Adjoint quotient maps for restricted Lie algebras Wn and Sn

TL;DR

This paper provides explicit descriptions of adjoint quotient maps for Jacobson-Witt and special Lie algebras, extending classical results and analyzing nilpotent elements in these restricted Lie algebras.

Contribution

It introduces explicit descriptions of adjoint quotient maps for Wn and Sn, including analogs of Kostant's criteria and slices, and generalizes Premet's results to Sn.

Findings

Explicit adjoint quotient maps for Wn and Sn

Kostant's regularity criterion analog for Wn

Analysis of nilpotent elements in Sn

Abstract

We give an explicit description of adjoint quotient maps for Jacobson-Witt algebra Wn and special algebra Sn. An analogue of Kostant's differential criterion of regularity is given for Wn. Furthermore, we describe the fiber of adjoint quotient map for Sn and construct the analogs of Kostant's transverse slice. In addition, we analysis the nilpotent elements and generalize some Premet's results to Sn.