Ring theoretical properties of affine cellular algebras

Abstract

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, Khovanov-Lauda-Rouquier 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 k-algebras 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 ideals is idempotent, generated by an idempotent or finitely generated.