On Krammer's Representation of the Braid Group

Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view of proving the linearity of these groups. We decompose this representation in irreducible components and compute its Zariski closure, as well as its restriction to para...

We establish isomorphisms between certain specializations of Birman-Murakami-Wenzl algebras and the symmetric squares of Temperley-Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an application, we compute the closed images of the irreducible braid group representations factoring over these specialized BMW algebras.

We construct representations of the braid groups B_n on n strands on free Z[q,q^-1,s,s^-1]-modules W_{n,l} using generic Verma modules for an integral version of quantum sl_2. We prove that the W_{n,2} are isomorphic to the faithful Lawrence Krammer Bigelow representations of B_n after appropriate identification of parameters of Laurent polynomial rings by constructing explicit integral bases and isomorphism. We also prove that the B_n-representations W_{n,l} are irreducible ...

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.

For any n>3, we give a family of finite dimensional irreducible representations of the braid group B_n. Moreover, we give a subfamily parametrized by 0<m<n of dimension the combinatoric number (n,m). The representation obtained in the case m=1 is equivalent to the Standard representation.

We introduce an Ariki-Koike like extension of the Birman-Murakami-Wenzl Algebra and show it to be semi-simple. This algebra supports a faithful Markov trace that gives rise to link invariants of closures of Coxeter type B braids.

The Lawrence-Krammer representation of the braid groups recently came to prominence when it was shown to be faithful by myself and Krammer. It is an action of the braid group on a certain homology module $H_2(\tilde{C})$ over the ring of Laurent polynomials in $q$ and $t$. In this paper we describe some surfaces in $\tilde{C}$ representing elements of homology. We use these to give a new proof that $H_2(\tilde{C})$ is a free module. We also show that the $(n-2,2)$ representat...

In this paper we discuss representations of the Birman-Wenzl-Murakami algebra as well as of its dilute extension containing several free parameters. These representations are based on superalgebras and their baxterizations permit us to derive novel trigonometric solutions of the graded Yang-Baxter equation. In this way we obtain the multiparametric $R$-matrices associated to the $U_q[sl(r|2m)^{(2)}]$, $U_q[osp(r|2m)^{(1)}]$ and $U_q[osp(r=2n|2m)^{(2)}]$ quantum symmetries. Tw...

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$, $VB_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}, t_1^{\pm1},t_2^{\pm1},\ldots, t_{n-1}^{\pm1}]\right)$ which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful repre...

We give a method to construct new self-adjoint representations of the braid group. In particular, we give a family of irreducible self-adjoint representations of dimension arbitrarily large. Moreover we give sufficient conditions for a representation to be constructed with this method.