Equivariant Chevalley–Eilenberg Cohomology

• Equivariant Chevalley–Eilenberg cohomology is a generalized theory that incorporates symmetry constraints to link Lie algebraic and geometric data.
• It adapts classical alternating cochains and Chevalley–Eilenberg differentials by imposing compatibility with equivariant maps like restriction, transfer, and conjugation.
• The framework unifies distinct approaches—such as Cₚ-Green functors and coset superspace methods—providing insights into singular extensions and invariant structures.

Equivariant Chevalley–Eilenberg cohomology is a family of cohomological constructions that extends classical Chevalley–Eilenberg theory from a single Lie algebra to settings with additional symmetry data. In the recent literature, the term appears in at least two technically distinct forms: as a cohomology theory for $C_p$-Green functors of Lie type, where equivariance is encoded by restriction, transfer, and conjugation maps between the two subgroup levels $e$ and $C_p$; and as the cohomology of basic Maurer–Cartan forms on a coset superspace $G/H$ with respect to a covariant differential $D$ (Anowar et al., 10 Jul 2025, Catenacci et al., 2020). In both settings, the classical ingredients—alternating cochains, a Chevalley–Eilenberg-type differential, and extension-theoretic or geometric interpretations of low-degree classes—are retained only after imposing compatibility with the relevant equivariant structure. Related work on left-invariant involutive structures on compact Lie groups develops the same general Chevalley–Eilenberg reduction principle, namely that geometric cohomology may, under suitable hypotheses, be computed from invariant or Lie-algebraic data (Jahnke, 2019).

1. Classical prototype and equivariant generalization

The classical Chevalley–Eilenberg cohomology of a Lie algebra $\mathfrak g$ with coefficients in a module $M$ is built from alternating multilinear maps

$$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$

equipped with the standard Lie-algebra differential. In that classical setting, $H^2$ classifies abelian or singular extensions of $\mathfrak g$ by $e$0 (Anowar et al., 10 Jul 2025).

The equivariant generalizations in the cited literature modify this picture in different ways. For $e$1-Green functors of Lie type, a cochain is not a single alternating map but a compatible pair of alternating maps at the subgroup levels $e$2 and $e$3, and the complex is constrained by Mackey/Frobenius compatibility. For coset superspaces $e$4, the relevant cochains are not arbitrary forms on $e$5 but basic forms, meaning forms that are both horizontal and $e$6-invariant, and the differential is the covariant differential $e$7 rather than the full de Rham operator (Catenacci et al., 2020).

A plausible implication is that “equivariant Chevalley–Eilenberg cohomology” should be read as a contextual term rather than the name of a single universal construction. What unifies these usages is the passage from ordinary Lie-algebraic cochains to complexes that encode symmetry constraints intrinsically.

2. $e$8-Green functors of Lie type

A $e$9-Mackey functor $C_p$0 is called a $C_p$1-Green functor of Lie type if $C_p$2 and $C_p$3 are Lie algebras, restriction and conjugation are Lie algebra homomorphisms, and transfer satisfies Frobenius relations with the bracket. Explicitly, for all $C_p$4 and $C_p$5,

$C_p$6

$C_p$7

The paper also introduces a graded version: if both values are graded Lie algebras and the structure maps preserve grading, one obtains a $C_p$8-Green functor of graded Lie type (Anowar et al., 10 Jul 2025).

Because $C_p$9 has only two subgroups, the equivariant data are concentrated in the pair

$G/H$0

together with the maps $G/H$1, $G/H$2, and $G/H$3. This two-level structure is the basic organizing device of the theory. It replaces a single Lie algebra by a Mackey/Green-functor-like object carrying Lie brackets at both levels.

The examples given in the paper show that the definition is meant to be broad rather than exceptional. They include the Heisenberg Lie algebra with a cyclic permutation action, $G/H$4 with $G/H$5-action $G/H$6, fixed-point constructions from a Lie algebra with $G/H$7-action, direct-sum or diagonal constructions, and derivation Lie algebras $G/H$8 and $G/H$9 (Anowar et al., 10 Jul 2025).

3. Equivariant cochains, differential, and cup products

To construct cohomology in the $D$0-Green-functor setting, tensor and exterior products are first defined equivariantly. For $D$1, the tensor product $D$2 is obtained by quotienting the direct sum of levelwise tensor products by relations expressing compatibility with transfer and restriction: $D$3

$D$4

Conjugation acts diagonally,

$D$5

and the resulting construction is again a Mackey functor. The exterior product is then the quotient of $D$6 by the diagonal ideal generated by equal factors, exactly paralleling the ordinary exterior quotient (Anowar et al., 10 Jul 2025).

For $D$7, the equivariant morphism space

$D$8

consists of pairs

$D$9

that commute with restriction, transfer, and conjugation. If $\mathfrak g$0 is a $\mathfrak g$1-Green functor of Lie type and $\mathfrak g$2 an $\mathfrak g$3-bimodule, the cochain groups are

$\mathfrak g$4

The differential is defined levelwise by the ordinary Chevalley–Eilenberg formula. At the $\mathfrak g$5-level,

$\mathfrak g$6

and analogously at the $\mathfrak g$7-level. The paper verifies the usual nilpotency relations

$\mathfrak g$8

so that

$\mathfrak g$9

The cohomology is therefore “levelwise” Chevalley–Eilenberg cohomology, but only after imposing Mackey-functor compatibility on the cochains themselves.

The theory also carries a cup product. For cochains

$M$0

the product is defined levelwise by the standard alternating shuffle expression with the Lie-module bracket in the target. It satisfies the usual Leibniz identity,

$M$1

and similarly at the $M$2-level. Consequently the cup product descends to cohomology, and the graded cohomology

$M$3

becomes a $M$4-Green functor of graded Lie type (Anowar et al., 10 Jul 2025).

4. Second cohomology and singular extensions

A singular extension of a $M$5-Green functor of Lie type $M$6 by an $M$7-bimodule $M$8 is an exact sequence of Mackey functors

$M$9

where $$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$0 is itself a $$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$1-Green functor of Lie type and the induced sequences at $$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$2 and $$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$3 are singular extensions of Lie algebras. An extension is $$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$4-split if there are Lie algebra sections

$$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$5

compatible with restriction, transfer, and conjugation. Two extensions are equivalent if there is a natural isomorphism of the middle terms commuting with the structure maps (Anowar et al., 10 Jul 2025).

The central extension-classification statement is that equivalence classes of singular extensions correspond to second cohomology classes. Concretely, given an $$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$6-split extension and chosen splittings, one defines cocycles by the defect of the sections from being Lie homomorphisms: $$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$7

$$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$8

The Jacobi identity yields the cocycle conditions, while compatibility with conjugation, restriction, and transfer forces the equivariant identities, including

$$C^n(\mathfrak g,M)=\operatorname{Hom}(\wedge^n\mathfrak g,M),$$9

$H^2$0

$H^2$1

$H^2$2

Conversely, a $H^2$3-cocycle $H^2$4 produces a singular extension by the semidirect-sum construction

$H^2$5

with brackets

$H^2$6

$H^2$7

and componentwise structure maps

$H^2$8

Changing the splitting changes the cocycle by a coboundary, so second cohomology classifies equivalence classes of singular extensions in the equivariant setting (Anowar et al., 10 Jul 2025).

5. Coset superspaces and the basic-subcomplex definition

A second major usage of equivariant Chevalley–Eilenberg cohomology appears for coset superspaces $H^2$9. One starts from a Lie supergroup $\mathfrak g$0, a Lie sub-supergroup $\mathfrak g$1, and a vector-space splitting

$\mathfrak g$2

with $\mathfrak g$3 an $\mathfrak g$4-invariant complement in the reductive case: $\mathfrak g$5 The Maurer–Cartan form on the quotient decomposes into coset vielbeins and subgroup connection forms,

$\mathfrak g$6

and the curvature of the $\mathfrak g$7-connection is

$\mathfrak g$8

The covariant differential is introduced as

$\mathfrak g$9

The paper emphasizes that this operator is not nilpotent on all forms, because the curvature obstructs nilpotency unless one restricts to the appropriate class of forms (Catenacci et al., 2020).

That restriction is the basic subcomplex. A form $e$00 is basic if it is both $e$01-invariant and horizontal: $e$02 for every generator $e$03. The equivariant cohomology of the coset is then defined as the cohomology of the basic subcomplex with respect to $e$04: $e$05

This framework is embedded in a broader comparison with ordinary and integral-form Chevalley–Eilenberg cohomology for Lie superalgebras. Ordinary cochains are identified with superforms

$e$06

while integral-form cochains are

$e$07

A central theorem gives an isomorphism, up to degree reversal,

$e$08

A notable feature of the coset theory is that exactness must be tested only against basic primitives. The paper’s model example is the bosonic coset $e$09, where

$e$10

but $e$11 is not basic, so the class remains nontrivial in equivariant cohomology. This sharply distinguishes equivariant closedness and exactness in the basic subcomplex from ordinary closedness and exactness on all forms (Catenacci et al., 2020).

6. Computed examples on super-cosets

The coset-superspace theory is accompanied by explicit computations, and these computations show that equivariant Chevalley–Eilenberg cohomology can be either finite-dimensional or infinite-dimensional depending on the embedding and the basic-form constraints (Catenacci et al., 2020).

Space	Nonzero $e$12	Distinguished classes or generators
$e$13	$e$14 for $e$15	$e$16
$e$17	$e$18 for $e$19	constants only
$e$20	$e$21 for $e$22	trivial above degree $e$23
$e$24	$e$25 for even $e$26, $e$27 for odd $e$28	$e$29
$e$30	$e$31 for even $e$32	$e$33
$e$34	$e$35 for $e$36 in one embedding; $e$37 for even $e$38 in the other	$e$39
$e$40	$e$41 in even degrees	infinite-dimensional
$e$42	$e$43 for $e$44, $e$45 for even $e$46	additional $e$47-form $e$48
$e$49	generated by $e$50, a $e$51-form, and bilinear towers	infinite towers from $e$52
$e$53	generated by $e$54 and $e$55	$e$56-form no longer basic
$e$57	$e$58 for $e$59	$e$60

Among these, $e$61 is singled out in the paper as the most physically important coset example. Its covariant Maurer–Cartan equations are

$e$62

and the nontrivial degree-$e$63 class is represented by

$e$64

The same paper stresses that ordinary Chevalley–Eilenberg cohomology of superalgebras underlies FDA formulations of supergravity and higher Wess–Zumino terms for super $e$65-branes, while the equivariant version is the algebraic object naturally adapted to coset superspaces such as $e$66 and $e$67 (Catenacci et al., 2020).

7. Relation to Chevalley–Eilenberg reduction on compact Lie groups

A related, though differently formulated, extension of the Chevalley–Eilenberg viewpoint appears in the study of left-invariant elliptic and hypocomplex involutive structures on compact Lie groups. There, the basic object is the $e$68-complex

$e$69

whose cohomology $e$70 generalizes both de Rham and Dolbeault cohomology. Restricting to left-invariant forms yields $e$71, and there is also a purely algebraic Chevalley–Eilenberg-type complex $e$72 with cohomology $e$73. The natural map

$e$74

is always injective (Jahnke, 2019).

Under further hypotheses, the geometric cohomology can be computed entirely from invariant or Lie-algebraic data. The paper proves, for example, that if $e$75 has property (K) in degree $e$76, then

$e$77

Under closed-orbit and splitting assumptions,

$e$78

and when $e$79 is compact semisimple, Bott’s theorem yields

$e$80

This does not define “equivariant Chevalley–Eilenberg cohomology” in the same sense as the $e$81-Green-functor or coset-superspace constructions. However, it displays the same structural program: start from a geometric or symmetry-constrained complex, isolate an invariant subcomplex, and identify conditions under which the full cohomology is recovered by a finite-dimensional Lie-algebraic model. This suggests a broad unifying theme across the cited literature: equivariance enters by restricting admissible cochains, primitives, or representatives so that the resulting cohomology encodes both Lie-algebraic and symmetry data rather than Lie-algebraic data alone.