# Generalized conformal pseudo-Galilean algebras and their Casimir operators

## TL;DR

This paper introduces a generalized class of conformal pseudo-Galilean algebras, constructs their Casimir operators, and provides methods to compute invariants, extending the understanding of their algebraic structure and invariants.

## Contribution

The paper defines a new family of conformal pseudo-Galilean algebras, derives explicit Casimir operators, and develops a systematic procedure for computing invariants.

## Key findings

- Explicit Casimir operators derived from polynomial matrices.
- Number of invariants calculated for the central factor algebra.
- Complete sets of invariants identified under the condition d ≤ 2ℓ+2.

## Abstract

A generalization $\mathfrak{Gal}_{\ell}(p,q)$ of the conformal Galilei algebra $\mathfrak{g}_{\ell}(d)$ with Levi subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(p,q)$ is introduced and a virtual copy of the latter in the enveloping algebra of the extension is constructed. Explicit expressions for the Casimir operators are obtained from the determinant of polynomial matrices. For the central factor $\overline{\mathfrak{Gal}}_{\ell}(p,q)$, an exact formula giving the number of invariants is obtained and a procedure to compute invariants functions that do not depend on variables of the Levi subalgebra is developed. It is further shown that such solutions determine complete sets of invariants provided that the relation $d\leq 2\ell+2$ is satisfied.