Christian Lomp (Porto), Ring theoretical properties of affine cellular algebras.

Abstract:

As a generalisation of Graham and Lehrerâ€™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 an 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.

Roughly speaking, an affine cell ideal of an algebra A with involution * is a *-ideal J that is isomorphic as an A-bimodule to a generalized matrix ring M_n(B) over some commutative affine k-algebra B, whose multiplication is deformed by some "sandwich" matrix ψ, i.e. the product of two matrices x and y is defined to be x ψ y. An affine cellular algebra is then in particular a *-algebra A that admit a chain of *-ideals J

In this talk I will exhibit some ring theoretical properties of affine cellular algebras. In particular we will 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

This is joined work with Paula Carvalho, Steffen Koenig and Armin Shalile