nLab Maurer-Cartan equation



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Equality and Equivalence



In dg-Lie algebras

For (𝔤,[,],)(\mathfrak{g}, [-,-],\partial) a dg-Lie algebra (the differential of degree -1), a Maurer-Cartan element in 𝔤\mathfrak{g} is

  • an element a𝔤 1a \in \mathfrak{g}_1 of degree -1

  • such that the Maurer-Cartan equation holds

    a+12[a,a]=0. \partial a + \frac{1}{2}[a,a] = 0 \,.

This is a special case of MC elements for L-∞ algebras, which we discuss next.

In L L_\infty-algebras

For 𝔤\mathfrak{g} an L-∞ algebra with brackets [,,] k[-,\cdots,-]_k, a Maurer-Cartan element is an element a𝔤a \in \mathfrak{g} such that

k=0 1k![a,,a] k=0. \sum_{k = 0}^\infty \frac{1}{k!} [a, \cdots, a]_k = 0 \,.


As L L_\infty-homomorphisms

For 𝔤\mathfrak{g} an L-∞ algebra and AA a dg-algebra, also the tensor product A𝔤A \otimes \mathfrak{g} naturally inherits the structure of an L-∞ algebra.

Let 𝔤\mathfrak{g} be of finite type and write CE(𝔤)CE(\mathfrak{g}) for the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g}. Then MC-elements in A𝔤A \otimes \mathfrak{g} correspond bijectively to dg-algebra homomorphisms ACE(𝔤)A \leftarrow CE(\mathfrak{g}):

MC(A𝔤)Hom dgAlg(CE(𝔤),A). MC(A \otimes \mathfrak{g}) \simeq Hom_{dgAlg}(CE(\mathfrak{g}), A) \,.

A reference for this is for instance around def. 3.1 in (Hain 1983).

We unwind in steps how this comes about:

First, the space of graded algebra homomorphisms ACE(𝔤)A \leftarrow CE(\mathfrak{g}) is a subspace of the space of linear maps of graded vector spaces, and since CE(𝔤)\mathrm{CE}(\mathfrak{g}) is freely generated as a graded algebra and is of finite type by assumption, this is isomorphic to the space of grading preserving homomorphisms

Hom Vect[](𝔤 *,A) \mathrm{Hom}_{\mathrm{Vect}[{\mathbb{Z}]}}(\mathfrak{g}^*,A)

of linear grading-preserving maps from the graded vector space 𝔤 *\mathfrak{g}^* of dual generators to AA. By the usual relation in Vect[]\mathrm{Vect}[\mathbb{Z}] for 𝔤\mathfrak{g} of finite type, this is isomorphic to the space of elements of total degree degree 1 in elements of AA tensored with 𝔤\mathfrak{g}:

(A𝔤) 1Hom Vect[](𝔤 *,A). (A \otimes \mathfrak{g})_1 \simeq \mathrm{Hom}_{\mathrm{Vect}[{\mathbb{Z}]}}(\mathfrak{g}^*,A) \,.

The dg-algebra homomorphism form the subspace of this space

Hom dgAlg(CE(𝔤),A)Hom grAlg(CE(𝔤),A)(A𝔤) 1 \mathrm{Hom}_{dgAlg}(\mathrm{CE}(\mathfrak{g}),A) \hookrightarrow \mathrm{Hom}_{grAlg}(\mathrm{CE}(\mathfrak{g}),A) \simeq (A \otimes \mathfrak{g})_1

on elements that respect the differential. Under the above equivalence this are elements aa in (A𝔤) 1(A \otimes \mathfrak{g})_1 satisfying a certain condition. By inspection one finds that this condition is precisely the MC equation

d AA+a+[a Aa] 2+[a Aa Aa] 3+=0. d_A A + \partial a + [a \cdot_A a]_2 + [a \cdot_A a \cdot_A a ]_3 + \cdots = 0 \,.

For instance if A=Ω (X)A = \Omega^\bullet(X) is the de Rham algebra of a smooth manifold XX, then MC(A𝔤)MC(A \otimes \mathfrak{g}) is the space of flat L-∞ algebra valued differential forms on XX. See there for more details.


For 𝔤\mathfrak{g} a Lie algebra, XX a smooth manifold, there is a canonical dg-Lie algebra structure on Ω (X)𝔤\Omega^\bullet(X) \otimes \mathfrak{g}.

A Maurer-Cartan element is then precisely a Lie algebra valued 1-form AA whose curvature 2-form vanishes

d dRA+[AA]=0. d_{dR} A + [A \wedge A] = 0 \,.

Range of versions and applications

Maurer–Cartan equation is a name for very many related equations in geometry, algebra, deformation theory, category theory and deformation quantization. Such equations express for example certain conditions in theory of isometric embedding of submanifolds into a euclidean space (‘structure equations’, with relations to the Lie groups O(n)O(n)), invariance of invariant differential forms (Maurer-Cartan forms) on Lie groups, flatness of connections on principal or associated fibre bundles, the solutions in some contexts parametrize infinitesimal deformations, or define twisting cochains. In the context of BV-quantization, a Maurer–Cartan equation has the role of classical master equation.

A Maurer–Cartan equation for A A_\infty-algebras is usually referred to as a generalized Maurer–Cartan equation as it has more summands than the one for dg-algebras. In some contexts like A A_\infty-categories, some authors prefer the geometric terminology ‘homological vector field’ as a datum on a formal geometric space which satisfies a Maurer–Cartan equation. Solutions to Maurer-Cartan equation for a dg- or A A_\infty algebra are called Maurer-Cartan elements.

Maurer-Cartan and Lie theory

Sophus Lie considered groups of transformations first and discovered Lie algebras only later (letter to Mayer, 1874). He has shown that infinitesimally one can solve the Maurer–Cartan equations for a given set of structure constants of a finite-dimensional Lie algebra. This means that one can construct a neighborhood with either the invariant differential form, or dually the invariant vector fields whose commutator corresponds to the commutator of the Lie algebra. This amounts to integrating the Lie algebra to a local Lie group. Only much later, Elie Cartan succeeded in proving the global version of integration, that is the Cartan–Lie theorem. J-P. Serre in an influential textbook called the Cartan–Lie theorem the “third Lie theorem”, which became a rather popular term in recent years, though one should correctly call so just the theorem on local solvability of Maurer–Cartan equation.


The original article:

  • Ludwig Maurer, Über allgemeinere Invarianten-Systeme, Münch. Ber. 18 (1888), 103-150.

Textbook accounts:

A MathOverflow entry about Maurer-Cartan forms for Lie groups: maurer-cartan-form

On Lie's third theorem from the point of view of Maurer–Cartan equations:

  • Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces

  • N. Bourbaki, Lie algebras and lie groups, historical appendix

  • F. Engel, P. Heegaard, Sophus Lie Samlede Avhandliger (Collected works)

In the generality of L L_\infty -algebras:

Also around def. 3.1 in

  • R. M. Hain, Twisting cochains and duality between minimal algebras and minimal Lie algebras, Trans. Amer. Math. Soc. 277 (1983), no. 1, 397–411.

In relation to equations of motion of Yang-Mills theory and gravity (by truncation of string field theory):

Last revised on October 22, 2021 at 09:38:53. See the history of this page for a list of all contributions to it.