Higher Jones Algebras and their simple Modules

Henning Haahr Andersen

Published 2018-02-23Version 1

Let $k$ be a field. Denote by $p$ the characteristic of $k$ and assume $p>0$. We prove that the group algebra $kS_r$ of the symmetric group $S_r$ on $r$ letters has some natural semisimple quotients, namely $\End_{GL(V)}(V^{\underline \otimes r})$. Here $V$ is any vector space over $k$ and $\underline \otimes$ is the reduced tensor product on the fusion category for $GL(V)$. We determine the dimensions of all simple modules for these quotients. Replace now $GL(V)$ by the corresponding quantum group $ U_q(\mathfrak{gl}_n)$. Assuming that $q$ is a root of unity in an arbitrary field $k$ we obtain in a similar way natural quotients of the Hecke algebras $H_r(q) = H_q(S_r)$ over $k$ and we prove analogous formulas for the dimensions of their simple modules. In the case where $n=2$ the quotients we get are also quotients of the Temperley-Lieb algebra $TL_r(q+q^{-1})$ and may in fact be identified with the Jones algebras, see \cite{A17} and \cite{ILZ}. Finally, we replace the general linear groups first by the symplectic and then by the orthogonal groups. In these cases our method leads to semisimple quotients of Brauer algebras and (in the quantum case) BMW-algebras.