Search results for " Codimension growth"

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 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…

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.

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.

Trace Codimensions of Algebras and Their Exponential Growth

2022

The trace codimensions give a quantitative description of the identities satisfied by an algebra with trace. Here we study the asymptotic behaviour of the sequence of trace codimensions c tr n(A) and of pure trace codimensions c ptr n (A) of a finite-dimensional algebra A over a field of characteristic zero. We find an upper and lower bound of both codimensions and as a consequence we get that the limits limn→∞ctrn(A)√n and limn→∞cptrn(A) √n always exist and are integers. This result gives a positive answer to a conjecture of Amitsur in this setting. Finally we characterize the varieties of algebras whose exponential growth is bounded by 2

On the growth of varieties of algebras

2009