As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced by Koenig and Xi 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. Since then several classes of algebras, like the Khovanov-Lauda-Rouquier algebras or Kleshchev's graded quasihereditary algebras have been shown to be affine cellular.
In this talk we will describe some ring theoretical properties of affine cellular algebras. In particular we will show that any affine cellular algebra A satisfies a polynomial identity, from which it follows that simple modules are finite dimensional, in case A is an affine k-algebra over a field k. Furthermore, we show that A can be embedded into its asymptotic algebra if the occurring commutative affine k-algebras Bj are reduced and the determinants of the sandwich matrices are non-zero divisors. As a consequence we show that the Gelfand-Kirillov dimension of A is less or equal to the largest Krull dimension of the algebras Bj and that equality hold in case all affine cell ideals are idempotent or if the Krull dimension of the algebras Bj is less 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.
