Dual contractions and algebraic families

TL;DR

This paper introduces a duality framework for Inönü-Wigner contractions of symmetric Lie algebras, unifying original and dual contractions within a single algebraic family to reveal their geometric relationship.

Contribution

It defines a dual real form inside the complexification of a symmetric Lie algebra and shows how both contractions are fibers of an algebraic family with an anti-holomorphic involution.

Findings

Original and dual contractions are fibers of a single algebraic family.

The duality connects contractions with algebraic-family methods.

The framework unifies geometric and algebraic perspectives on contractions.

Abstract

We introduce a duality for In\"{o}n\"{u}-Wigner contractions attached to real symmetric Lie algebras. Starting from a symmetric pair $(\mathfrak{g},\theta)$, we define a dual real form $\mathfrak{g}^{*}$ inside the complexification of $\mathfrak{g}$ and consider the corresponding contraction with respect to the common fixed-point subalgebra $\mathfrak{g}^{\theta}$. The main result shows that the original contraction and its dual appear as real fibers of a single algebraic family of complex Lie algebras equipped with an anti-holomorphic involution. This places the two contractions in one geometric framework and connects them with the algebraic-family methods developed in recent work on contractions, real forms, and hidden symmetries.