Mathematics

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of Z 2 for an abelian group G . We prove that there are only two forms of them. Using such description together with some other techniques, we get a complete list of all universal R -matrices on some Hopf algebras.

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils, Lauve and Witherspoon and we call the restricted Jordan plane. In this paper the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given.

We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of Γ -structured elliptic curves with marked points, where Γ=Z/MZ×Z/NZ , and M,N≥1 are two integers. It restricts to a flat connection on Γ -twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subgroups of the pure braid group on the torus. We show that the universal ellipsitomic KZB connection realizes as the usual KZB connection associated with elliptic dynamical r -matrices with spectral parameter, and finally, also produces representations of cyclotomic Cherednik algebras.

We study webs in quantum type C , focusing on the rank three case. We define a linear pivotal category Web( sp 6 ) diagrammatically by generators and relations, and conjecture that it is equivalent to the category FundRep( U q ( sp 6 )) of quantum sp 6 representations generated by the fundamental representations, for generic values of the parameter q . We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor Web( sp 6 )→FundRep( U q ( sp 6 )) , that all Hom -spaces in Web( sp 6 ) are finite-dimensional, and that the endomorphism algebra of the monoidal unit in Web( sp 6 ) is 1 -dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum sp 6 link invariants, akin to the Kauffman bracket description of the Jones polynomial.

This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and O -operators. In general, solutions of the nhacYBe are characterized in terms of generalized O -operators. The characterization can be given by the classical O -operators precisely when the solutions satisfy an invariant condition. When the invariant condition is compatible with a Frobenius algebra, such solutions have close relationships with Rota-Baxter operators on the Frobenius algebra. In general, solutions of the nhacYBe can be produced from Rota-Baxter operators, and then from O -operators when the solutions are taken in semi-direct product algebras. In the other direction, Rota-Baxter operators can be obtained from solutions of the nhacYBe in unitizations of algebras. Finally a classifications of solutions of the nhacYBe satisfying the mentioned invariant condition in all unital complex algebras of dimensions two and three are obtained. All these solutions are shown to come from Rota-Baxter operators.

A generalised orbifold of a defect TQFT Z is another TQFT Z A obtained by performing a state sum construction internal to Z . As an input it needs a so-called orbifold datum A which is used to label stratifications coming from duals of triangulations and is subject to conditions encoding the invariance under Pachner moves. In this paper we extend the construction of generalised orbifolds of 3 -dimensional TQFTs to include line defects. The result is a TQFT acting on 3-bordisms with embedded ribbon graphs labelled by a ribbon category W A that we canonically associate to Z and A . We also show that for special orbifold data, the internal state sum construction can be performed on more general skeletons than those dual to triangulations. This makes computations with Z A easier to handle in specific examples.

We discuss a graph complex formed by directed acyclic graphs with external legs. This complex comes in particular with a map to the ribbon graph complex computing the (compactly supported) cohomology of the moduli space of points M g,n , extending an earlier result of Merkulov-Willwacher. It is furthermore quasi-isomorphic to the hairy graph complex computing the weight 0 part of the compactly supported cohomology of M g,n according to Chan-Galatius-Payne. Hence we can naturally connect the works Chan-Galatius-Payne and of Merkulov-Willwacher and the ribbon graph complex and obtain a fairly satisfying picture of how all the pieces and various graph complexes fit together, at least in weight zero.

We construct a new family of irreducible representations of U q ( g R ) and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the generators of U q ( g R ) act by positive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations which corresponds to induction by the minimal parabolic (i.e. Borel) subgroup. We also study in detail the special case of type A n acting on L 2 ( R n ) with minimal functional dimension, and establish the properties of its central characters and universal R operator. We construct a positive version of the evaluation module of the affine quantum group U q ( sl ˆ n+1 ) modeled over this minimal positive representation of type A n .

In this paper we construct combinatorial bases of parafermionic spaces associated with the standard modules of the rectangular highest weights for the untwisted affine Lie algebras. Our construction is a modification of G. Georgiev's construction for the affine Lie algebra sl ˆ (n+1,C) ---the constructed parafermionic bases are projections of the quasi-particle bases of the principal subspaces, obtained previously in a series of papers by the first two authors. As a consequence we prove the character formula of A. Kuniba, T. Nakanishi and J. Suzuki for all non-simply-laced untwisted affine Lie algebras.

In this work we define partial (co)actions on multiplier Hopf algebras, we also present examples and properties. From a partial comodule coalgebra we construct a partial smash coproduct generalizing the constructions made by the L. Delvaux, E. Batista and J. Vercruysse.