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This paper deals with left non-degenerate set-theoretic solutions to the Yang-Baxter equation (=LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution is associated a shelf…

Additive deformations of bialgebras in the sense of J. Wirth, i.e. deformations of the multiplication map fulfilling a certain compatibility condition w.r.t. the coalgebra structure, can be generalized to braided bialgebras. The theorems…

We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase…

As one knows, for every Poisson manifold $M$ there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector. Let a Lie…

Using the language of vertex operator algebras (VOAs) and vector-valued modular forms we study the modular group representations and spaces of 1-point functions associated to intertwining operators for Virasoro minimal model VOAs. We…

We consider all genus two correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of a sequence of…

For a vertex operator algebra $V$ with conformal vector $\omega$, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi-conformal vectors of…

We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp.…

We develop criteria to decide if an $N=2$ or $N=4$ super conformal algebra is a subalgebra of a super vertex operator algebra in general, and of a super lattice theory in particular. We give some specific examples.

We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its corresponding small-time heat kernel…

Let $\mathcal{O}$ be the category of representations of the Borel subalgebra of a quantum affine algebra introduced by Jimbo and the first author. We show that the Grothendieck ring of a certain monoidal subcategory of $\mathcal{O}$ has the…

We study the abelianization of Kontsevich's Lie algebra associated with the Lie operad and some related problems. Calculating the abelianization is a long-standing unsolved problem, which is important in at least two different contexts:…

Motivated by a question of A.~Skalski and P.M.~So{\l}tan about inner faithfulness of the S.~Curran's map, we revisit the results and techniques of T.~Banica and J.~Bichon's Crelle paper and study some group-theoretic properties of the…

This is a brief note on a numerical computation of the dimension of the center of the small quantum group u_q(sl_3) at the 5th root of unity. The obtained dimension is 57. Using the description of the regular block of the center for…

Given a vertex Lie algebra $\mathscr L$ equipped with an action by automorphisms of a cyclic group $\Gamma$, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over…

To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of…

Given a skew left brace $B$, a method is given to construct all the non-degenerate set-theoretic solutions $(X,r)$ of the Yang Baxter equation such that the associated permutation group $\mathcal{G}(X,r)$ is isomorphic, as a skew left…

We study a natural q-analogue of a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory, (called Manin Matrices in [5]) . These matrices we shall call…

The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson…