Quantum folding

Abstract

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.