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Register a new userOn a category of $gl_{infty}$-modules
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We study a particular category ${cal{C}}$ of $gl_{infty}$-modules and a subcategory ${cal{C}}_{int}$ of integrable $gl_{infty}$-modules. As the main results, we classify the irreducible modules in these two categories and we show that every module in category ${cal{C}}_{int}$ is semi-simple. Furthermore, we determine the decomposition of the tensor products of irreducible modules in category ${cal{C}}_{int}$.
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In this paper, we present a canonical association of quantum vertex algebras and their $phi$-coordinated modules to Lie algebra $gl_{infty}$ and its 1-dimensional central extension. To this end we construct and make use of another closely related infinite-dimensional Lie algebra.
Haisheng Li showed that given a module (W,Y_W(cdot,x)) for a vertex algebra (V,Y(cdot,x)), one can obtain a new V-module W^{Delta} = (W,Y_W(Delta(x)cdot,x)) if Delta(x) satisfies certain natural conditions. Li presented a collection of such Delta-operators for V=L(k,0) (a vertex operator algebra associated with an affine Lie algebras, k a positive integer). In this paper, for each irreducible L(k,0)-module W, we find a highest weight vector of W^{Delta} when Delta is associated with a miniscule coweight. From this we completely determine the action of these Delta-operators on the set of isomorphism equivalence classes of L(k,0)-modules.
We endow a non-semisimple category of modules of unrolled quantum sl(2) with a Hermitian structure. We also prove that the TQFT constructed in arXiv:1202.3553 using this category is Hermitian. This gives rise to projective representations of the mapping class group in the group of indefinite unitary matrices.
We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p,1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an textit{entwined} category (the one with monodromy, but not with braiding).
Let $lambda in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_{lambda}$, where $W_{lambda}$ is the stabilizer of $lambda$ in a finite Weyl group $W$. In this paper, we give a module $mathbb{K}_{w}(lambda)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = infty$ of the nonsymmetric Macdonald polynomial $E_{w lambda}(q,,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $mathbb{K}_{w}(lambda)$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_{w}^{-}(lambda)$ by the sum of the submodules $V_{z}^{-}(lambda)$ for all those coset representatives $z$ of minimal length for the cosets in $W/W_{lambda}$ such that $z > w$ in the Bruhat order $<$ on $W$.
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