Search results for "Codimension growth"
showing 10 items of 15 documents
Varieties of algebras with pseudoinvolution and polynomial growth
2017
Let A be an associative algebra with pseudoinvolution (Formula presented.) over an algebraically closed field of characteristic zero and let (Formula presented.) be its sequence of (Formula presented.) -codimensions. We shall prove that such a sequence is polynomially bounded if and only if the variety generated by A does not contain five explicitly described algebras with pseudoinvolution. As a consequence, we shall classify the varieties of algebras with pseudoinvolution of almost polynomial growth, i.e. varieties of exponential growth such that any proper subvariety has polynomial growth and, along the way, we shall give also the classification of their subvarieties. Finally, we shall de…
Varieties with at most cubic growth
2019
Abstract Let V be a variety of non necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions c n ( V ) , n = 1 , 2 , … , and here we study varieties of polynomial growth. We classify all possible growth of varieties V of algebras satisfying the identity x ( y z ) ≡ 0 such that c n ( V ) C n α , with 1 ≤ α 3 , for some constant C. We prove that if 1 ≤ α 2 then c n ( V ) ≤ C 1 n , and if 2 ≤ α 3 , then c n ( V ) ≤ C 2 n 2 , for some constants C 1 , C 2 .
Varieties with at most quadratic growth
2010
Let V be a variety of non necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions cn(V); n = 1; 2, … and here we study varieties of polynomial growth. Recently, for any real number a, 3 < a < 4, a variety V was constructed satisfying C1n^a < cn(V) < C2n^a; for some constants C1;C2. Motivated by this result here we try to classify all possible growth of varieties V such that cn(V) < Cn^a; with 0 < a < 2, for some constant C. We prove that if 0 < a < 1 then, for n large, cn(V) ≤ 1, whereas if V is a commutative variety and 1 < a < 2, then lim logn cn(V) = 1 o…
Graded algebras with polynomial growth of their codimensions
2015
Abstract Let A be an algebra over a field of characteristic 0 and assume A is graded by a finite group G . We study combinatorial and asymptotic properties of the G -graded polynomial identities of A provided A is of polynomial growth of the sequence of its graded codimensions. Roughly speaking this means that the ideal of graded identities is “very large”. We relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free G -graded algebra in the variety generated by A . We describe the irreducible modules that can appear in the decomposition, we show that their multiplicities are eventually constant depending on the shape obtaine…
On algebras of polynomial codimension growth
2016
Let A be an associative algebra over a field F of characteristic zero and let $$c_n(A), n=1, 2, \ldots $$ , be the sequence of codimensions of A. It is well-known that $$c_n(A), n=1, 2, \ldots $$ , cannot have intermediate growth, i.e., either is polynomially bounded or grows exponentially. Here we present some results on algebras whose sequence of codimensions is polynomially bounded.
Varieties of almost polynomial growth: classifying their subvarieties
2007
Let G be the infinite dimensional Grassmann algebra over a field F of characteristic zero and UT2 the algebra of 2 x 2 upper triangular matrices over F. The relevance of these algebras in PI-theory relies on the fact that they generate the only two varieties of almost polynomial growth, i.e., they grow exponentially but any proper subvariety grows polynomially. In this paper we completely classify, up to PI-equivalence, the associative algebras A such that A is an element of Var(G) or A is an element of Var(UT2).
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…
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.
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.
On Almost Nilpotent Varieties of Linear Algebras
2020
A variety \(\mathcal {V}\) is almost nilpotent if it is not nilpotent but all proper subvarieties are nilpotent. Here we present the results obtained in recent years about almost nilpotent varieties and their classification.