# QHWM of the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators

## TL;DR

This paper classifies irreducible quasifinite highest weight modules over orthogonal and symplectic Lie subalgebras of matrix quantum pseudo differential operators, linking them to modules of infinite matrices and classical Lie subalgebras.

## Contribution

It provides a classification and realization of these modules, connecting quantum pseudo differential operator Lie subalgebras with classical and infinite matrix Lie algebras.

## Key findings

- Classification of irreducible quasifinite highest weight modules achieved.
- Realization of modules in terms of infinite matrix Lie algebra modules.
- Connections established between quantum pseudo differential operators and classical Lie subalgebras.

## Abstract

In this paper we classify the irreducible quasifinite highest weight modules over the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators. We also realize them in terms of the irreducible quasifinite highest weight modules of the Lie algebras of infinite matrices with finitely many nonzero diagonals and its classical Lie subalgebras of types B, C and D.