Mathematics

Let V be a vertex operator algebra and g=(1 2 ⋯k) be a k -cycle which is viewed as an automorphism of the vertex operator algebra V ⊗k . It is proved that Dong-Li-Mason's associated associative algebra A g ( V ⊗k ) is isomorphic to Zhu's algebra A(V) explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible g -twisted V ⊗k -modules and irreducible V -modules.

We study permutation orbifolds of the 2 -fold and 3 -fold tensor product for the Virasoro vertex algebra V c of central charge c . In particular, we show that for all but finitely many central charges ( V ⊗3 c ) Z 3 is a W -algebra of type (2,4,5, 6 3 ,7, 8 3 , 9 3 , 10 2 ) . We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine W -algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated (2,5) -minimal vertex algebra L − 22 5 is presented.

We discuss the deformed function algebra of a simply connected reductive Lie group G over the complex numbers using a basis consisting of matrix elements of finite dimensional representations. This leads to a preferred deformation, meaning one where the structure constants of comultiplication are unchanged. The structure constants of multiplication are controlled by quantum 3j symbols. We then discuss connections earlier work on preferred deformations that involved Schur-Weyl duality.

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q -Hopf fibration on the standard q -sphere. We also construct the Poisson level of the spin connection on a principal bundle.

We bring forward the notions of large quantum groups and their relatives. The starting point is the concept of distinguished pre-Nichols algebra arXiv:1405.6681 belonging to a one-parameter family; we call such an object a \emph{large} quantum unipotent subalgebra. By standard constructions we introduce \emph{large} quantum groups and \emph{large} quantum Borel subalgebras. We first show that each of these three large quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of arXiv:math/0201042. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras U q ⊃ U ⩾ q ⊃ U + q are also described. Multiparameter quantum super groups at roots of unity fit in ou context as well as quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. All steps of our approach are applicable in wider generality and are carried out using general constructions with restricted and non-restricted integral forms and Weyl groupoid actions. Our approach provides new proofs to results in the literature without reductions to rank two cases.

We prove that sufficiently many intertwining operators of type F 4 unitary affine VOAs satisfy polynomial energy bounds. This finishes the Wassermann type analysis of intertwining operators for all WZW-models.

Hexagon relations are algebraic realizations of four-dimensional Pachner moves. `Constant' -- not depending on a 4-simplex in a triangulation of a 4-manifold -- hexagon relations are proposed, and their polynomial-valued cohomology is constructed. This cohomology yields polynomial mappings defined on the so called `coloring homology space', and these mappings can, in their turn, yield piecewise linear manifold invariants. These mappings are calculated explicitly for some examples. It is also shown that `constant' hexagon relations can be obtained as a limit case of already known `nonconstant' relations, and the way of taking the limit is not unique. This non-uniqueness suggests the existence of an additional structure on the `constant' coloring homology space.

In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an O-operator, which gives rise to a pair of ON-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra g are corresponding to ON-structures on the bimodule ( g ∗ ; ad ∗ ,− R ∗ ) , and KVΩ -structures are corresponding to solutions of the strong Maurer-Cartan equation on a twilled pre-Lie algebra associated to an s -matrix.

We first prove an analogue of Lagrange theorem for global dimensions of fusion categories, then we give a complete classifications of pre-modular fusion categories of integer global dimensions less than or equal to 10 .