Lowest dimensional example on non-universality of generalized In\"on\"u-Wigner contractions

TL;DR

This paper demonstrates that generalized In"on"u-Wigner contractions are not universal in four-dimensional Lie algebras, providing the lowest-dimensional example where such contractions are insufficient, and identifies bounds on parameters for these contractions.

Contribution

It presents the first explicit example of non-universality of generalized IW-contractions in four-dimensional Lie algebras and determines bounds on contraction parameters.

Findings

Identifies a unique pair of complex four-dimensional Lie algebras with non-equivalent contraction properties.

Shows that generalized IW-contractions are not sufficient for all contractions in four dimensions.

Establishes a lower bound of three on the exponents needed for all generalized IW-contractions.

Abstract

We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in conventional algebraic terms). Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra. The example we constructed demonstrates that even in the dimension four generalized IW-contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized IW-contractions are not universal. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra is not characteristically nilpotent and, therefore, admits nontrivial diagonal derivations. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized IW-contractions of four-dimensional Lie algebras is also found.