Mathematics

We study the meromorphic open-string vertex algebras and their modules over the two-dimensional Riemannian manifolds that are complete, connected, orientable, and of constant sectional curvature K≠0 . Using the parallel tensors, we explicitly determine a basis for the meromorphic open-string vertex algebra, its modules generated by eigenfunctions of the Laplace-Beltrami operator, and their irreducible quotients. We also study the modules generated by lowest weight subspace satisfying a geometrically interesting condition. It is showed that every irreducible module of this type is generated by some (local) eigenfunction on the manifold. A classification is given for modules of this type admitting a composition series of finite length. In particular and remarkably, if every composition factor is generated by eigenfunctions of eigenvalue p(p−1)K for some p∈ Z + , then the module is completely reducible.

In this paper, we continue with the ideas presented in [GVR17]. In this opportunity, we apply the fermionic action concept to classify in cohomology terms the minimal modular extensions of a super-Tannakian category. For this goal, we study some properties of equivariantization and de-equivariantization processes and cohomology data for the fermionic case.

The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra sl(2|1) . We generalize this fact to the context of unrolled quantum groups and construct the associated topological ribbon Hopf algebras. Then we use such an algebra, the discrete Fourier transforms, a symmetrized graded integral and a modified trace to define a modified graded Hennings invariant. Finally, we use the notion of a modified integral to extend this invariant to empty manifolds and show that it recovers the CGP-invariant.

In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of S L 2 (Z) associated with transitive modular categories are proven to be minimal and irreducible. Together with the Verlinde formula, we characterize prime transitive modular categories as the Galois conjugates of the adjoint subcategory of the quantum group modular category C( sl 2 ,p−2) for some prime p>3 . As a consequence, we completely classify transitive modular categories. Transitivity of super-modular categories can be similarly defined. A unique factorization of any transitive super-modular category into s-simple transitive factors is obtained, and the split transitive super-modular categories are completely classified.

The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on G and G ˆ act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group SU q (2) ˆ and the quantum az+b group.

Let k be an algebraically closed field of characteristic 0 or p>2 . Let G be an affine supergroup scheme over k . We classify the indecomposable exact module categories over the tensor category sCoh f (G) of (coherent sheaves of) finite dimensional O(G) -supermodules in terms of (H,Ψ) -equivariant coherent sheaves on G . We deduce from it the classification of indecomposable {\em geometrical} module categories over $\sRep(\mathcal{G})$. When G is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\sRep(\mathcal{G})$. In particular, we obtain a classification of twists for the supergroup algebra kG of a finite supergroup scheme G , and then combine it with \cite[Corollary 4.1]{EG3} to classify finite dimensional triangular Hopf algebras with the Chevalley property over k .

In this paper, we investigate the Lie algebra structures of weight one subspaces of C 2 -cofinite vertex operator superalgebras. We also show that for any positive integer k , vertex operator superalgebras L sl(1|n+1) (k,0) and L osp(2|2n) (k,0) have infinitely many irreducible admissible modules. As a consequence, we give a proof of the fact that L g (k,0) is C 2 -cofinite if and only if g is either a simple Lie algebra, or g=osp(1|2n) , and k is a nonnegative integer. As an application, we show that L G(3) (1,0) is a vertex operator superalgebra such that the category of L G(3) (1,0) -modules is semisimple but L G(3) (1,0) is not C 2 -cofinite.

For C a finite tensor category we consider four versions of the central monad, A 1 ,…, A 4 on C . Two of them are Hopf monads, and for C pivotal, so are the remaining two. In that case all A i are isomorphic as Hopf monads. We define a monadic cointegral for A i to be an A i -module morphism 1→ A i (D) , where D is the distinguished invertible object of C . We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case C is braided, to an integral for the braided Hopf algebra L= ∫ X X ∨ ⊗X in C studied by Lyubashenko (1995). Our main motivation stems from the application to finite dimensional quasi-Hopf algebras H . For the category of finite-dimensional H -modules, we relate the four monadic cointegrals (two of which require H to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called γ -symmetrised cointegrals in the pivotal case, for γ the modulus of H . For (not necessarily semisimple) modular tensor categories C , Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of C , in particular of SL(2,Z) on C(L,1) . In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of S and T which uses the monadic cointegral.

Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the H 0 -Poisson structures of Crawley-Boevey. We derive from our results a representation theoretic description of action-angle duality for several classical integrable systems.

The 3-categories of semisimple 2-categories and of multifusion categories are shown to be equivalent. A weakening of the notion of fusion 2-category is introduced. Multifusion 2-categories are defined, and the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories is described.