Codimensions of algebras and growth functions

Journal of Algebra, 1999

Advances in Applied Mathematics, 2006

Let 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 n n β .

Proceedings of the American Mathematical Society, 2000

Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with involution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.

2011

Advances in Mathematics, 1998

Let A be a PI-algebra over a field F. We study the asymptotic behavior of the sequence of codimensions c n (A) of A. We show that if A is finitely generated over F then Inv(A)=lim n Ä n c n (A) always exists and is an integer. We also obtain the following characterization of simple algebras: A is finite dimensional central simple over F if and only if Inv(A)=dim A.

Advances in Applied Mathematics, 2011

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. addresses: [email protected] (A. Giambruno), [email protected] (I. Shestakov), [email protected] (M. Zaicev).

Journal of Algebra, 2001

By the Giambruno-Zaicev theorem for associative p.i. algebras, the exponential rate of growth of the codimensions of such a p.i. algebra is always a positive integer. Here we calculate that integer for various generic p.i. algebras which are given by a single identity. These include Capelli-type identities and the various powers of the standard polynomials.

2006

Advances in Mathematics, 1999

Let A be an associative PI-algebra over a field F of characteristic zero. By studying the exponential behavior of the sequence of codimensions [c n (A)] of A, we prove that Inv(A)=lim n Ä n c n (A) always exists and is an integer. We also give an explicit way for computing such integer: let B be a finite dimensional Z 2-graded algebra whose Grassmann envelope G(B) satisfies the same identities of A; then Inv(A)=Inv(G(B))=dim C (0) +dim C (1) where C (0) +C (1) is a suitable Z 2-graded semisimple subalgebra of B.

2014

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [33], [34]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8].