A Lie-Theoretic Construction of Spherical Symplectic Reflection Algebras

We propose a construction of the spherical subalgebra of a symplectic reection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semisimple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra.     

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with gr(A) in RG(p), where gr(A) is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_±(p), A(q), or H(32), constructed in [5] and [8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.     

An algebraic extension of the MacMahon Master Theorem

We present a new algebraic extension of the classical MacMahon Master Theorem. The basis of our extension is the Koszul duality for non-quadratic algebras defined by Berger. Combinatorial implications are also discussed.

     

Central extensions of current groups in two dimensions

The paper is devoted to generalization of the theory of loop groups to the two-dimensional case. To every complex Riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex Lie group G by the Jacobian of this surface. This extension is topologically nontrivial, as in the loop group case. Orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalG-bundles over the surface. When the surface is the torus (elliptic curve), classification of coadjoint orbits is related to linear difference equations in one variable, and to classification of conjugacy classes in the loop group. We study integral orbits, for which the Kirillov-Kostant form is a curvature form for some principal torus bundle. The number of such orbits for a given level is finite, as in the loop group case; conjecturedly, they correspond to analogues of integrable modules occurring in conformal field theory.     

Reductions of Tensor Categories Modulo Primes

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category 𝒞 if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V|² of any simple object V of 𝒞. We show, using the Ito–Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.     

Affine Gelfand-Dickey brackets and holomorphic vector bundles

We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem leads to the problem of classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a finite but (unlike the traditional case of operators on the circle) arbitrarily large codimension, and compute it explicitly.