Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLow-dimensional representations of the three component loop braid group
63
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.
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 conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the $q$-Racah polynomials which appear as $6j$-symbols in the Racah problem for $U_q(mathfrak{sl}_2)$. Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.
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 $mathfrak{g}$ acts on $D^b(mathcal{C})$ by Rickard complexes, providing a triangulated equivalence $Theta_{w_0}:D^b(mathcal{C}_mu) to D^b(mathcal{C}_{w_0(mu)})$, where $mu$ is a weight of $V$ and $Theta_{w_0}$ is a positive lift of the longest element of the Weyl group. We prove that this equivalence is t-exact up to shift when $V$ is isotypic, generalising a fundamental result of Chuang and Rouquier in the case $mathfrak{g}=mathfrak{sl}_2$. For general $V$, we prove that $Theta_{w_0}$ is a perverse equivalence with respect to a Jordan-Holder filtration of $mathcal{C}$. Using these results we construct, from the action of $B$ on $V$, an action of the cactus group on the crystal of $V$. This recovers the cactus group action on $V$ defined via generalised Schutzenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.
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 $K_t(mathfrak{g})$ of Hernandez-Leclercs category $C_{mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors ${mathscr{S}_i}_{iin mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{infty}$. Under an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t({A^{(1)}_{N-1}})$, the functors ${mathscr{S}_i}_{1le ile N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
