Parabolic twists for linear algebras $A_{n-1}$

V. D. Lyakhovsky

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 20, Tome 347 (2007), pp. 187-213

Résumé

New solutions of twist equations for universal enveloping algebras $U(A_{n-1})$ are found. They can be presented 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 the generalizations of Jordanian twists (with carrier $b^2$) for special deformed extensions of the Hopf algebra $U(b^2)$. 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 $A_{n-1}$, $g_{\mathcal P}\cap\mathbb N_g^-\ne\varnothing$. The parabolic twisting elements $\mathcal F_{\mathcal P}$ are obtained in the explicit form. The details of the construction are illustrated by considering the examples $n=4$ and $n=11$.