Lawrence-Bigelow representations, bases and duality

前幂零Lawrence-Krammer-Bigelow表示

The pro-nilpotent Lawrence-Krammer-Bigelow representation

本文构造了辫群LKB表示的3变量富化，作为以前幂零表示塔为极限且以原始LKB表示为底层的方法，并推广至曲面辫群与环辫群。

Lawrence-Krammer-Bigelow representations and dual Garside length of braids

We show that the span of the variable $q$ in the Lawrence-Krammer-Bigelow representation matrix of a braid is equal to the twice of the dual Garside length of the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow's geometric approach. The key observation is that the dual Garside length of a braid can be read off a certain labeling of its curve diagram.

A note on the Lawrence-Krammer-Bigelow representation

A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this monodromy representation using standard tools from the theory of hyperplane arrangements. In particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but that it cannot b…

The Lawrence-Krammer-Bigelow Representations of the Braid Groups via Quantum SL_2

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 over the fractional field Q (q,s).

运动群与映射类群的拓扑表示——一种统一的函子构造

Topological representations of motion groups and mapping class groups -- a unified functorial construction

本文通过函子方法统一构造同调表示，引入定义在包含映射类群和运动群的范畴上的同调表示函子，统一了Lawrence-Bigelow等已知构造并得到新表示。

On the image of the Lawrence-Krammer representation

A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negative-definite Hermitian. Since unitary matrices diagonalize, the conjugacy class of a matrix in the unitary group is determined by its eigenvalues. It is shown that the eigenvalues of a Lawrence-Krammer matrix satisfy some symmetry relations. Using the fact that non-invertible knots exist, the symmetry …