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Register a new userA Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules
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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.
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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 ${cal{C}}_{int}$.
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 build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).
These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects, assuming only superficial familiarity with closed TQFTs in terms of commutative Frobenius algebras. The generalisation of this relation is a construction of pivotal 2-categories from defect TQFTs. We review this construction in detail, flanked by a range of examples. Furthermore we explain how open/closed TQFTs are equivalent to Calabi-Yau categories and the Cardy condition, and how to extract such data from pivotal 2-categories.
We equip Ellis and Brundans version of the odd categorified quantum group for sl(2) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible decategorifications of the associated dg-2-category. One version gives rise to a categorification of quantum sl(2) at a fourth root of unity, while the other version produces a subalgebra of quantum gl(1|1) defined over the integers. Both of these algebras appear in connection with quantum algebraic approaches to the Alexander polynomial.
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