# Group theoretical aspects of $L^2(\mathbb{R}^+)$, $L^2(\mathbb{R}^2)$ and associated Laguerre polynomials

## TL;DR

This paper explores the algebraic structures underlying certain function spaces and associated Laguerre polynomials, revealing Lie algebra connections and representation theories relevant to mathematical physics.

## Contribution

It introduces a ladder algebraic structure for $L^2(
eal^+)$, constructs quadratic generators for various Lie algebras, and links these to representations in $L^2(
eal^2)$.

## Key findings

- Ladder algebraic structure for $L^2(
eal^+)$ with Lie algebra closure
- Construction of quadratic generators for $so(3)$, $so(2,1)$, $so(3,2)$
- Unitary irreducible representations in $L^2(
eal^2)$ similar to spherical harmonics

## Abstract

A ladder algebraic structure for $L^2(\mathbb{R}^+)$ which closes the Lie algebra $h(1)\oplus h(1)$, where $h(1)$ is the Heisenberg-Weyl algebra, is presented in terms of a basis of associated Laguerre polynomials. Using the Schwinger method the quadratic generators that span the alternative Lie algebras $so(3)$, $so(2,1)$ and $so(3,2)$ are also constructed. These families of (pseudo) orthogonal algebras also allow to obtain unitary irreducible representations in $L^2(\mathbb{R}^2)$ similar to those of the spherical harmonics.