Lie Pairs: Bridging Algebra, Geometry, and Homotopy

• Lie pairs are algebraic structures that bridge Lie groups and Lie algebras in formal geometry, providing a framework for studying singular spaces and deformations.
• They also manifest as Lie algebroid pairs, where an inclusion of Lie algebroids over a manifold underpins the deformation theory and cohomological analyses pertinent to foliations and complex structures.
• Extensions of Lie pairs include dual and matched pair constructions that foster developments in symplectic geometry, solvable Lie algebra theory, and homotopical algebra.

A Lie pair is a term that appears in several, fundamentally distinct but interrelated mathematical contexts, each blending the algebraic, differential, and homotopical structures underlying Lie theory. The concept encompasses formal-geometry constructs (as a bridge between Lie groups and Lie algebras in the formal manifold context), inclusions of Lie algebroids (providing a framework for the geometry of singular spaces, foliations, and connections), algebraic and coalgebraic dualities (matched pairs, dual pairs), as well as the construction blocks for solvable Lie algebras and certain semifield or semiring–based generalizations.

1. Lie Pairs in Formal Geometry and Equivalence with Formal Lie Groups

In the formal geometric setting, a Lie pair is defined as a quadruple $(\mathfrak q, L, \rho, \iota)$, where $\mathfrak q$ is a finite-dimensional complex Lie algebra, $L$ is a real Lie group, $\rho: L \to \mathrm{Aut}(\mathfrak q)$ is a smooth action by Lie algebra automorphisms, and $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$ is an injective Lie algebra map satisfying:

• $L$-equivariance of $\iota$ (equivariance under the respective adjoint and induced actions),
• Compatibility between the infinitesimal action $d\rho$ and the adjoint action via $\iota$.

The category $\mathrm{LiePair}$ has as objects such quadruples, and morphisms are pairs $\mathfrak q$0 comprising a Lie algebra homomorphism $\mathfrak q$1 and a group homomorphism $\mathfrak q$2 compatible with the defining data.

A central theorem establishes a categorical equivalence between Lie pairs and formal Lie groups—group objects in the category of formal manifolds (possibly infinite-dimensional, with structure sheaf locally isomorphic to $\mathfrak q$3). The functor $\mathfrak q$4 (taking the tangent Lie algebra at the identity and the reduced group) provides one direction, and a construction using the universal enveloping algebra and formal group action provides the inverse. This establishes Lie pairs as the algebraic avatars of formal group objects, embedding their cohomological and deformation theory in the formal-geometry setting (Chen et al., 28 Apr 2026).

2. Lie Algebroid Pairs: Structure, Deformation Theory, and Applications

In differential geometry and singular foliation theory, a Lie algebroid pair (or "Lie pair" for short) is an inclusion $\mathfrak q$5 of Lie algebroids over the same base manifold $\mathfrak q$6. These are vector bundles with bracket and anchor structure maps satisfying the Lie-Rinehart conditions. The inclusion $\mathfrak q$7 encapsulates both subalgebroid and corresponding quotient structures, with the quotient bundle $\mathfrak q$8 inheriting an $\mathfrak q$9-module structure via the Bott connection.

Infinitesimal and formal deformation theory of such pairs is governed by $L$0-algebras. For a fixed splitting $L$1, the Chevalley–Eilenberg complex $L$2 acquires a cubic $L$3-structure, and in the case where $L$4 is a matched sum of $L$5 and $L$6, this reduces to a dg Lie algebra controlling the deformation functor. The general deformation theory recovers and unifies classical results such as the Kodaira–Spencer dg Lie algebra for deformations of complex structures and the deformation theory of transversely holomorphic foliations (Ni et al., 3 Dec 2025).

The theory is further deepened via the construction of dg-manifolds and dg-Lie algebroids associated to Lie pairs. For matched pairs, there is a quasi-isomorphism linking the Atiyah class of the dg-Lie algebroid to that of the original Lie pair, and this underpins the identification of geometric and algebraic curvature invariants (Batakidis et al., 2016).

3. Algebraic and Coalgebraic Dual Pair Structures

Within symplectic and Poisson geometry, a dual pair—sometimes referred to as "Lie-Weinstein dual pair"—is a configuration where a symplectic manifold $L$7 carries Hamiltonian commuting actions of two Lie groups $L$8, $L$9, with corresponding equivariant momentum maps $\rho: L \to \mathrm{Aut}(\mathfrak q)$0, $\rho: L \to \mathrm{Aut}(\mathfrak q)$1, such that their fibers are symplectically orthogonal. This formalism, in its linear and matrix group incarnations, connects to fluid mechanics (linear EPDiff and ideal fluid dual pairs) and allows for powerful orbit-reduction and Clebsch variable constructions in finite dimensions (Skerritt et al., 2018). Here, "Lie pairs" identify configurations where mutual centralizers and Poisson-centralizing subalgebras yield a correspondence between coadjoint orbits, providing linear "toy models" for infinite-dimensional dual pairs.

4. Lie Pairs in Algebraic Bracketing: Construction Blocks for Solvable Lie Algebras

A complementary algebraic notion, sometimes also called a "Lie pair," is a pair $\rho: L \to \mathrm{Aut}(\mathfrak q)$2 consisting of a linear map $\rho: L \to \mathrm{Aut}(\mathfrak q)$3 and an eigenvector $\rho: L \to \mathrm{Aut}(\mathfrak q)$4, yielding a canonical Lie bracket on the dual space $\rho: L \to \mathrm{Aut}(\mathfrak q)$5. The bracket $\rho: L \to \mathrm{Aut}(\mathfrak q)$6 (with $\rho: L \to \mathrm{Aut}(\mathfrak q)$7) produces solvable and, in the nilpotent $\rho: L \to \mathrm{Aut}(\mathfrak q)$8 case, nilpotent Lie algebras. It is proven that arbitrary real finite-dimensional solvable Lie algebras can be assembled from suitable collections of such Lie pairs, making them constructive "atoms" for the structure theory of solvable Lie algebras (Dobrogowska et al., 2023).

5. Matched Pairs, Bicrossed Products, and Generalizations

Matched pairs generalize the notion of direct sum by coupling two Lie algebras $\rho: L \to \mathrm{Aut}(\mathfrak q)$9 with mutually interacting module structures. Specifically, linear maps $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$0 and $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$1 are required to satisfy coupled compatibility conditions, ensuring closure under the so-called bicrossed bracket. Matched pairs arise naturally in the study of Rota–Baxter Lie algebras, Manin triples, and cocycle twists of Hopf and Lie algebraic structures (Wang, 1 Oct 2025, Zhang, 2021). In the context of Lie algebroids, "matched pair" refers to a decomposition $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$2 by two subalgebroids each acting (by flat connections) on the other, yielding simplifications in cohomological and dg-Lie constructions (Batakidis et al., 2016).

6. Homotopical and $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$3 Extensions

Lie pairs serve as organizing data for enhanced algebraic structures:

• In the $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$4-context, to every inclusion of Lie algebroids $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$5 one canonically attaches a minimal $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$6-algebra structure on $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$7, providing a universal enveloping algebra for the corresponding $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$8-algebroid $$\iota: \mathrm{Lie}(L) \otimes_{\mathbb{R}} \mathbb{C} \hookrightarrow \mathfrak q$$9 (Stiénon et al., 2022).
• Infinitesimal deformations and homological properties of Lie-Rinehart pairs, their strong homotopy versions, and bar–cobar adjunctions are developed within the framework of $L$0-categories, where the formal existence and uniqueness up to homotopy of BV-type resolutions is established (Pištalo, 6 Jan 2026).
• For strongly homotopy Lie pairs $L$1, the Atiyah class $L$2 for an $L$3-module $L$4 produces a Lie structure on Chevalley–Eilenberg cohomology, and its invariance under $L$5-compatible deformations is demonstrated (Chen et al., 2016).

7. Generalizations: Semialgebraic Lie Pairs and Compatibility Structures

The "Lie pair" notion is extended to semialgebraic and semiring contexts, as in the definition of Lie semialgebra pairs $L$6 over a commutative semiring $L$7. Here, axioms mirror the usual alternation, antisymmetry, and Jacobi identity, but only modulo a null set $L$8 replacing the role of zero. Three categories of morphisms—strict, weak, and surpassing-relation-preserving—support analogues of the fundamental Poincaré-Birkhoff-Witt theorem, highlighting the flexibility of Lie pair constructions in generalized algebraic settings (Gatto et al., 2023).

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References to Key Papers

• Formal Lie groups and Lie pairs equivalence: (Chen et al., 28 Apr 2026)
• Lie algebroid pairs, deformation theory, and dg-manifolds: (Ni et al., 3 Dec 2025, Batakidis et al., 2016)
• Dual pairs for matrix Lie groups: (Skerritt et al., 2018)
• Lie pairs as structure blocks for solvable algebras: (Dobrogowska et al., 2023)
• Matched pairs, bicrossed products, and Rota-Baxter structures: (Wang, 1 Oct 2025, Zhang, 2021)
• $L$9 structures and universality: (Stiénon et al., 2022)
• Homotopical algebra of Lie-Rinehart pairs: (Pištalo, 6 Jan 2026)
• Strongly homotopy Lie pairs and Atiyah classes: (Chen et al., 2016)
• Lie semialgebra pairs and PBW theory for semirings: (Gatto et al., 2023)