Chains of twists for symplectic Lie algebras

TL;DR

This paper develops a universal method for constructing chains of twists for symplectic Lie algebras, enabling explicit quantizations of classical r-matrices and overcoming previous difficulties with improper chains.

Contribution

It introduces a recursive algorithm for regular chains of twists in U(sp(N)), generalizing previous results and demonstrating the universality of the primitivization effect.

Findings

Existence of a twist F_{B,k} for U(sp(N)) composed of maximal extended Jordanian twists.

Deformation of the primitive subalgebra sp(N-1) via twisting.

Explicit example of the full chain of extended twists for U(sp(3)).

Abstract

Serious difficulties arise in the construction of chains of twists for symplectic Lie algebras. Applying the canonical chains of extended twists to deform the Hopf algebras U(sp(N)) one is forced to deal only with improper chains (induced by the U(sl(N)) subalgebras). In the present paper this problem is solved. For chains of regular injections the sets of maximal extended jordanian twists F_{E,k} are considered. We prove that there exists for U(sp(N)) the twist F_{B,k} composed of the factors F_{E,k}. It is demonstrated that the twisting procedure deforms the space of the primitive subalgebra sp(N-1). The recursive algorithm for such deformation is found. This construction generalizes the results obtained for orthogonal classical Lie algebras and demonstrates the universality of primitivization effect for regular chains of subalgebras. For the chain of maximal length the twists F_{B,k,max} become full, their carriers contain the Borel subalgebra B(sp(N)). Using such twisting procedures one can obtain the explicit quantizations for a wide class of classical r-matrices. As an example the full chain of extended twists for U(sp(3)) is considered.