Mathematics

There is an embedding of affine vertex algebras V k ( gl n )↪ V k ( sl n+1 ) , and the coset C k (n)=Com( V k ( gl n ), V k ( sl n+1 )) is a natural generalization of the parafermion algebra of sl 2 . It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter W ∞ -algebra of type W(2,3,…) . In this paper, we consider an analogous structure of orthogonal type, namely D k (n)=Com( V k ( so 2n ), V k ( so 2n+1 ) ) Z 2 . We realize this algebra as a one-parameter quotient of the two-parameter even spin W ∞ -algebra of type W(2,4,…) , and we classify all coincidences between its simple quotient D k (n) and the algebras W ℓ ( so 2m+1 ) and W ℓ ( so 2m ) Z 2 . As a corollary, we show that for the admissible levels k=−(2n−2)+ 1 2 (2n+2m−1) for so ˆ 2n the simple affine algebra L k ( so 2n ) embeds in L k ( so 2n+1 ) , and the coset is strongly rational. As a consequence, the category of ordinary modules of L k ( so 2n+1 ) at such a level is a braided fusion category.

We introduce Hopf algebroid covariance on Woronowicz's differential calculus. Using it, we develop quite a general framework of noncommutative complex geometry that subsumes the one in [2]. We present transverse complex and Kähler structures as examples and discuss several other examples. Relation with past literature is described.

Let a reductive group G act on a smooth affine complex algebraic variety X. Let g be the Lie algebra of G and μ: T ∗ (X)→g be the moment map. If the moment map is flat, and for a generic character χ:g→C , the action of G on μ −1 (χ) is free, then we show that for very generic characters χ the corresponding quantum Hamiltonian reduction of the ring of differential operators D(X) is simple.

We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.

We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in Kähler quantisation, with polarisations coming from the nonabelian Hodge hyper-Kähler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.

Geometric vertex algebras are a simplified version of Huang's geometric vertex operator algebras. We give a self-contained account of the equivalence of geometric vertex algebras with Z-graded vertex algebras.

We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided fusion category C . We describe such Hopf algebras as orbits for the action of a reductive group on an affine variety. We then show that the closed orbits are precisely the orbits of Nichols algebras, and that all other algebras are therefore deformations of Nichols algebras. For the case where the category C is the category G G YD of Yetter-Drinfeld modules over a finite group G , this reduces the question of generation by group-like elements and skew-primitives to a geometric question about rigidity of orbits. Comparing the results of Angiono Kochetov and Mastnak, this gives a new proof for the generation of finite dimensional pointed Hopf algebras with abelian groups of group-like elements by skew-primitives and group-like elements. We show that if V is a simple object in C and B(V) is finite dimensional, then B(V) must be rigid. We also show that a non-rigid Nichols algebra can always be deformed to a pre-Nichols algebra or a post-Nichols algebra which is isomorphic to the Nichols algebra as an object of the category C .

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra T p for any integer p>2 which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of T p , as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.

Let U and V be vertex operator algebras with module (sub)categories U and V , respectively, satisfying suitable assumptions which hold for example if U and V are semisimple rigid braided (vertex) tensor categories with countably many inequivalent simple objects. If τ is a map from the set of inequivalent simple objects of U to the objects V with τ(U)=V , then we glue U and V along U⊠V via τ to obtain the object A=⨁X⊗τ(X) where the sum is over all inequivalent simple objects of U . Assuming U and V form a commuting pair in A in the sense that the multiplicity of V is U , our main theorem is that there is a braid-reversed equivalence between U and V mapping X to τ(X ) ∗ if and only if A can be given the structure of a simple conformal vertex algebra that (conformally) extends U⊗V .

We study glued tensor and free products of compact matrix quantum groups with cyclic groups -- so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing.