Representations of the braid group B_n and the highest weight modules of U(sl_{n-1}) and U_q(sl_{n-1})

TL;DR

This paper constructs a broad family of braid group representations using q-deformations and links them to highest weight modules of quantum groups, extending previous results and providing new insights into their structure.

Contribution

It generalizes earlier braid group representations by connecting them with highest weight modules of U_q(sl_{n-1}) for arbitrary n, offering a unified framework.

Findings

Constructed a family of irreducible braid group representations using q-deformation.

Established a connection between these representations and highest weight modules of quantum groups.

Extended previous classifications of braid group representations to arbitrary dimensions.

Abstract

In [1] we have constructed a [n+1/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension using a $q-$deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries (2000), who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand (2000) obtained an equivalent representation of B_3 by considering two special operators in the space ${\mathbb C}^n[X].$ Slightly more general representations were given by I. Tuba and H. Wenzl (2001). They involve [n+1/2] parameters (and also use the classical Pascal's triangle). The latter authors also gave the complete classification of all simple representations of $B_3$ for dimension $n\leq 5$. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and equivalence of the constructed representations. In the present article we show that all representations constructed in [1] may be obtained by taking exponent of the highest weight modules of U(sl}_2 and U_q(sl_2). We generalize these connections between the representation of the braid group $B_n$ and the highest weight modules of the U_q(sl_{n-1}) for arbitrary} n using the well-known reduced Burau representation.