$W$-algebras and higher analogs of the Kniznik-Zamolodchikov equations

TL;DR

This paper explores the potential for constructing higher analogs of the Knizhnik-Zamolodchikov equations using higher order central elements in Lie algebras, successfully deriving such an equation for certain cases.

Contribution

It investigates the possibility of extending KZ equations via higher order central elements, deriving a new equation for specific Lie algebra series.

Findings

Higher order central elements can be used to construct analogs of KZ equations.

Construction is not possible for all Lie algebra series.

A new higher analog of the KZ equation is derived for series B and D.

Abstract

The key role in the derivation of the Knizhnik-Zamolodchikov equations in the $WZW$-theory is played by the energy-momentum tensor, that is constructed from a central Casimir element of the second order in a universal enveloping algebra of a corresponding Lie algebra. In the paper a possibility of construction of analogs of Knizhnik-Zamolodchikov equations using higher order central elements is investigated. The Gelfand elements of the third order for a simple Lie algebra of series $A$ and Capelli elements of the fourth order for the a simple Lie algebra of series $B$, $D$ are considered. In the first case the construction is not possible a the second case the desired equation is derived.