The Birman–Murakami–Wenzl Algebras of Type D_n

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 \( {{{\mathbb{Z}\left[ {{\delta^{\pm 1}},{l^{\pm 1}},m} \right]}} \left/ {{\left( {m\left( {1 - \delta } \right) - \left( {l - {l^{ - 1}}} \right)} \right)}} \right.} \) 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 \( \mathbb{Z}\left[ {{\delta^{\pm 1}}} \right] \). 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 E n share many structural properties with the classical ones (of type A n ) and those of type D n .