Canonical forms for matrices of Saletan contractions

TL;DR

This paper characterizes Saletan contractions of Lie algebras through canonical matrix forms determined by partitions of the Fitting component's dimension, providing classifications in low dimensions.

Contribution

It introduces a canonical form for Saletan contractions based on Fitting component partitions and classifies all such contractions in three dimensions.

Findings

Canonical forms are determined by partitions of the Fitting component.

Maximal Fitting component contractions are studied and classified.

All three-dimensional contractions of this type are fully classified.

Abstract

We show that each Saletan (linear) contraction can be realized, up to change of bases of the initial and the target Lie algebras, by a matrix-function that is completely defined by a partition of the dimension of Fitting component of its value at the limit value of the contraction parameter. The codimension of the Fitting component and this partition constitute the signature of the Saletan contraction. We study Saletan contractions with Fitting component of maximal dimension and trivial one-part partition. All contractions of such kind in dimension three are completely classified.