Noncommutative symmetric functions and Laplace operators for classical Lie algebras

TL;DR

This paper introduces new Laplace (Casimir) operators for orthogonal and symplectic Lie algebras, utilizing noncommutative symmetric functions and graph-based path methods, extending known results for gl(N).

Contribution

It constructs novel Laplace operators for classical Lie algebras using noncommutative symmetric functions and determinant decompositions, providing new insights and alternative proofs.

Findings

Constructed new Laplace operators for orthogonal and symplectic Lie algebras.

Expressed operators via paths in graphs related to Lie algebra generators.

Provided an alternative proof for the gl(N) case using quantum determinant decomposition.

Abstract

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).