# Higher Spin Symmetries and Deformed Schr\"odinger Algebra in Conformal Mechanics

## TL;DR

This paper explores the complex symmetry structures of 1+1-dimensional matrix PDEs with Calogero potentials, revealing higher spin superalgebras and deformed Schrödinger algebra connections, enriching the understanding of conformal mechanics.

## Contribution

It uncovers new invariant superalgebras and deformed symmetry structures in Calogero-based conformal mechanics, including higher spin and deformed Schrödinger algebras.

## Key findings

- Identification of $osp(2|2)$ superalgebra invariance.
- Construction of invariant algebras from differential operators.
- Equivalence of symmetries in pure and DFF Calogero PDEs.

## Abstract

The dynamical symmetries of $1+1$-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial De Alfaro-Fubini-Furlan, DFF, damping term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the $sl(2)\oplus u(1)$ invariance of the Calogero Conformal Mechanics, an $osp(2|2)$ invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the universal enveloping algebra of the deformed Heisenberg algebra, which is shown to be equivalent to a deformed version of the Schr\"odinger algebra. This vector space also gives rise to a higher spin (gravity) superalgebra. We furthermore prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical symmetries, being related by a similarity transformation and a redefinition of the time variable.