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In the present paper we introduce a quantum analogue of the classical folding of a simply-laced Lie algebra g to the non-simply-laced algebra g^sigma along a Dynkin diagram automorphism sigma of g For each quantum folding we replace g^sigma by its Langlands dual g^sigma^v and construct a nilpotent Lie algebra n which interpolates between the nilpotnent parts of g and (g^sigma)^v, together with its quantized enveloping algebra U_q(n) and a Poisson structure on S(n). Remarkably, for the pair (g, (g^sigma)^v)=(so_{2n+2},sp_{2n}), the algebra U_q(n) admits an action of the Artin braid group Br_n and contains a new algebra of quantum n x n matrices with an adjoint action of U_q(sl_n), which generalizes the algebras constructed by K. Goodearl and M. Yakimov in [12]. The hardest case of quantum folding is, quite expectably, the pair (so_8,G_2) for which the PBW presentation of U_q(n) and the corresponding Poisson bracket on S(n) contain more than 700 terms each.
The goal of this paper is to construct quantum analogues of Chevalley groups inside completions of quantum groups or, more precisely, inside completions of Hall algebras of finitary categories. In particular, we obtain pentagonal and other identities
We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been p
The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii} of the corr
These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the CMSA Math
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integ