Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan-Weyl realizations for quantum affine algebras

TL;DR

This paper explores the twisting of quantum (super)algebras via the universal R-matrix, establishing connections between different realizations and demonstrating isomorphisms of q-deformations for superalgebras with varying Cartan matrices.

Contribution

It provides explicit relations between Drinfeld's second realization and Cartan-Weyl generators, and shows how Drinfeld's comultiplication is a twist of the standard one using the universal R-matrix.

Findings

Universal R-matrix factors generate twistings of Hopf structures.

Isomorphic superalgebras with different Cartan matrices have isomorphic q-deformations.

Drinfeld's comultiplication is a twisting of the standard comultiplication.

Abstract

We show that some factors of the universal R-matrix generate a family of twistings for the standard Hopf structure of any quantized contragredient Lie (super)algebra of finite growth. As an application we prove that any two isomorphic superalgebras with different Cartan matrices have isomorphic q-deformations (as associative superalgebras) and their standard comultiplications are connected by such twisting. We present also an explicit relation between the generators of the second Drinfeld's realization and Cartan-Weyl generators of quantized affine nontwisted Kac-Moody algebras. Further development of the theory of quantum Cartan-Weyl basis, closely related with this isomorphism, is discussed. We show that Drinfeld's formulas of a comultiplication for the second realization are a twisting of the standard comultiplication by factors of the universal R-matrix. Finally, properties of the Drinfeld's comultiplication are considered.