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Motivated by physical and topological applications, we study representations of the group $mathcal{LB}_3$ of motions of $3$ unlinked oriented circles in $mathbb{R}^3$. Our point of view is to regard the three strand braid group $mathcal{B}_3$ as a subgroup of $mathcal{LB}_3$ and study the problem of extending $mathcal{B}_3$ representations. We introduce the notion of a emph{standard extension} and characterize $mathcal{B}_3$ representations admiting such an extension. In particular we show, using a classification result of Tuba and Wenzl, that every irreducible $mathcal{B}_3$ representation of dimension at most $5$ has a (standard) extension. We show that this result is sharp by exhibiting an irreducible $6$-dimensional $mathcal{B}_3$ representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible $2$-dimensional $mathcal{LB}_3$ representations (2) extensions of irreducible $3$-dimensional $mathcal{B}_3$ representations and (3) irreducible $mathcal{LB}_3$ representations whose restriction to $mathcal{B}_3$ has abelian image.
We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by th
We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $hat{mathfrak{gl}}_N$.
The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(mathfrak{sl}_2)$ are considered. It is shown that these intermediate Casimirs are related by a co
Let $mathfrak{g}$ be a semisimple simply-laced Lie algebra of finite type. Let $mathcal{C}$ be an abelian categorical representation of the quantum group $U_q(mathfrak{g})$ categorifying an integrable representation $V$. The Artin braid group $B$ of
Let $mathfrak{g}_0$ be a simple Lie algebra of type ADE and let $U_q(mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(mathfrak{g}_0)$ on the quantum Grothendieck ring $K_t(