Complete dg Lie Algebras

• Complete dg Lie algebras are graded Lie algebras equipped with a differential and a topology that ensures the convergence of infinite sums.
• They serve as essential models in deformation theory and rational homotopy, typically constructed via inverse limits of finite-dimensional nilpotent quotients.
• Integration to dg Lie groups and derived bracket constructions extend their applications to advanced topics in mathematical physics and higher homotopical structures.

Complete differential graded Lie algebras (complete dg Lie algebras, abbreviated as cdgls) are central objects in modern algebra, topology, and theoretical physics, providing the algebraic substrate for deformation theory, rational homotopy, and the paper of higher and derived structures. A complete dg Lie algebra is a graded Lie algebra equipped with a differential and a topology ensuring convergence of infinite sums (usually defined via an inverse system of finite-dimensional nilpotent quotients or with respect to a filtration). This completeness is crucial in contexts such as deformation theory, where solutions and parameter spaces may be infinite-dimensional or formal.

1. Foundational Structure and Definitions

A differential graded Lie algebra (dgLA) consists of a graded vector space $L = \bigoplus_{i \in \mathbb{Z}} L^i$ with a bilinear bracket $[\cdot, \cdot]: L^i \times L^j \to L^{i+j}$ of degree 0 satisfying graded antisymmetry and the graded Jacobi identity, and a differential $d: L^i \to L^{i+1}$ of degree 1 obeying $d^2=0$ and the graded Leibniz rule $d([x, y]) = [d x, y] + (-1)^{|x|}[x, d y]$. A cdgl further requires that $L$ is complete with respect to a topology, typically the inverse limit of finite-dimensional nilpotent quotients, i.e., $L \cong \varprojlim_\alpha L_\alpha$ (Félix et al., 2022).

Completeness ensures well-definedness for infinite formal sums, convergence in deformation problems, and robustness under operations such as forming Maurer–Cartan moduli spaces.

2. Differential Operators and Chevalley–Eilenberg Calculus

In non-associative settings, such as Lie and graded Lie algebras, differential operators are not defined via partial derivatives but are built recursively. For graded Lie algebras, zero-order operators are $A$-module endomorphisms compatible with the grading, first-order operators are sums of graded derivations and zero-order operators, and higher-order operators are compositions of first-order ones. The graded derivations satisfy the graded Leibniz rule

$$d(a \cdot b) = d(a) \cdot b + (-1)^{[d][a]} a \cdot d(b)$$

where $[d], [a]$ denote degrees (Sardanashvily, 2010). The algebra of differential operators organizes itself around these graded commutators and derivations.

The Chevalley–Eilenberg differential calculus defines a differential graded algebra associated to the Lie algebra of derivations. The complex $C^*(DA;A)$ consists of $K$-multilinear, skew-symmetric maps from $DA^{\otimes k}$ to $A$ and the coboundary operator

$$(\delta c)(E_0, \dots, E_k) = \sum_{i=0}^k (-1)^i E_i(c(E_0, …, \hat{E}_i, …, E_k)) + \sum_{i < j} (-1)^{i+j} c([E_i, E_j], E_0, …, \hat{E}_i, …, \hat{E}_j, …, E_k)$$

(Sardanashvily, 2010).

This calculus captures the cohomological and geometric aspects of complete dg Lie algebras, precisely in situations where all derivations are inner and the center is trivial, reducing to the classical Maurer–Cartan complex.

3. Homotopy Theoretic Framework and Rational Realization

The theory of complete dg Lie algebras is deeply intertwined with rational homotopy theory. There exists a pair of adjoint functors between the category of simplicial sets (SSet) and complete dg Lie algebras (cDGL). The model functor $X \mapsto \mathcal{L}_X$ assigns to a finite simplicial set a complete dg Lie algebra, typically constructed as $\mathcal{L}_X = L(s^{-1} NC^+(X))$, where $NC^+(X)$ is the normalized chain complex and $s^{-1}$ is desuspension (Buijs et al., 2016).

The realization functor, defined as $$(L) = \mathrm{Hom}_{\mathrm{cDGL}}(\mathcal{L}_\Delta, L)$$, serves as a simplicial set encoding the homotopy type realized by the Maurer–Cartan solutions of $L$. This establishes a Quillen equivalence—a model category structure—where fibrations are surjective maps in nonnegative degrees, and weak equivalences are those inducing bijections on Maurer–Cartan sets and quasi-isomorphisms on twisted components.

The homotopy type of the realization of a Lie model agrees with the Bousfield–Kan completion of the corresponding simplicial set, cementing the role of cdgls as algebraic models for rational homotopy types, including for non-simply connected spaces (Félix et al., 2021).

A central technical result is that all known realization functors for cdgls (e.g., classical Quillen, Deligne–Getzler–Hinich, representable via cosimplicial free cdgls) are homotopy equivalent (Félix et al., 2022), and the realization functor factors through crossed modules, relating two-stage Lie algebra structures directly to classifying spaces of these modules (Félix et al., 2021).

4. Derived Bracket Constructions and Leibniz Theory

Complete dg Lie algebras also serve as ambient spaces for the representation and cohomology of more general algebraic structures, such as Leibniz algebras. The derived bracket construction,

$[x_1, x_2] := (d x_1, x_2)$

embeds an odd Leibniz structure inside a dg Lie algebra (Uchino, 2013, Mostovoy, 2019). Universal constructions (such as the universal derived bracket or universal enveloping dg Lie algebra) permit the paper of Leibniz (co)homology via subcomplexes of the full Leibniz complex, reflecting the anti-cyclicity of the Leibniz operad. These universal models are significant in deformation theory and in constructing and analyzing complete dg Lie algebras via inverse limits or completions.

Minimal edgl models, built as completions of free Lie algebras with differentials landing in commutator ideals, provide analogues of minimal Sullivan models for cdgas and facilitate the paper of rational homotopy types, cell attachments, and inert maps in topological spaces (Félix et al., 2022).

5. Integration to Differential Graded Lie Groups

An essential aspect of the theory is the integration of cdgls to differential graded Lie groups (dg Lie groups, DGLGs). The integration utilizes graded Hopf algebras and Harish–Chandra pairs, aligning the algebra of functions on a graded (or dg) group with a graded Hopf algebra structure compatible with the differential (Jubin et al., 2019). For a cdgl $\mathfrak{g}$, one integrates the degree 0 part to a Lie group $G_0$ and packages higher graded data (and the differential) in the corresponding graded Harish–Chandra pair. The multiplicative homological vector field $Q$ on $G$ guarantees compatibility:

$$(Q \otimes \mathrm{id} + \mathrm{id} \otimes Q) \circ m^* = m^* \circ Q$$

with $Q$ globalizing the differential via an explicit cocycle construction:

$\mathcal{X}(g) = \mathrm{Ad}_{G_0}(g, d) - d$

and

$$(Qf)(u, g) = (-1)^{\deg(u)} f(u)(g) - f(\mathcal{X}(u), g)$$

The category of dg Lie groups is equivalent to that of differential graded Harish–Chandra pairs, making integration and differentiation formally dual.

Completeness is essential—constructions require convergence and functional analytic control over power series, especially in infinite-dimensional cases or when handling formal deformations.

6. Applications in Deformation Theory, Topology, and Physics

Complete dg Lie algebras play a key role in deformation theory. The solutions to the Maurer–Cartan equation

$d a + \frac{1}{2}[a, a] = 0$

parametrize infinitesimal deformations, and their gauge equivalence classes—encoded by the Deligne groupoid—classify non-abelian extensions and deformation types (Gouray, 2018).

In rational homotopy theory, cdgls model not only the rational homotopy groups but also capture topological structures such as 2-types via crossed modules (Félix et al., 2021), and realize rationalizations of spaces—extending Sullivan's approach to all path-connected spaces (Félix et al., 2022). Minimal edgl models provide explicit computations for Whitehead products and LS category invariants, bridging the gap between algebraic and topological classification.

In physics, tensor hierarchies in supergravity and exceptional field theory are formalized via infinity-enhanced Leibniz structures, which can be reinterpreted as complete dg Lie algebras upon appropriate suspension and extension. The DGLA formalism unifies gauge transformations, Bianchi identities, and first-order duality relations, encoding the entire hierarchy and the dynamical equations of fields, including scalar potentials and gauge-covariant dynamics (Bonezzi et al., 2019).

7. Connections to Higher and Derived Structures

Recent developments include the framework for locally finite $\infty$-modules over a dg Lie algebra, giving rise to "almost" model categories for representations and providing homotopy liftings of classical constructions—such as Loday–Pirashvili modules—to the context of Leibniz$_\infty$ algebras enriched over the Chevalley–Eilenberg dg algebra (Chen et al., 2022). The completeness and local finiteness ensure good homotopical and deformation-theoretic properties for parameter spaces and moduli.

In the context of vertex algebras, dg vertex Lie algebras admit universal enveloping constructions via left adjoint functors, thereby yielding explicit dg vertex algebras. This enhances the homotopical framework for vertex algebra theory and extends classical (Virasoro, Neveu–Schwarz, affine) algebras to the dg and complete setting (Caradot et al., 2023).

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Complete differential graded Lie algebras serve as a comprehensive algebraic framework connecting deformation theory, rational homotopy, representation theory, and the underlying geometric and physical structures. Complete cdgls, through their homotopical, cohomological, and integration properties, unify various approaches to modeling and solving problems across mathematics and mathematical physics. Their completeness—expressed categorically, topologically, or via formal inverse limits—provides the necessary infrastructure for convergence, explicit computation, and functoriality in broad-ranging applications.