Ring theoretical properties of affine cellular algebras

TL;DR

This paper investigates the ring-theoretic properties of affine cellular algebras, showing they satisfy polynomial identities, can embed into their asymptotic algebra under certain conditions, and analyzing their Gelfand-Kirillov dimension.

Contribution

It establishes fundamental ring-theoretic properties of affine cellular algebras, including polynomial identities and embedding conditions, expanding understanding of their structural characteristics.

Findings

Affine cellular algebras satisfy polynomial identities.

They can be embedded into their asymptotic algebra under specific conditions.

The Gelfand-Kirillov dimension is bounded by the Krull dimension of associated algebras.

Abstract

As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine versions of diagram algebras like affine Hecke algebras of type A and affine Temperley-Lieb algebras in a unifying fashion. Affine cellular algebras include Kleshchev's graded quasihereditary algebras, KLR algebras and various other classes of algebras. In this paper we will study ring theoretical properties of affine cellular algebras. We show that any affine cellular algebra $A$ satisfies a polynomial identity. Furthermore, we show that $A$ can be embedded into its asymptotic algebra if the occurring commutative affine algebra $B_j$ are reduced and the determinants of the swich matrices are non-zero divisors. As a consequence, we show that the Gelfand-Kirillov dimension of $A$ is less than or equal to the largest Krull dimension of the algebras $B_j$ and that equality hold, in case all affine cell ideals are idempotent or if the Krull dimension of the algebras $B_j$ is less than or equal to $1$. Special emphasis is given to the question when an affine cell ideal is idempotent, generated by an idempotent or finitely generated.