Lie pairs and formal Lie groups

Abstract: In a previous paper, we introduce and study formal manifolds, which generalize smooth manifolds. In this paper, we establish the basic theory of formal Lie groups, which are group objects in the category of formal manifolds. In particular, extending the classical formal Lie theory theorem, we prove that the category of formal Lie groups is equivalent to the category of Lie pairs.

• The paper introduces a categorical equivalence between Lie pairs, Hopf formal (co)algebras, and formal Lie groups, linking algebraic and geometric structures.
• It employs rigorous functorial constructions and compatibility conditions to extend classical Lie theory into the realm of formal geometry.
• The framework offers practical tools for representation theory and applications to automorphic forms and non-compact group actions.

Lie Pairs and Formal Lie Groups: Equivalences, Structures, and Implications

Introduction

The paper "Lie pairs and formal Lie groups" (2604.25616) provides a comprehensive framework that systematically unifies the concepts of finite-dimensional complex Lie algebras, real Lie groups, and their interaction within the context of formal geometry and topological Hopf algebras. The central result is a categorical equivalence between the notions of Lie pairs, Hopf formal (co)algebras, and formal Lie groups. The theory extends the classical formal Lie theory theorem to a new, broader setting, explicitly characterizing the relationship between complexified algebraic and smooth geometric objects through the formalism of formal geometry. This provides a precise foundation for the representation theory of real algebraic groups as well as applications in the analysis on formal manifolds, particularly with respect to automorphic forms and non-compact group actions.

Lie Pairs: Definition and Motivation

A Lie pair is defined as a pair $(\mathfrak{q},L)$, where $\mathfrak{q}$ is a finite-dimensional complex Lie algebra, $L$ is a (possibly disconnected) real Lie group, equipped with a compatible algebraic action $L\curvearrowright \mathfrak{q}$ and an injective, $L$-equivariant complex Lie algebra homomorphism $\iota: \mathfrak{l} \to \mathfrak{q}$, where $\mathfrak{l}$ is the complexified Lie algebra of $L$. The compatibility conditions enforce that the differential of the group action coincides with the adjoint action via $\iota$. This structure encapsulates simultaneously the algebraic (infinitesimal) symmetry via $\mathfrak{q}$ and the global (possibly continuous or discrete) symmetry via $\mathfrak{q}$0.

The category of Lie pairs generalizes both the classical categories of finite-dimensional complex Lie algebras (as $\mathfrak{q}$1) and Lie groups (as $\mathfrak{q}$2). The morphisms in this category consist of compatible pairs of group and algebra homomorphisms, subjected to strict commutativity diagrams ensuring the intertwining of algebraic and geometric structures.

Formal Manifolds and Formal Lie Groups

Formal manifolds, as established in [CSW1, CSW2, CSW4], provide a functorial generalization of smooth manifolds by working with sheaves of formal power series locally modeled on $\mathfrak{q}$3 over underlying paracompact Hausdorff topological spaces. Infinitesimal formal manifolds correspond to single-point spaces, whose structure sheaves are isomorphic to formal power series rings.

A formal Lie group is a group object in the category of formal manifolds, characterized by morphisms encoding group multiplication, unit, and inverse, with the usual group-theoretic commutativity and associativity constraints diagrammatically enforced at the level of sheaves. This definition strictly generalizes the classical concept of finite-dimensional real or complex Lie groups and Bochner's formal group laws.

Importantly, every finite-dimensional complex Lie algebra $\mathfrak{q}$4 (via its universal enveloping algebra and continuous dual Hopf algebra structure) defines an infinitesimal formal Lie group. Conversely, every Lie group is naturally a formal Lie group.

Topological Hopf Formal Algebras and Coalgebras

The paper leverages the structure of formal power series and smooth function algebras by equipping them with suitable locally convex topologies, distinguishing between projective ($\mathfrak{q}$5) and inductive ($\mathfrak{q}$6) tensor algebra topologies. The algebras of global sections of structure sheaves on formal manifolds become formal algebras, while compactly supported formal distributions (topological duals) become formal coalgebras. Hopf structures arise naturally via the functors induced by the group object morphisms, with duality ensuring equivalence between the algebraic and coalgebraic perspectives.

This setup enables, for example, the recovery of group-theoretic information from topological and algebraic data (smooth functions, distributions), as well as the precise algebraic description of infinitesimal objects.

Main Equivalence Theorem

The central result is the equivalence of four categories:

1. Lie pairs.
2. Hopf formal coalgebras.
3. The opposite category of Hopf formal algebras.
4. Formal Lie groups.

For any formal Lie group $\mathfrak{q}$7, spaces of global functions and distributions inherit topological Hopf algebra and coalgebra structures respectively. The categorical machinery developed shows that the adjoint representation and algebraic embeddings reconstruct the Lie pair $\mathfrak{q}$8, where $\mathfrak{q}$9 is the complexified Lie algebra of $L$0 and $L$1 is its reduction to a real Lie group.

This equivalence extends and generalizes the classical formal Lie theory theorem—which identifies finite-dimensional complex Lie algebras with finite-dimensional formal group laws and finitely generated Hopf algebras—by incorporating group actions and continuous symmetries. The manifold-level perspective thus subsumes both algebraic and smooth geometric cases.

Subgroup Structures, Actions, and Quotients

The theory introduces and systematically categorizes formal Lie subgroups, including their closedness, normality, and their functor-of-points realizations. Actions of formal Lie groups on formal manifolds are defined in terms of morphisms satisfying the natural associativity and identity constraints.

Key structural results include the decomposition theorems for semidirect products, the existence and uniqueness of stabilizer formal Lie subgroups, and the precise criteria for regularity and normality of subgroups in the formal context. Quotients in the formal category are constructed with rigorous universal property arguments, paralleling the properties of homogeneous spaces in the smooth setting, and are shown to be formal manifolds themselves when the subgroups involved are closed and regular.

Explicit Constructions and Functoriality

A notable constructive result is the functorial association of a formal Lie group to any Lie pair as a quotient of an explicit semidirect product by an infinitesimal subgroup. Concretely, for a Lie pair $L$2, one constructs the formal Lie group $L$3, where $L$4 and $L$5 are the infinitesimal formal Lie groups associated to $L$6 and the complexified Lie algebra $L$7, and $L$8 acts on $L$9 via the adjoint representation. The functor mapping a formal Lie group $L\curvearrowright \mathfrak{q}$0 to its associated Lie pair $L\curvearrowright \mathfrak{q}$1 is quasi-inverse to this construction, establishing the categorical equivalence.

Representation-Theoretic and Geometric Implications

This categorical equivalence has significant implications for representation theory, particularly in the analytic representation theory of non-compact real groups and the study of smooth representations realized on formal or infinite-dimensional manifolds. The theory also underpins geometric structures arising in the theory of automorphic forms, where non-compactness and infinite-dimensional features are inescapable.

The explicit description of structure sheaves, their duals, and the functoriality of group actions in the formal category yield a robust toolkit for extending classical Lie theory results to settings involving formal neighbourhoods, infinitesimal symmetries, or generalized distributional algebras.

Conclusion

This work establishes a highly structured and robustly functorial equivalence between the categories of Lie pairs, formal Lie groups, and topological Hopf formal (co)algebras. The methodology extends beyond the classical context of finite-dimensional smooth and algebraic symmetries, enabling systematic passage between algebraic, analytic, and geometric viewpoints via formal geometry. Future directions include detailed analysis of the smooth representation theory of Lie pairs in the formal context, applications to automorphic forms, and further exploration of the interplay between topological Hopf algebras and representation theory for infinite-dimensional or non-compact settings.