Dual contractions and algebraic families

Abstract

We introduce a duality for Inönü-Wigner contractions attached to real symmetric Lie algebras. Starting from a symmetric pair $(\mathfrak{g},\theta )$, we define a dual real form ${\mathfrak{g}}^{\ast }$ 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.