Cohomological rigidity of conformal Galilei algebras and their central extensions

TL;DR

This paper investigates the second adjoint cohomology of complexified conformal Galilei algebras and their central extensions, establishing conditions for their cohomological rigidity across various dimensions and spin values.

Contribution

It provides a complete cohomological analysis of conformal Galilei algebras, identifying when they are rigid or deformable, especially for half-integer spins in dimensions other than two.

Findings

Conformal Galilei algebras are cohomologically rigid for all dimensions except two when spin is half-integer.

The second adjoint cohomology space is computed explicitly using Hochschild-Serre theorem.

Rigidity results help classify possible deformations of these non-semisimple Lie algebras.

Abstract

In this paper, we study the second adjoint cohomology of the compexification of the real conformal Galilei algebras \(\mathfrak{cga}_\ell(d,\mathbb{R})\) and their central extensions. These algebras are non-semisimple Lie algebras that appear as non-relativistic analogues of conformal Lie algebras. Using cohomological methods, including the Hochschild-Serre factorization theorem, we compute the space \(H^2(\mathfrak{g}, \mathfrak{g})\) and examine conditions under which these algebras are cohomologically rigid. Our main result shows that the conformal Galilei algebras are rigid for all spatial dimensions \(d \neq 2\) when the spin \(\ell\) is a half-integer. This provides a complete characterization of the formal rigidity of these algebras in the specified parameter range.