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 $(\mathfrak{g},θ)$, 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 $\mathfrak{g}^θ$. 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.

• The paper establishes a duality framework that links Inönü–Wigner contractions with their dual counterparts within symmetric Lie algebras.
• It employs algebraic family constructions and anti-holomorphic involutions to interpolate Lie algebra structures across different real fibers.
• The results enable transferring representation and spectral properties across real forms, impacting studies in mathematical physics.

Dual Contractions and Algebraic Families: A Detailed Analysis

Introduction and Motivation

This paper establishes a formal duality between Inönü–Wigner contractions derived from real symmetric Lie algebras and their duals. By introducing a dual real form $g^*$ of a given real symmetric Lie algebra $(g, \theta)$ within the complexification $g(\mathbb{C})$, and analyzing their respective contractions with respect to the fixed-point subalgebra $g^{\theta}$, the authors demonstrate that both the original contraction and its dual emerge naturally as real fibers of a unifying algebraic family of complex Lie algebras. This geometric linkage facilitates a systematic understanding of the interplay between real forms, their contractions, and symmetric pair duality, thereby amplifying the relevance of algebraic families and real structures in representation theory and mathematical physics.

Contractions and Simple Inönü–Wigner Contractions

The paper begins by rigorously formalizing the notion of Lie algebra contractions as degenerations via parameterized families of automorphisms. When the contraction is orchestrated via a direct sum decomposition $g = k \oplus p$ such that $k$ is a subalgebra, the limiting algebra is $k \ltimes p$ with $p$ abelianized. For symmetric Lie algebras $(g, \theta)$, with $\theta$ an involutive automorphism, the simple Inönü–Wigner contraction crucially hinges on the $(g, \theta)$0 and $(g, \theta)$1 eigenspaces $(g, \theta)$2 and $(g, \theta)$3:

$(g, \theta)$4

Here, $(g, \theta)$5 acts as the identity on $(g, \theta)$6 and scales $(g, \theta)$7 by $(g, \theta)$8. The paper clarifies that this procedure is well-defined up to isomorphism classes that depend only on the subalgebra $(g, \theta)$9, formalized by an explicit bijection between equivalence classes of subalgebras and simple contraction classes.

Algebraic Families and Real Structures

Central to the framework is the concept of an algebraic family of complex Lie algebras parametrized over the affine line $g(\mathbb{C})$0. Such families allow the interpolation of Lie algebra structures as the parameter varies, seamlessly connecting different real forms and contractions. Each fiber at $g(\mathbb{C})$1 corresponds to a specialized Lie algebra determined by polynomially varying structure constants.

The real structure on such a family is enforced by an anti-holomorphic involution, yielding a family of real Lie algebras as fixed-point subalgebras. This enables the conceptual and technical unification of disparate real forms (parametrized by $g(\mathbb{C})$2, $g(\mathbb{C})$3, $g(\mathbb{C})$4) within a single algebraic entity.

Dual Symmetric Lie Algebras and Dual Contractions

Given a symmetric Lie algebra $g(\mathbb{C})$5, the dual real form $g(\mathbb{C})$6 is constructed inside the complexification as:

$g(\mathbb{C})$7

$g(\mathbb{C})$8 inherits a symmetric structure from $g(\mathbb{C})$9 via extension of $g^{\theta}$0 and complex conjugation, forming a new symmetric pair. The dual contraction is then defined as the Inönü–Wigner contraction of $g^{\theta}$1 with respect to the same subalgebra $g^{\theta}$2, realized by scaling the $g^{\theta}$3 summand.

The critical algebraic-geometric insight shown in the paper is that both the original contraction $g^{\theta}$4 and its dual $g^{\theta}$5, as well as the two non-contracted real forms, all appear as fibers in a single algebraic family, explicitly constructed and possessing a canonical anti-holomorphic involution.

Main Theorem and Explicit Family Construction

The main result establishes the existence of an algebraic family $g^{\theta}$6 of complex Lie algebras over $g^{\theta}$7, together with an anti-holomorphic involution $g^{\theta}$8, such that:

• For $g^{\theta}$9, the real fiber $g = k \oplus p$0,
• For $g = k \oplus p$1, the real fiber $g = k \oplus p$2,
• For $g = k \oplus p$3, the real fiber $g = k \oplus p$4.

This construction is made explicit via matrix embeddings, leveraging Ado's theorem and compatible real forms, ensuring that interpolation between fibers remains within the category of real Lie algebras.

The paper provides explicit examples highlighting the behavior in classical types, such as the $g = k \oplus p$5 case, and recovers real versus split orthogonal forms and their contractions as fibers.

Implications and Theoretical Significance

By situating dual contractions as fibers in a unified algebraic-geometric family, this work elucidates deep connections between real forms, contractions, and dualities (notably, it aligns with Cartan or $g = k \oplus p$6-duality perspectives). The results imply that spectral, structural, or representation-theoretic information can be transferred across the family: for example, in quantum mechanical applications (e.g., hydrogen atom symmetry algebras), results about spectrum or symmetry in one regime ($g = k \oplus p$7 or $g = k \oplus p$8) extend, via analyticity, to the entire family (2604.10162).

The framework promotes the use of algebraic families as a tool for categorifying and systematizing relationships among Lie algebras, with ramifications in representation theory, real group theory, and mathematical physics, potentially influencing future advances in the description of hidden symmetries, real form interpolation, and duality phenomena.

Conclusion

The paper rigorously demonstrates that dual pairs of Inönü–Wigner contractions associated with real symmetric Lie algebras and their duals are not isolated constructs but are naturally realized as fibers of a single algebraic family over the affine line. This geometric unification clarifies longstanding dualities and contraction phenomena in Lie theory and opens new avenues for the transference of algebraic and analytic information within and across real forms. The explicit realization of these algebraic families equips researchers with concrete tools for further investigations into symmetric pairs, contractions, and their roles in mathematical physics and representation theory.