We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of osp(1|2) and the N=1 super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.

We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra g^, from g-module homomorphisms. When g=sl_2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to g^, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.

A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories C of modules for the fixed-point vertex operator subalgebra V^G of a vertex operator (super)algebra V with finite automorphism group G. The main results are that every V^G-module in C with a unital and associative V-action is a direct sum of g-twisted V-modules for possibly several g∈G, that the category of all such twisted V-modules is a braided G-crossed (super)category, and that the G-equivariantization of this braided G-crossed (super)category is braided tensor equivalent to the original category C of V^G-modules. This generalizes results of Kirillov and Müger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether V^G is strongly rational if V is strongly rational. We show that V^G is indeed strongly rational if V is strongly rational, G is any finite automorphism group, and V^G is C_2-cofinite.

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed construction of the canonical algebra in C⊠C^rev: if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C) X′⊠X is a commutative algebra, with X′ a representing object for Hom_C(∙⊗_C X,1_C). Conversely, let A=⨁i∈I U_i ⊠ V_i be a simple commutative algebra in U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. If the unit objects of U and V form a commuting pair in A, we show there is a braid-reversed equivalence between subcategories of U and V sending U_i to (V_i)*. When U and V are module categories for simple vertex operator algebras U and V, we glue U and V along U⊠V via a map τ: Irr(U)→Obj(V) such that τ(U)=V to create A=⨁X∈Irr(U) X′⊗τ(X). Thus under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A is a simple conformal vertex algebra extending U⊗V. As examples, we glue Kazhdan-Lusztig categories at generic levels to obtain new vertex algebras extending the tensor product of two affine vertex algebras, and we prove braid-reversed equivalences between certain module categories for affine vertex algebras and W-algebras at admissible levels.

Suppose V^G is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V. We show that if all irreducible V^G-modules contained in V live in some braided tensor category of V^G-modules, then they generate a tensor subcategory equivalent to the category Rep G of finite-dimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V. Additionally, we show that if the fusion rules for the irreducible V^G-modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of V^G-modules. These results do not require rigidity on any tensor category of V^G-modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when V^G is C_2-cofinite but not necessarily rational. When V^G is both C_2-cofinite and rational and V is a vertex operator algebra, we use the equivalence between Rep G and the corresponding subcategory of V^G-modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to Rep SU(2), up to modification by an abelian 3-cocycle.

Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V. We employ tensor categories to study untwisted (also called local) A-modules in C, using results of Huang-Kirillov-Lepowsky showing that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V-modules to A-modules is a vertex tensor functor. We give two applications. First, we derive Verlinde formulae for regular vertex operator superalgebras and regular (1/2)ℤ-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and ℤ-graded components, respectively. Second, we analyze parafermionic cosets C=Com(V_L,V) where L is a positive definite even lattice and V is regular. If the category of either V-modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.

Let e be an arbitrary even nilpotent element in the general linear Lie super- algebra glM|N and let We be the associated finite W-superalgebra. Let Ym|n be the super Yangian associated to the Lie superalgebra glm|n. A subalgebra of Ym|n, called the shifted super Yangian and denoted by Ym|n(σ), is defined and studied. Moreover, an explicit iso- morphism between We and a quotient of Ym|n(σ) is established.

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Narain lattices are unimodular lattices {\it in} $\R^{r, s} $, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2, 2} $ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2, 2} $, a new interpretation of the Gauss product of binary forms brings new light to a number of related objects--rank 4 rational Narain lattices, over-lattices, rank 2 primitive sublattices of an abstract rank 4 even unimodular lattice U^ 2, and isomorphisms of discriminant groups of rank 2 lattices.

Based on the orthogonal Labastida-Mario-Ooguri-Vafa conjecture made by L. Chen and Q. Chen (2012), we derive an infinite product formula for Chern-Simons partition functions, which generalizes Liu and Peng's recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discussed.

Based on large N Chern-Simons/topological string duality, in a series of papers, Labastida, Mario, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

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For each simple Lie algebra g, we construct an algebra embedding of the quantum group Uq(g) into certain quantum torus algebra Dg via the positive representations of split real quantum group. The quivers corresponding to Dg is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.

In our previous work, we studied positive representations of split real quantum groups U q q~ (g R ) restricted to their Borel part and showed that they are closed under taking tensor products. But the tensor product decomposition was only constructed abstractly using the GNS representation of a C*-algebraic version of the Drinfeld–Jimbo quantum groups. Here, using the recently discovered cluster realization of quantum groups, we write the decomposition explicitly by realizing it as a sequence of cluster mutations in the corresponding quiver diagram representing the tensor product.

We study the tensor product decomposition of the split real quantum group Uqq~(sl(2,R)) from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive representations of Uqq~(sl(2,R)) is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch-Gordan coefficients of the tensor product decomposition of finite dimensional representations of the compact quantum group Uq(sl2) by solving certain functional equations and using normalization arising from tensor products of canonical basis. We propose a general strategy to deal with the tensor product decomposition for the higher rank split real quantum group Uqq~(gR)

We study the positive representations Pλ of split real quantum groups Uqq (gℝ) restricted to the Borel subalgebra Uqq (bℝ). We prove that the restriction is independent of the parameter λ. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra UqqC ∗ (b ℝ) defined earlier, which allows us to decompose their tensor product using the theory of the “multiplicative unitary”. In particular, the quantum mutation operator can be constructed from the multiplicity module, which will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim.

The universal R operator for the positive representations of split real quantum groups is computed, generalizing the formula of compact quantum groups forumla by Kirillov–Reshetikhin and Levendorskiĭ–Soibelman, and the formula in the case of forumla by Faddeev, Kashaev, and Bytsko-Teschner. Several new functional relations of the quantum dilogarithm are obtained, generalizing the quantum exponential relations and the pentagon relations. The quantum Weyl element and Lusztig's isomorphism in the positive setting are also studied in detail. Finally, we introduce a C*-algebraic version of the split real quantum group in the language of multiplier Hopf algebras, and consequently the definition of R is made rigorous as the canonical element of the Drinfeld's double U of certain multiplier Hopf algebra Ub. Moreover, a ribbon structure is introduced for an extension of U.

We found an explicit construction of a representation of the positive quantum group and its modular double by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group parametrized by the standard decomposition of the longest element . Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the q-tori by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an space in the von Neumann setting can also be defined.

We construct the positive principal series representations for Uq(gR) where g is of type Bn, Cn, F4 or G2, parametrized by Rn where n is the rank of g. We show that under the representations, the generators of the Langlands dual group Uq(LgR) are related to the generators of Uq(gR) by the transcendental relations. This gives a new and very simple analytic relation between the Langlands dual pair. We define the modified quantum group Uqq(gR)=Uq(gR) ⊗ Uq(LgR) of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the q-tori polynomials.

We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the noncompact quantum dilogarithm at |$b=1$|⁠.Communicated by Michio Jimbo

We prove that certain parabolic Kazhdan-Lusztig polynomials calculate the graded decomposition matrices of v-Schur algebras given by the Jantzen filtration of Weyl modules, confirming a conjecture of Leclerc and Thibon.

We categorify a coideal subalgebra of the quantum group of S[2r+ 1 by introducing a 2-category analogous to the one defined by Khovanov一Lauda-Rouquier， and show that self-dual indecomposable l-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the category ð of type B/C.

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

We prove a conjecture of Miemietz and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case in Varagnolo and Vasserot (in press) [15].