This note tells you how to construct a k(n)-dimensional family of (isomorphism classes of) irreducible representations of dimension n for the three string braid group B_3, where k(n) is an admissible function of your choosing; for example take k(n) = [ n/2 ] +1 as in arXiv:0803.2778 and arXiv:0803.2785.
We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.
The necklace braid group $\mathcal{NB}_n$ is the motion group of the $n+1$ component necklace link $\mathcal{L}_n$ in Euclidean $\mathbb{R}^3$. Here $\mathcal{L}_n$ consists of $n$ pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group $\mathcal{NB}_n$, especially those obtained as extensions of representations of the braid group $\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. We show that any irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in …
Representations of the braid group B_n and the highest weight modules of U(sl_{n-1}) and U_q(sl_{n-1})
在[1]中，我们利用Pascal三角形的$q-$变形，构造了任意维辫群B_3的[n+1/2]+1参数不可约表示族.这种构造推广了S. P. Humphries（2000）的结果，后者利用经典的Pascal三角形构造了任意维辫群B_3的表示. E. Ferrand（2000）通过考虑空间${\mathbb C}^n[X].$中的两个特殊算子得到了B_3的等价表示。Tuba和H.温兹尔（2001）。它们涉及[n+1/2]个参数（也使用经典的帕斯卡三角形）。后者的作者也给出了完整的分类的所有简单表示$B_3$为尺寸$n\leq 5$。我们的构造推广了所有的结果，并对其中的一些结果给出了新的解释。我们还研究了所构造的表示的不可约性和等价性。本文证明了文[1]中构造的所有表示都可以通过对U（sl）_2和U_q（sl）_2的最高权模取指数而得到。本文利用约化Burau表示推广了辫群$B_n$的表示与U_q（sl_{n-1}）的最高权模之间的关系.
