Lowest Dimensional Example on Non-universality of Generalized Inönü–Wigner 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 Inönü–Wigner 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 Inönü– Wigner contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized Inönü–Wigner contractions are not universal. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized Inönü–Wigner contractions of four-dimensional Lie algebras is also found.

Cite this paper

@inproceedings{Popovych2009LowestDE, title={Lowest Dimensional Example on Non-universality of Generalized Inönü–Wigner Contractions}, author={Dmytro Popovych and Roman O. Popovych}, year={2009} }