The Birman-Murakami-Wenzl algebras of type E_n

Abstract

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E_ n for n = 6, 7, 8 are shown to be semisimple and free over the integral domain Z[δ^(±1),l^(±1),m]/(m1−δ)(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 E_n is a homomorphic ring image and is also semisimple and free of the same rank as an algebra over the ring Z[δ^(±1)]. 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 E_n turns out to be a subalgebra of the BMW algebra of the same type. So, the BMW algebras of type E_n share many structural properties with the classical ones (of type A_n) and those of type D_n .

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