Search results for "Colength"
showing 6 items of 6 documents
Some Numerical Invariants of Multilinear Identities
2017
We consider non-necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe most of the results obtained in recent years on two numerical sequences that can be attached to the multilinear identities satisfied by an algebra: the sequence of codimensions and the sequence of colengths.
Varieties of algebras with pseudoinvolution: Codimensions, cocharacters and colengths
2022
Abstract Let A be a finitely generated superalgebra with pseudoinvolution ⁎ over an algebraically closed field F of characteristic zero. In this paper we develop a theory of polynomial identities for this kind of algebras . In particular, we shall consider three sequences that can be attached to Id ⁎ ( A ) , the T 2 ⁎ -ideal of identities of A: the sequence of ⁎-codimensions c n ⁎ ( A ) , the sequence of ⁎-cocharacter χ 〈 n 〉 ⁎ ( A ) and the ⁎-colength sequence l n ⁎ ( A ) . Our purpose is threefold. First we shall prove that the ⁎-codimension sequence is eventually non-decreasing, i.e., c n ⁎ ( A ) ≤ c n + 1 ⁎ ( A ) , for n large enough. Secondly, we study superalgebras with pseudoinvoluti…
Characterizing varieties of colength ≤4
2009
Let A be an associative algebra over a field F of characteristic zero, and let χ n (A), n = 1,2,…, be the sequence of cocharacters of A. For every n ≥ 1, let l n (A) denote the nth colength of A, counting the number of S n -irreducibles appearing in χ n (A). In this article, we classify the algebras A such that the sequence of colengths l n (A), n = 1,2,…, is bounded by four. Moreover we construct a finite number of algebras A 1,…, A d , such that l n (A) ≤ 4 if and only if A 1,…, A d ∉ var(A).
Minimal star-varieties of polynomial growth and bounded colength
2018
Abstract Let V be a variety of associative algebras with involution ⁎ over a field F of characteristic zero. Giambruno and Mishchenko proved in [6] that the ⁎-codimension sequence of V is polynomially bounded if and only if V does not contain the commutative algebra D = F ⊕ F , endowed with the exchange involution, and M , a suitable 4-dimensional subalgebra of the algebra of 4 × 4 upper triangular matrices , endowed with the reflection involution. As a consequence the algebras D and M generate the only varieties of almost polynomial growth. In [20] the authors completely classify all subvarieties and all minimal subvarieties of the varieties var ⁎ ( D ) and var ⁎ ( M ) . In this paper we e…
On multiplicities of cocharacters for algebras with superinvolution
2021
Abstract In this paper we deal with finitely generated superalgebras with superinvolution, satisfying a non-trivial identity, whose multiplicities of the cocharacters are bounded by a constant. Along the way, we prove that the codimension sequence of such algebras is polynomially bounded if and only if their colength sequence is bounded by a constant.
Algebras with intermediate growth of the codimensions
2006
AbstractLet F be a field of characteristic zero and let A be an F-algebra. The polynomial identities satisfied by A can be measured through the asymptotic behavior of the sequence of codimensions and the sequence of colengths of A. For finite dimensional algebras we show that the colength sequence of A is polynomially bounded and the codimension sequence cannot have intermediate growth. We then prove that for general nonassociative algebras intermediate growth of the codimensions is allowed. In fact, for any real number 0<β<1, we construct an algebra A whose sequence of codimensions grows like nnβ.