Full chains of twists for symplectic algebras

TL;DR

This paper develops a method to construct explicit full twist deformations of symplectic and other classical Lie algebras using chains of extended Jordanian twists, providing explicit formulas for universal R-matrices.

Contribution

It introduces a canonical form for full chains of twists in simple Lie algebras and constructs explicit universal R-matrices for classical series, advancing deformation theory.

Findings

Explicit full chains of twists are constructed for classical Lie algebras.

Universal R-matrices are explicitly derived for these twists.

The structure of twist chains relates to the algebra's Dynkin diagram.

Abstract

The problem of constructing the explicit form for full twist deformations of simple Lie algebras g with twist carriers containing the maximal nilpotent subalgebra N^+(g) is studied. Our main tool is the sequence of regular subalgebras g_i in U(g) that become primitive under the action of extended jordanian twists F_E: U(g) -> U_E(g). It is demonstrated that the structure of the sequence {g_i} is defined by the extended Dynkin diagram of algebra g. To construct the injection of g_i in U_E(g) the special deformations of algebras U_E(g) are performed. They are reduced to the (cohomologically trivial) twists F_s. Thus it is proved that full chains of twists can be written in the canonical form F_B = F'_N ... F'_2F'_1. The links F'_i in such chains must contain not only the extended twists F_E but also the factors F_s whose form depend on the type of the series of classical algebra g. The explicit forms of universal R-matrices (and the R-matrices in the fundamental representations) corresponding to full chains of twists for classical simple Lie algebras are found. The properties of the construction are illustrated by the example of full chain of extended twists for algebra sp(3).