Twist deformations in dual coordinates

TL;DR

This paper explores twist deformations in dual coordinates, analyzing their algebraic structures, constructing explicit transformations for simple Lie algebras, and introducing new solutions and realizations of twist deformations.

Contribution

It provides explicit forms of transformations for simple Lie algebras, constructs a family of deformations for U(sl(3)), and introduces new realizations of parabolic twists.

Findings

Explicit transformation formulas for simple Lie algebras g and g#.

A parametrized family of extended Jordanian deformations for U(sl(3)).

New realizations of the parabolic twist.

Abstract

Twist deformation U_F(g) is equivalent to the quantum group Fun_d(G#) and has two preferred bases: the one originating from U(g) and that of the coordinate functions on the dual Lie group G#. The costructure of the Hopf algebra U_F(g) is analized in terms of group G#. The weight diagram of the adjoint representation of the algebra g# is constructed in terms of the root system L(g). The explicit form of the g --> g# transformation can be obtained for any simple Lie algebra g and the factorizable chain F of extended Jordanian twists. The dual group approach is used to find new solutions of the twist equations. The parametrized family of extended Jordanian deformations for U(sl(3)) is constructed and studied in terms of SL(3)#. New realizations of the parabolic twist are found.