Mathematics

We study involutive non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation on a finite set X. The emphasis is on the case where (X,r) is indecomposable, so the associated permutation group acts transitively on X. One of the major problems is to determine how such solutions are built from the imprimitivity blocks; and also how to characterize these blocks. We focus on the case of so called simple solutions, which are of key importance. Several infinite families of such solutions are constructed for the first time. In particular, a broad class of simple solutions of order p^2, for any prime p, is completely characterized.

In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum A in a modular fusion category C was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum A in C , we introduce a ribbon category C A and show that it is again a modular fusion category. The definition of C A is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when A is given by a simple commutative Δ -separable Frobenius algebra A in C ; (ii) when A is an orbifold datum in C=Vect , built from a spherical fusion category S . We show that in case (i), C A is ribbon-equivalent to the category of local modules of A , and in case (ii), to the Drinfeld centre of S . The category C A thus unifies these two constructions into a single algebraic setting.

This paper is a contribution to the construction of non-semisimple modular categories. We establish when Müger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which relative monoidal centers give (non-semisimple) modular categories, and we also show that examples include representation categories of small quantum groups. We further derive conditions under which representations of more general quantum groups, braided Drinfeld doubles of Nichols algebras of diagonal type, give (non-semisimple) modular categories.

In recent work, Damiolini-Gibney-Tarasca showed that for a C 2 -cofinite rational CFT-type vertex operator algebra V , sheaves of conformal blocks are locally free and satisfy the factorization property. In this article, we use analytic methods to prove that sewing conformal blocks is convergent, solving a conjecture proposed by Zhu and Huang.

We generalize the mathematical definition of Coulomb branches of 3 -dimensional N=4 SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type BCFG as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type ADE .

Concepts of first order differential calculus (FODC) on a monoidal Hom-algebra and left-covariant FODC over a left Hom-quantum space with respect to a monoidal Hom-Hopf algebra are presented. Then, extension of the universal FODC over a monoidal Hom-algebra to a universal Hom-differential calculus is described. Next, concepts of left(right)-covariant and bicovariant FODC over a monoidal Hom-Hopf algebra are studied in detail. Subsequently, notion of quantum Hom-tangent space associated to a bicovariant Hom-FODC is introduced and equipped with an analogue of Lie bracket (commutator) through Woronowicz' braiding. Finally, it is proven that this commutator satisfies quantum versions of the antisymmetry relation and Hom-Jacobi identity.

We explore the relationship between the classical constructions of cumulants and Koszul brackets, showing that the former are an expontial version of the latter. Moreover, under some additional technical assumptions, we prove that both constructions are compatible with standard homological perturbation theory in an appropriate sense. As an application of these results, we provide new proofs for the homotopy transfer Theorem for L ??and IB L ??algebras based on the symmetrized tensor trick and the standard perturbation Lemma, as in the usual approach for A ??algebras. Moreover, we prove a homotopy transfer Theorem for commutative B V ??algebras in the sense of Kravchenko which appears to be new. Along the way, we introduce a new definition of morphism between commutative B V ??algebras.

We characterize cyclic algebras over the associative and the framed little disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a homotopy coherent way. When the symmetric monoidal bicategory is specified to consist of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little disks algebras, respectively, are equivalent to pivotal Grothendieck-Verdier categories and balanced braided Grothendieck-Verdier categories, a type of category that was introduced by Boyarchenko-Drinfeld based on Barr's notion of a ∗ -autonomous category. We use these results and Costello's derived modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any balanced braided Grothendieck-Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko's mapping class group representations. II) We establish a Grothendieck-Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing a result of Bakalov-Kirillov.

In previous work of the first and third authors, we proposed a conjecture that the Kauffman bracket skein module of any knot in S 3 carries a natural action of the rank 1 double affine Hecke algebra S H q, t 1 , t 2 depending on 3 parameters q, t 1 , t 2 . As a consequence, for a knot K satisfying this conjecture, we defined a three-variable polynomial invariant J K n (q, t 1 , t 2 ) generalizing the classical colored Jones polynomials J K n (q) . In this paper, we give explicit formulas and provide a quantum group interpretation for the generalized Jones polynomials J K n (q, t 1 , t 2 ) . Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by K.\ Habiro: as in the classical case, they imply the integrality of J K n (q, t 1 , t 2 ) and, in fact, make sense for an arbitrary knot K independent of whether or not it satisfies our earlier conjecture. When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of J K n (q, t 1 ) are determined by Macdonald orthogonal polynomials of type A 1 .

We give a skein-theoretic realization of the gl n double affine Hecke algebra of Cherednik using braids and tangles in the punctured torus. We use this to provide evidence of a relationship we conjecture between the classical skein algebra of the punctured torus and the elliptic Hall algebra of Burban and Schiffmann.