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Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $mathfrak{sl}_2$ at generic $q$. We show that for any $rge 1$, the endomorphism algebra of $Motimes V^{otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.
We use analogues of Enrights and Arkhipovs functors to determine the quiver and relations for a category of $mathfrak{sl}_2 ltimes L(4)$-modules which are locally finite (and with finite multiplicities) over $mathfrak{sl}_2$. We also outline serious
For an irreducible module $P$ over the Weyl algebra $mathcal{K}_n^+$ (resp. $mathcal{K}_n$) and an irreducible module $M$ over the general liner Lie algebra $mathfrak{gl}_n$, using Shens monomorphism, we make $Potimes M$ into a module over the Witt a
This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. In this part, we categorify all tensor
Let $n>1$ be an integer, $alphain{mathbb C}^n$, $bin{mathbb C}$, and $V$ a $mathfrak{gl}_n$-module. We define a class of weight modules $F^alpha_{b}(V)$ over $sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector
We classify the simple bounded weight modules of ${mathfrak{sl}(infty})$, ${mathfrak{o}(infty)}$ and ${mathfrak{sp}(infty)}$, and compute their annihilators in $U({mathfrak{sl}(infty}))$, $U({mathfrak{o}(infty))}$, $U({mathfrak{sp}(infty))}$, respectively.