Parabolic twists for the linear algebras A _n−1

Abstract

New solutions of twist equations for the universal enveloping algebras U (A_n−1) are found. These solutions can be represented as products of full chains of extended Jordanian twists $\mathcal{F}_{\widehat{ch}}$ Abelian factors (“rotations”) $\mathcal{F}^R$, and sets of quasi-Jordanian twists $\mathcal{F}^{\widehat{J}}$. The latter are generalizations of Jordanian twists (with carrier b²) for special deformed extensions of the Hopf algebra U (b²). The carrier subalgebra $g_\mathcal{P}$ for the composition $\mathcal{F}_\mathcal{P} = \mathcal{F}^{\widehat{J}} \mathcal{F}^R \mathcal{F}_{\widehat{ch}}$ is a nonminimal parabolic subalgebra in A _n−1 such that $g_\mathcal{P} \cap \mathbb{N}_g^ - \ne \emptyset$. The parabolic twisting elements $\mathcal{F}_\mathcal{P}$ are obtained in an explicit form. Details of the construction are illustrated by considering the examples n = 4 and n = 11. Bibliography: 21 titles.