The Birman-Murakami-Wenzl algebras of type E n

Abstract

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 Z[δ ±1 , l ±1 , m]/(m(1− 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 En 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 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.

Brauer algebras of multiply laced Weyl type

“…At the beginning of this paper, we first recall necessary classical knowledge about Coxeter groups, Coxeter diagrams, their root systems.We also introduce several recent results about a partial order on some mutually orthogonal root sets associated to Coxeter groups of simply laced type from [10], and also the corresponding Brauer algebras. Subsequently we introduce the classical Brauer algebra and prove some results similar to those in [15]. In Section 5, we introduce the definition of Brauer algebras of Weyl type and some of their basic properties, which generalizes some results in [12], [13], [14] and [20].…”

The Dieck-Temperley-Lieb algebras of type B and C in Brauer algebras

“…Similarly, we replace DTLM(Q) by TLM(Q). When Q is of type A n , D n , E 6 , E 7 and E 8 , the algebra TL(Q) is a subalgebra of Br(Q), and the monomials of height 0 form a basis of TL(Q)( [8]).…”

“…action of the corresponding Weyl groups, all admissible root sets of type A n , D n , E 6 , E 7 , E 8 have appeared in [10], [12] and [8], and are listed in Table 2.…”

“…Remark 5.9. There is a more general version for simply laced types in [8]. We keep notations as in […”

“…By ψ, we see that ψ(e i ) = E i+2 , for 1 ≤ i ≤ n − 1, which are Temperley-Lieb generators of TL(D n+1 ). By [8,Proposition 4], the subalgebra of TL(D n+1 )generated by {E i+2 } n−1 i=1 is isomorphic to TL(A n−1 ). Now we have…”