Search results for " Codimensions"

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…

Graded polynomial identities and exponential growth

2009

Let $A$ be a finite dimensional algebra over a field of characteristic zero graded by a finite abelian group $G$. Here we study a growth function related to the graded polynomial identities satisfied by $A$ by computing the exponential rate of growth of the sequence of graded codimensions of $A$. We prove that the $G$-exponent of $A$ exists and is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of $A$.

On the asymptotics for $ast$-Capelli identities

Let Fbe the free associative algebra with involution ∗ over a field of characteristic zero. If L and M are two natural numbers let Γ∗_M+1,L+1 denote theT∗-idealofFgenerated by the∗-capellipolynomialsCap+M+1,Cap−L+1 alternanting on M+1 symmetric variables and L+1skew variables,respectively.It is well known that, if F is an algebraic closed field, every finite dimensional ∗-simple algebra is isomorphic to one of the following algebras (see [4], [2]):· (Mk(F),t) with the transpose involution; · (M2m(F),s) with the symplectic involution; · (Mk(F)⊕Mk(F)op,∗) with the exchange involution. The aim of this talk is to show a relation among the asymptotics of the∗-codimensions of the finite dimensional ∗…

Varieties of algebras of polynomial growth

2008

Let V be a proper variety of associative algebras over a field F of characteristic zero. It is well-known that V can have polynomial or exponential growth and here we present some classification results of varieties of polynomial growth. In particular we classify all subvarieties of the varieties of almost polynomial growth, i.e., the subvarieties of var(G) and var(UT 2), where G is the Grassmann algebra and UT2 is the algebra of 2 x 2 upper triangular matrices.