The FRT-Construction via Quantum Affine Algebras and Smash Products

TL;DR

This paper constructs affine type FRT-Construction algebras using smash products of quantum affine algebras, revealing new algebraic structures related to quantum groups and their deformations.

Contribution

It introduces a novel method of constructing affine type FRT-Construction algebras via smash products of De Concini-Kac-Procesi algebras, extending previous finite-type results.

Findings

Constructed affine type FRT-Construction algebras using smash products.

Demonstrated twisting of algebra multiplication by cocycles.

Connected constructions to Faddeev-Reshetikhin-Takhtajan bialgebras.

Abstract

For every element w in the Weyl group of a simple Lie algebra g, De Concini, Kac, and Procesi defined a subalgebra U_q^w of the quantized universal enveloping algebra U_q(g). The algebra U_q^w is a deformation of the universal enveloping algebra U(n_+\cap w.n_-). We construct smash products of certain finite-type De Concini-Kac-Procesi algebras to obtain ones of affine type; we have analogous constructions in types A_n and D_n. We show that the multiplication in the affine type De Concini-Kac-Procesi algebras arising from this smash product construction can be twisted by a cocycle to produce certain subalgebras related to the corresponding Faddeev-Reshetikhin-Takhtajan bialgebras.