(In)finite extensions of algebras from their Inonu-Wigner contractions

TL;DR

This paper demonstrates that Inonu-Wigner contractions of algebras can be understood as infinite towers of abelian extensions, providing a unified framework for various non-relativistic and ultra-relativistic algebraic structures.

Contribution

It introduces a novel perspective that contractions correspond to infinite extensions, applicable to both central and non-central cases, unifying several algebraic structures.

Findings

Inonu-Wigner contraction corresponds to an infinite tower of abelian extensions.

The method applies to central and non-central extensions, including the Bargmann and Weyl algebras.

Examples include Galilean, Carrollian, and Newton-Hooke algebras.

Abstract

The way to obtain massive non-relativistic states from the Poincare algebra is twofold. First, following Inonu and Wigner the Poincare algebra has to be contracted to the Galilean one. Second, the Galilean algebra is to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any Inonu-Wigner contraction of one algebra to another, corresponds to an infinite tower of abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apart from the Bargmann (non-zero mass) extension of the Galilean algebra, our list of examples includes the Weyl algebra obtained from an extension of the contracted SO(3) algebra, the Carrollian (ultra-relativistic) contraction of the Poincare algebra, the exotic Newton-Hooke algebra and some others. The paper is dedicated to the memory of Laurent Houart (1967-2011).