Brauer algebras of type B

Abstract

For each n 1, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type B n . It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type D nC1 and point out a cellular structure in it. This work is a natural sequel to the introduction of Brauer algebras of type C n , which are subalgebras of classical Brauer algebras of type A 2n 1 and differ from the current ones for n > 2. A novel feature is the failure of admissible root sets to describe the tops and bottoms of the diagrams corresponding to monomials in the Brauer algebra of type B n ; instead of these sets we use extended admissible sets in order to find normal forms for monomials in the algebra.

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

“…Then by the diagram version of [6, Theorem 1.1], we find that the four morphisms in the commutative diagrams of Theorem 4.5 are injective. e 0 = 1 n n + 1 2n 1 n n + 1 2n As in [6,7,21,22], The following can be verified. As in [6,7,21,22], we define STL(Q) is the subalgebra of TL(Q) generated by the σ-invariant submomoid of TLM(Q), where Q can be A 2n−1 , D n+1 or E 6 .…”

“…Remark 3.3. In classical finite Weyl groups, we can define automorphisms on Dynkin diagrams of simply-laced Weyl groups to obtain the Weyl groups of non-simply laced types listed in Figure 3, and we have already applied these automorphisms on simply-laced Brauer algebras to define and study Brauer algebras of non-simply laced types, which can be found in [6] for type C n from A 2n−1 , [7] for type B n from D n+1 , [21] for type F 4 from E 6 , [22] for type G 2 from D 4 . These conclusions are contained in [20] for completing the project of obtaining Brauer algebras of non-simply laced type from simply-laced types.…”

Brauer algebras of multiply laced Weyl type

“…From the proof of Lemma 5.6, by Lemma 9.2 and Proposition 5.3, we see that r 1 r 0 r 1 r 0 r 1 e 0 = e 0 , r 0 r 1 r 0 r 1 r 0 e 1 = e 1 r 0 r 1 r 0 r 1 r 0 . Therefore our lemma holds by the analogous argument in Proposition [14,Lemma 4.8], [21,Lemma 4.2], and [13, Lemma 6.5] through writing an element as the product of one element in D i and another in N i . Lemma 9.6.…”

“…Here we apply the diagram representation of Br(C n ) from [13]. In fact, the algebra Br(B n ) has a diagram representation inherited from Br(D n+1 ) ( [14]). In view of induction on n − |J| and restriction to connected components of J, it suffices to prove the result for J = {1, .…”

Brauer algebras of type $$\mathrm{I}_2^n$$ I 2 n ( $$n\ge 5$$ n ≥ 5 )

“…These are motivated by considering the invariant subgroups under non-trivial automorphisms by action on their Dynkin diagrams. We have already utilized it to obtain Brauer algebras of type C n ( [5]), B n ( [7]), and F 4 ([14]). This paper can be regarded as a part of the project of finding Brauer algebras of non-simply laced types from simply laced types.…”