Search results for "Polynomial Identity"
showing 10 items of 57 documents
Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions
2018
Let A be a superalgebra with graded involution or superinvolution ∗ and let $c_{n}^{*}(A)$, n = 1,2,…, be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of $\mathbb {Z}_{2}$ and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varie…
Codimensions of star-algebras and low exponential growth
2020
In this paper we prove that if A is any algebra with involution * satisfying a non-trivial polynomial identity, then its sequence of *-codimensions is eventually non-decreasing. Furthermore, by making use of the *-exponent we reconstruct the only two *-algebras, up to T*-equivalence, generating varieties of almost polynomial growth. As a third result we characterize the varieties of algebras with involution whose exponential growth is bounded by 2.
Classifying Algebras with Graded Involutions or Superinvolutions with Multiplicities of their Cocharacter Bounded by One
2020
Let A be superalgebra over a field of characteristic zero and let ∗ be either a graded involution or a superinvolution defined on A. In this paper we characterize the ∗-algebras whose ∗-cocharacter has multiplicities bounded by one, showing a set of ∗-polynomial identities satisfied by such algebras.
Star-fundamental algebras: polynomial identities and asymptotics
2020
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 A is attached a numerical sequence c n ∗ ( A ) c_n^*(A) , n ≥ 1 n\ge 1 , called the sequence of ∗ * -codimensions of A A . Its asymptotic is an invariant giving a measure of the ∗ * -polynomial identities satisfied by A A . It is well known that for a PI-algebra such a sequence is exponentially bounded and exp ∗ ( A ) = lim n → ∞ c n ∗ ( A ) n \exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)} can be explicitly compute…
Codimensions of algebras and growth functions
2008
Abstract Let A be an algebra over a field F of characteristic zero and let c n ( A ) , n = 1 , 2 , … , be its sequence of codimensions. We prove that if c n ( A ) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α > 1 , an F-algebra A α such that lim n → ∞ c n ( A α ) n exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.
Cocharacters of Bilinear Mappings and Graded Matrices
2012
Let Mk(F) be the algebra of k ×k matrices over a field F of characteristic 0. If G is any group, we endow Mk(F) with the elementary grading induced by the k-tuple (1,...,1,g) where g ∈ G, g2 ≠ 1. Then the graded identities of Mk(F) depending only on variables of homogeneous degree g and g − 1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g − 1 and we find an explicit decomposition of the …
Multialternating Jordan polynomials and codimension growth of matrix algebras
2007
Abstract Let R be the Jordan algebra of k × k matrices over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables of order k 2 and we prove that f is not a polynomial identity of R . We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f , we are able to prove that the exponential rate of growth of the sequence of Jordan codimensions of R in precisely k 2 .
Gradings on the algebra of upper triangular matrices of size three
2013
Abstract Let UT 3 ( F ) be the algebra of 3 × 3 upper triangular matrices over a field F . On UT 3 ( F ) , up to isomorphism, there are at most five non-trivial elementary gradings and we study the graded polynomial identities for such gradings. In case F is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . We finally compute the multiplicities in the graded cocharacter sequence for every elementary G -grading on UT 3 ( F ) .
Standard polynomials and matrices with superinvolutions
2016
Abstract Let M n ( F ) be the algebra of n × n matrices over a field F of characteristic zero. The superinvolutions ⁎ on M n ( F ) were classified by Racine in [12] . They are of two types, the transpose and the orthosymplectic superinvolution. This paper is devoted to the study of ⁎-polynomial identities satisfied by M n ( F ) . The goal is twofold. On one hand, we determine the minimal degree of a standard polynomial vanishing on suitable subsets of symmetric or skew-symmetric matrices for both types of superinvolutions. On the other, in case of M 2 ( F ) , we find generators of the ideal of ⁎-identities and we compute the corresponding sequences of cocharacters and codimensions.
Varieties of Superalgebras of Polynomial Growth
2017
∗ The author was partially supported by MIUR of Italy.