Contractions from Grading

TL;DR

This paper explores algebraic contractions relevant to physics, using $ ext{Z}_m$-gradings and introducing partial gradings to unify various algebraic limits like Galilean and Carroll, providing a natural framework for these transformations.

Contribution

It introduces a systematic algebraic approach to contractions via gradings and partial gradings, unifying multiple physical algebra limits.

Findings

Algebraic contractions can be understood through $ ext{Z}_m$-gradings.

Partial gradings naturally arise in the context of algebra contractions.

The framework encompasses flat space, Carroll, Galilean, and conformal algebra limits.

Abstract

We note that large classes of contractions of algebras that arise in physics can be understood purely algebraically, via identifying appropriate $\mathbb{Z}_m$-gradings (and their generalizations) on the parent algebra. This includes various types of flat space/Carroll limits of finite and infinite dimensional (A)dS algebras, as well as Galilean and Galilean Conformal algebras. Our observations can be regarded as providing a natural context for the Grassmann approach of arXiv:1312.2941. We also introduce a related notion, which we call partial grading, that arises naturally in this context.