Search results for "Polynomial Identity"
showing 10 items of 57 documents
Polynomial growth of the codimensions: a characterization
2009
Let A A be a not necessarily associative algebra over a field of characteristic zero. Here we characterize the T-ideal of identities of A A in case the corresponding sequence of codimensions is polynomially bounded.
On almost nilpotent varieties of subexponential growth
2015
Abstract Let N 2 be the variety of left-nilpotent algebras of index two, that is the variety of algebras satisfying the identity x ( y z ) ≡ 0 . We introduce two new varieties, denoted by V sym and V alt , contained in the variety N 2 and we prove that V sym and V alt are the only two varieties almost nilpotent of subexponential growth.
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).
Varieties of superalgebras of almost polynomial growth
2011
Abstract Let V gr be a variety of superalgebras and let c n gr ( V gr ) , n = 1 , 2 , … , be its sequence of graded codimensions. Such a sequence is polynomially bounded if and only if V gr does not contain a list of five superalgebras consisting of a commutative superalgebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and natural Z 2 -gradings. In this paper we completely classify all subvarieties of the varieties generated by these five superalgebras, by giving a complete list of finite dimensional generating superalgebras.
Proper identities, Lie identities and exponential codimension growth
2008
Abstract The exponent exp ( A ) of a PI-algebra A in characteristic zero is an integer and measures the exponential rate of growth of the sequence of codimensions of A [A. Giambruno, M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998) 145–155; A. Giambruno, M. Zaicev, Exponential codimension growth of P.I. algebras: An exact estimate, Adv. Math. 142 (1999) 221–243]. In this paper we study the exponential rate of growth of the sequences of proper codimensions and Lie codimensions of an associative PI-algebra. We prove that the corresponding proper exponent exists for all PI-algebras, except for some algebras of exponent two strictly related to t…
Almost polynomial growth: Classifying varieties of graded algebras
2015
Let G be a finite group, V a variety of associative G-graded algebras and c (V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all sub…
Classifying G-graded algebras of exponent two
2019
Let F be a field of characteristic zero and let $$\mathcal{V}$$ be a variety of associative F-algebras graded by a finite abelian group G. If $$\mathcal{V}$$ satisfies an ordinary non-trivial identity, then the sequence $$c_n^G(\mathcal{V})$$ of G-codimensions is exponentially bounded. In [2, 3, 8], the authors captured such exponential growth by proving that the limit $$^G(\mathcal{V}) = {\rm{lim}}_{n \to \infty} \sqrt[n]{{c_n^G(\mathcal{V})}}$$ exists and it is an integer, called the G-exponent of $$\mathcal{V}$$ . The purpose of this paper is to characterize the varieties of G-graded algebras of exponent greater than 2. As a consequence, we find a characterization for the varieties with …
Anomalies on codimension growth of algebras
2015
Abstract This paper deals with the asymptotic behavior of the sequence of codimensions c n ( A ) ${c_{n}(A)}$ , n = 1 , 2 , … , ${n=1,2,\ldots,}$ of an algebra A over a field of characteristic zero. It is shown that when such sequence is polynomially bounded, then lim sup n → ∞ log n c n ( A ) ${\limsup_{n\to\infty}\log_{n}c_{n}(A)}$ and lim inf n → ∞ log n c n ( A ) ${\liminf_{n\to\infty}\log_{n}c_{n}(A)}$ can be arbitrarily distant. Also, in case the codimensions are exponentially bounded, we can construct an algebra A such that exp ( A ) = 2 ${\exp(A)=2}$ and, for any q ≥ 1 ${q\geq 1}$ , there are infinitely many integers n such that c n ( A ) > n q 2 n ${c_{n}(A)&…
Differential Identities and Varieties of Almost Polynomial Growth
2022
Let V be an L-variety of associative L-algebras, i.e., algebras where a Lie algebra L acts on them by derivations, and let c(n)(L) (V), n >= 1, be its Lcodimension sequence. If V is generated by a finite-dimensional L-algebra, then such a sequence is polynomially bounded only if V does not contain UT2, the 2 x 2 upper triangular matrix algebra with trivial L-action, and UT2 epsilon where L acts on UT2 as the 1-dimensional Lie algebra spanned by the inner derivation epsilon induced by e11. In this paper we completely classify all the L-subvarieties of var(L)(UT2) and var(L)(UT2 epsilon) by giving a complete list of finite-dimensional L-algebras generating them.
On the graded identities and cocharacters of the algebra of 3×3 matrices
2004
Abstract Let M2,1(F) be the algebra of 3×3 matrices over an algebraically closed field F of characteristic zero with non-trivial Z 2 -grading. We study the graded identities of this algebra through the representation theory of the hyperoctahedral group Z 2 ∼S n . After splitting the space of multilinear polynomial identities into the sum of irreducibles under the Z 2 ∼S n -action, we determine all the irreducible Z 2 ∼S n -characters appearing in this decomposition with non-zero multiplicity. We then apply this result in order to study the graded cocharacter of the Grassmann envelope of M2,1(F). Finally, using the representation theory of the general linear group, we determine all the grade…