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Register a new userOn a symmetry of the category of integrable modules
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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.
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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 ${cal{C}}_{int}$.
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 C_n denote the representation category of a finite supergroup generated by purely odd n-dimensional vector space. We compute the Brauer-Picard group BrPic(C_n) of C_n. This is done by identifying BrPic(C_n) with the group of braided tensor autoequivalences of the Drinfeld center of C_n and studying the action of the latter group on the categorical Lagrangian Grassmannian of C_n. We show that this action corresponds to the action of a projective symplectic group on a classical Lagrangian Grassmannian.
We address the question whether the condition on a fusion category being solvable or not is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the $S$-matrix of the Drinfeld center and show that this invariant does determine the solvability of a fusion category provided it is group-theoretical.
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