^[3] Univ Estadual Campinas, Dept Math, BR-13083859 Campinas, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: ALGEBRAS AND REPRESENTATION THEORY; JUL 2021.
• Citações Web of Science: 0

Resumo

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 x H, we define the R-hull of A as the G x H-graded algebra given by R(A) = circle plus((g,h)is an element of G) (x) (H)A((g,h)) circle times 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 x H-graded algebra A is known. In particular, for any a, b is an element of N, we find a basis for the graded identities and the graded central polynomials for the algebra M-a,M-b(E), graded by the group G x Z(2). Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural Z(2)-grading and the matrix algebra Ma+b(F) is equipped with an elementary grading by the group G x Z(2), so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on M-a,M-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 M-a,M-b(E) circle times M-r,M-s(E) is PI-equivalent to M-ar+bs,M-as+br(E). Finally, when the grading group is Z(3) x Z(2) (resp. Z x Z(2)), we present a complete description of a basis for the graded central polynomials for the algebra M-2,M-1(E) (resp. M-a,M-b(E) in the case b = 1). (AU)

• Processo FAPESP: 19/12498-0 - Identidades polinomiais graduadas e com traço, e teoria dos invariantes
• Beneficiário: Claudemir Fideles Bezerra Júnior
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado
• Processo FAPESP: 18/23690-6 - Estruturas, representações e aplicações de sistemas algébricos
• Beneficiário: Ivan Chestakov
• Modalidade de apoio: Auxílio à Pesquisa - Temático