STAR-FUNDAMENTAL ALGEBRAS: POLYNOMIAL IDENTITIES AND ASYMPTOTICS

Autor(es)

Giambruno, Antonio ^[1] ; La Mattina, Daniela ^[1] ; Milies, Cesar Polcino ^{[2, 3]}

Afiliação do(s) autor(es)

^[1] Univ Palermo, Dipartimento Matemat & Informat, Via Archirafi 34, I-90123 Palermo - Italy

^[2] Univ Sao Paulo, Inst Matemat & Estat, Caixa Postal 66281, BR-05315970 Sao Paulo - Brazil

^[3] Univ Fed ABC, Av Estados 5001, Santo Andre, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY; v. 373, n. 11, p. 7869-7899, NOV 2020.
• Citações Web of Science: 1

Resumo

We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution {*}. To any star-algebra A is attached a numerical sequence c(n){*}(A), n >= 1, called the sequence of {*}- codimensions of A. Its asymptotic is an invariant giving a measure of the {*}- polynomial identities satisfied by A. It is well known that for a PI-algebra such a sequence is exponentially bounded and exp{*}(A) = limn(n ->infinity) n root c(n){*}(A) can be explicitly computed. Here we prove that if A is a star-fundamental algebra, C(1)n(t) exp{*} (A)n <= c(n){*}(A) <= C(2)n(t) exp{*} (A)(n), where C-1 > 0, C-2, t are constants and t is explicitly computed as a linear function of the dimension of the skew semisimple part of A and the nilpotency index of the Jacobson radical of A. We also prove that any finite dimensional star-algebra has the same {*}- identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in {[}J. Algebra 383 (2013), pp. 144-167] we get that if A is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, lim(n ->infinity) log(n) c(n){*} (A)/exp{*}(A)(n) exists and is an integer or half an integer. (AU)

• Processo FAPESP: 15/09162-9 - Álgebra não comutativa e aplicações
• Beneficiário: Francisco Cesar Polcino Milies
• Modalidade de apoio: Auxílio à Pesquisa - Temático