Search results for "algebra"

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

*-Graded Capelli polynomials and their asymptotics

2022

Let [Formula: see text] be the free *-superalgebra over a field [Formula: see text] of characteristic zero and let [Formula: see text] be the [Formula: see text]-ideal generated by the set of the *-graded Capelli polynomials [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] alternating on [Formula: see text] symmetric variables of homogeneous degree zero, on [Formula: see text] skew variables of homogeneous degree zero, on [Formula: see text] symmetric variables of homogeneous degree one and on [Formula: see text] skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of *-graded codimensions of [Formula: se…

On central polynomials and codimension growth

2022

Let A be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of A. In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.

MR2966998 Aljadeff, Eli; Kanel-Belov, Alexei Hilbert series of PI relatively free G-graded algebras are rational functions. Bull. Lond. Math. Soc. 44…

2013

Polynomial codimension growth of graded algebras

2009

We study associative $G$-graded algebras with 1 of polynomial $G$-codimension growth, where $G$ is a finite group. For any fixed $k\geq 1,$ we construct associative $G$-graded algebras of upper triangular matrices whose $G$-codimension sequence is given asymptotically by a polynomial of degree $k$ whose leading coefficient is the largest or smallest possible.

Graded involutions on upper-triangular matrix algebras

2009

THE HILBERT FUNCTION OF BIGRADED ALGEBRAS IN k[P1x P1]

2020

We classify the Hilbert functions of bigraded algebras in k[x1, x2, y1, y2] by introducing a numerical function called a Ferrers function.

Braided and symmetric internal groupoids

2011

Braided and symmetric internal groupoids in semi-abelian categories are discussed.

Capelli identities on algebras with involution or graded involution

2022

We present recent results about Capelli polynomials with involution or graded involution and their asymptotics. In the associative case, the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial of rank k2 + 1 and the codimensions of the matrix algebra Mk(F) was proved. This result was extended to superalgebras. Recently, similar results have been determined by the authors in the case of algebras with involution and superalgebras with graded involution.

A Survey on Just-Non-X Groups

2010

Let be a class of groups. A group which does not belong to but all of whose proper quotient groups belong to is called just-non- group. The present note is a survey of recent results on the topic with a special attention to topological groups.