# The Birman-Murakami-Wenzl algebras of type En

A.M. Cohen, D.B. Wales

The Birman-Murakami-Wenzl algebras (BMW algebras) of type En for n¿=¿6, 7, 8 are shown to be semisimple and free over the integral domain $\mathbb{Z}[d^{\pm1},l^{\pm1}, m]/(m(1-\delta)-(l-l^{-1}))$ of ranks 1,440,585; 139,613,625; and 53,328,069,225. We also show they are cellular over suitable rings. The Brauer algebra of type En is a homomorphic ring image and is also semisimple and free of the same rank as an algebra over the ring $\mathbb{Z}[d^{\pm1}]$. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. The generalized Temperley-Lieb algebra of type En turns out to be a subalgebra of the BMW algebra of the same type. So, the BMW algebras of type En share many structural properties with the classical ones (of type An) and those of type Dn.