Quantum folding

TL;DR

This paper introduces a quantum analogue of classical Lie algebra folding, constructing new algebraic structures that interpolate between different Lie algebras and reveal novel symmetries and actions, especially in complex cases like (so_8,G_2).

Contribution

It develops a quantum folding framework replacing classical algebra pairs with duals, constructing interpolating nilpotent Lie algebras and their quantizations, including new algebraic structures with braid group actions.

Findings

Constructed quantum analogues of Lie algebra foldings.

Identified new algebra of quantum matrices with braid group symmetry.

Detailed the complex structure of the (so_8,G_2) case with over 700 terms.

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.