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The Birman-Murakami-Wenzl algebras of type E n

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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 \( {{{\mathbb{Z}\left[ {{\delta^{\pm 1}},{l^{\pm 1}},m} \right]}} \left/ {{\left( {m\left( {1 - \delta } \right) - \left( {l - {l^{ - 1}}} \right)} \right)}} \right.} \) 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 .

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, POBox 513, 5600 MB, Eindhoven, The Netherlands

    Arjeh M. Cohen

  2. Mathematics Department, Sloan Lab, Caltech, Pasadena, CA, 91125, USA

    David B. Wales

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  1. Arjeh M. Cohen
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  2. David B. Wales
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Correspondence to Arjeh M. Cohen.

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Dedicated to professor T. A. Springer on the occasion of his 85th birthday

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Cohen, A.M., Wales, D.B. The Birman-Murakami-Wenzl algebras of type E n . Transformation Groups 16, 681–715 (2011). https://doi.org/10.1007/s00031-011-9150-9

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