Mathematics

By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level k⩾1 and type A (1) 1 . We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers-Ramanujan-type identities.

This monograph, along with a self-consistent presentation of the theory of q-W-algebras including the construction of algebraic group analogues of Slodowy slices, contains a description of q-W-algebras in terms of Zhelobenko type operators introduced in the book. This description is applied to prove the De Concini-Kac-Procesi conjecture on the dimensions of irreducible modules over quantum groups at roots of unity.

We relate extensions of completely unitary VOAs and (commutative) Q-systems. As an application, we show that any unitary extension of a completely unitary VOA is completely unitary.

The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of n -ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.

We find the free field construction of the basic W -current and screening currents for the deformed W -superalgebra W q,t (A(M,N)) associated with Lie superalgebra of type A(M,N) . Using this free field construction, we introduce the higher W -currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin-diagrams for the Lie superalgebra A(M,N) , though the screening currents are not. This allows us to define W q,t (A(M,N)) by generators and relations.

In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.

We describe a method for quantization of Poisson Hopf algebras in Q -linear symmetric monoidal categories. It is compatible with tensor products and can also be used to produce braided Hopf algebras. The main idea comes from the fact that nerves of groups are symmetric simplicial sets. Nerves of Hopf algebras then turn out to be braided rather than symmetric and nerves of Poisson Hopf algebras to be infinitesimally braided. The problem is thus solved via the standard machinery of Drinfeld associators.

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove that any continuum Kac-Moody algebra is canonically endowed with a non-degenerate invariant bilinear form. The positive and negative Borel subalgebras form a Manin triple with respect to this pairing, inducing on the continuum Kac-Moody algebra a topological quasi-triangular Lie bialgebra structure. We then construct an explicit quantization, which we refer to as a continuum quantum group, and we show that the latter is similarly realized as an uncountable colimit of Drinfeld-Jimbo quantum groups.

Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster X -varieties were introduced in [BFMNC18]. In this paper we show that the two constructions are compatible -- we extend the Fock-Goncharov quantization of X -varieties to the families of [BFMNC18]. As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A -varieties ([BZ05]). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in [LLRZ14], we compute a counter-example to quantum positivity of the quantum theta basis.

We introduce a new quantized enveloping superalgebra U q p n attached to the Lie superalgebra p n of type P . The superalgebra U q p n is a quantization of a Lie bisuperalgebra structure on p n and we study some of its basic properties. We also introduce the periplectic q -Brauer algebra and prove that it is the centralizer of the U q p n -module structure on C(n|n ) ?�l . We end by proposing a definition for a new periplectic q -Schur superalgebra.