Mathematics

We introduce the notion of irregular vertex (operator) algebras. The irregular versions of fundamental properties, such as Goddard uniqueness theorem, associativity and operator product expansions are formulated and proved. We also give some elementary examples of irregular vertex operator algebras.

We show that the quantum Berezinian which gives a generating function of the integrals of motions of XXX spin chains associated to super Yangian Y( gl m|n ) can be written as a ratio of two difference operators of orders m and n whose coefficients are ratios of transfer matrices corresponding to explicit skew Young diagrams. In the process, we develop several missing parts of the representation theory of Y( gl m|n ) such as q -character theory, Jacobi-Trudi identity, Drinfeld functor, extended T-systems, Harish-Chandra map.

We analyze the structure of the infinite jet algebra, or arc algebra, associated to level one Feigin-Stoyanovsky's principal subspaces. For A -series, we show that their Hilbert series can be computed either using the quantum dilogarithm or as certain generating functions over finite-dimensional representations of A -type quivers. In particular, we obtain new fermionic character formulas for level one A -type principal subspaces, which implies that they are classically free. We also analyze the infinite jet algebra of the principal subspace of the affine vertex algebra L 1 (so(5)) and of L 1 (so(8)) . We give new conjectural expressions for their characters and provide strong evidence that they are also classically free.

We study quantization schemes on a Kähler manifold and relate several interesting structures. We first construct Fedosov's star products on a Kähler manifold X as quantizations of Kapranov's L ∞ -algebra structure. Then we investigate the Batalin-Vilkovisky (BV) quantizations associated to these star products. A remarkable feature is that they are all one-loop exact, meaning that the Feynman weights associated to graphs with two or more loops all vanish. This leads to a succinct cochain level formula for the algebraic index.

We introduce the Kashiwara-Vergne bigraded Lie algebra associated with a finite abelian group and give its mould theoretic reformulation. By using the mould theory, we show that it includes the Goncharov's dihedral Lie algebra, which generalizes the result of Raphael and Schneps.

We study the representation theory of the Kazama-Suzuki coset vertex operator superalgebra associated with the pair of a complex simple Lie algebra and its Cartan subalgebra. In the case of type A 1 , B.L. Feigin, A.M. Semikhatov, and this http URL. Tipunin introduced another coset construction, which is "inverse" of the Kazama-Suzuki coset construction. In this paper we generalize the latter coset construction to arbitrary type and establish a categorical equivalence between the categories of certain modules over an affine vertex operator algebra and the corresponding Kazama-Suzuki coset vertex operator superalgebra. Moreover, when the affine vertex operator algebra is regular, we prove that the corresponding Kazama-Suzuki coset vertex operator superalgebra is also regular and the category of its ordinary modules carries a braided monoidal category structure by the theory of vertex tensor categories.

A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions obtained by Rump in [36] and answer in a positive way to a question posed by Cedó, Jespers, and Verwimp [19, Question 4.2]. Moreover, we develop a theory of extensions for left non-degenerate set-theoretic solutions of the Yang-Baxter equation that allows one to construct new families of set-theoretic solutions.

We construct a level − 1 2 vertex representation of the quantum N-toroidal algebra for type C n , which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type C n as a by-product.

We prove that the 4 D ± calculi on the quantum group S U q (2) satisfy a metric-independent sufficient condition for the existence of a unique bicovariant Levi-Civita connection corresponding to every bi-invariant pseudo-Riemannian metric.

We study covariant derivatives on a class of centered bimodules E over an algebra A. We begin by identifying a Z(A) -submodule X(A) which can be viewed as the analogue of vector fields in this context; X(A) is proven to be a Lie algebra. Connections on E are in one to one correspondence with covariant derivatives on X(A). We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.