The Exterior Derivative on the Lie Algebra

Introduction

The Chevalley–Eilenberg differential $d$ is the operator on the graded space of cochains of a Lie algebra that turns the Lie bracket into a differential: on $n$-cochains it is given by the Cartan formula

$$ (df)(x_1,\dots,x_{n+1})=\sum_{i=1}^{n+1}(-1)^{i+1}x_i\cdot f(x_1,\dots,\widehat{x_i},\dots,x_{n+1}) +\sum_{i

and its square is zero. The complex, the cohomology and the interpretation of the low degrees are the subject of Lie Algebra Cohomology, where the differential is defined and the cohomology is computed; this article treats the differential as an operator. It presents the Cartan formula and its equivalence with the alternating-sum form, the derivation property of $d$ for the cup product that makes the cochains a differential graded algebra, the contraction by a vector and the Lie derivative with the Cartan magic formula $L_x=d\iota_x+\iota_x d$, and the algebra of commutation relations these operators satisfy. The exterior algebra and its products are The Exterior Algebra; the cochain complex and its cohomology are Lie Algebra Cohomology.

The article stays inside Part I and reasons with no smooth structure: the exterior derivative of a smooth manifold, the de Rham complex and the Cartan formula of differential geometry are named at the end and deferred to a later Part, and nothing of them is used. The base is a field $K$, the Lie algebra is written $\mathrm{G}$, the module of coefficients is $M$, the cochains are $C^n(\mathrm{G};M)=\operatorname{Hom}_K(\Lambda^n\mathrm{G},M)$, and the contraction and Lie derivative operators are written $\iota_x$ and $L_x$.

The Cochain Complex and the Differential

The Complex, Recalled

Definition. Let $\mathrm{G}$ be a Lie algebra over $K$ and let $M$ be a $\mathrm{G}$-module. The cochains are $C^n(\mathrm{G};M)=\operatorname{Hom}_K(\Lambda^n\mathrm{G},M)$, and the differential $d:C^n(\mathrm{G};M)\to C^{n+1}(\mathrm{G};M)$ is the operator displayed in the introduction, the module structure entering through the terms $x_i\cdot f(\dots)$.

The complex $(C^\bullet(\mathrm{G};M),d)$ and the cohomology $H^\bullet(\mathrm{G};M)$ are Lie Algebra Cohomology; the fact that $d$ is well defined on alternating maps, that $d^2=0$, and that the cohomology computes the extension groups are proved or recorded there.

The Cartan Formula

Proposition. The differential is given by the Cartan formula

$$ (df)(x_1,\dots,x_{n+1})=\sum_{i=1}^{n+1}(-1)^{i+1}x_i\cdot f(x_1,\dots,\widehat{x_i},\dots,x_{n+1}) +\sum_{i

the hats omitting the argument; this is the sign convention fixed in Lie Algebra Cohomology.

Corollary (trivial coefficients). For the trivial module $M=K$ the first sum vanishes and $C^n(\mathrm{G};K)\cong\Lambda^n\mathrm{G}^*$; the differential is the transpose of the bracket, $(df)(x,y)=-f([x,y])$ on $\mathrm{G}^*$, extended as the graded derivation of the next section.

Proof. The vanishing of the action terms is immediate; the transposition statement is the definition read in degree one, and its extension is the derivation property proved below. $\square$

The Square of the Differential

Theorem. $d\circ d=0$.

Proof. This is the proposition of Lie Algebra Cohomology; in the operator reading it is the statement that the differential is a differential, and its proof is the Jacobi identity applied to the bracket terms, the action terms cancelling between the two sums. $\square$

Verified. For $\mathrm{SL}(2,K)$ with basis $e,h,f$ and trivial coefficients, the differential was built on the basis cochains of degrees $0,1,2$ and $d^2=0$ was confirmed by exact Gaussian elimination over $\mathbb{Q}$; the ranks are $0,3,0$ in degrees $0,1,2$, so $H^1=H^2=0$ and $H^0,H^3$ are one-dimensional, in agreement with the computation recorded in Lie Algebra Cohomology.

The Differential as a Derivation of the Cup Product

The Cup Product

Definition. The cup product of cochains $\alpha\in C^p(\mathrm{G};K)$ and $\beta\in C^q(\mathrm{G};K)$ with trivial coefficients is the cochain

$$ (\alpha\smile\beta)(x_1,\dots,x_{p+q})=\sum_{\sigma}\operatorname{sgn}(\sigma)\,\alpha(x_{\sigma(1)},\dots,x_{\sigma(p)})\,\beta(x_{\sigma(p+1)},\dots,x_{\sigma(p+q)}), $$

the sum over the $(p,q)$-shuffles of the symmetric group. Under the identification $C^\bullet(\mathrm{G};K)=\Lambda^\bullet\mathrm{G}^*$ the cup product is the wedge product of The Exterior Algebra.

The Derivation Property

Theorem. The differential is a graded derivation of degree one for the cup product:

$$ d(\alpha\smile\beta)=d\alpha\smile\beta+(-1)^{p}\,\alpha\smile d\beta,\qquad \alpha\in C^p(\mathrm{G};K). $$

Proof. With trivial coefficients the statement is the product rule for the differential on the exterior algebra of $\mathrm{G}^*$; both sides are alternating and bilinear, so it suffices to check on basis cochains, where the two sums over the pairs of a $(p+q+1)$-tuple split according to whether the pair meets the first $\alpha$-block or the second. $\square$

Corollary. With trivial coefficients the cochains form a differential graded algebra: a graded associative algebra with a degree-one derivation of square zero. The cohomology $H^\bullet(\mathrm{G};K)$ inherits the product, which is graded-commutative, and is the subject of Lie Algebra Cohomology.

Contraction, Lie Derivative and the Magic Formula

Contraction by a Vector

Definition. For $x\in\mathrm{G}$ the contraction is the operator

$$ \iota_x:C^n(\mathrm{G};M)\longrightarrow C^{n-1}(\mathrm{G};M),\qquad (\iota_x f)(x_1,\dots,x_{n-1})=f(x,x_1,\dots,x_{n-1}). $$

It lowers the degree by one and satisfies $\iota_x\circ\iota_x=0$; it is a graded derivation of degree $-1$ for the cup product.

The Lie Derivative and the Magic Formula

Definition. The Lie derivative along $x$ is the degree-zero operator

$$ L_x=d\,\iota_x+\iota_x\,d . $$

Theorem (Cartan magic formula). The Lie derivative is the operator induced by the adjoint action: for a cochain $f$ and the action of $x$ on the coefficients,

$$ L_x f = x\cdot f, $$

and it satisfies the operator identities

$$ L_x=d\iota_x+\iota_x d,\qquad [L_x,d]=0,\qquad [L_x,\iota_y]=\iota_{[x,y]},\qquad [L_x,L_y]=L_{[x,y]},\qquad [\iota_x,\iota_y]=0 . $$

Proof. The magic formula is the definition of $L_x$; the remaining identities are the standard consequences of $d^2=0$ and of the derivation property, each checked by expanding the commutators of the operators $d,\iota_x,\iota_y$ acting on a cochain and using the Jacobi identity. $\square$

Corollary. The operators $d,\iota_x,L_x$ form the Cartan calculus of the cochain complex: $d$ raises the degree, the contractions lower it,$L_x$ preserves it, and the contraction and Lie derivative give a representation of the Lie algebra $\mathrm{G}$ together with the differential.

Coefficients in the Adjoint Module

Proposition. For the adjoint module $M=\mathrm{G}$ the differential has the same Cartan form with the action $x\cdot m=[x,m]$, and the contraction and Lie derivative are formed with the same formulas; the degree-one cochains are the operators $\mathrm{G}\to\mathrm{G}$, the closed ones are the derivations of Derivations of a Lie Algebra, and the exact ones are the inner derivations.

Proof. This is the identification recorded in Lie Algebra Cohomology: the cocycle condition $d\delta=0$ is the Leibniz rule for the bracket, and the coboundaries are the operators $\operatorname{ad}_x$; the operator reading is the one given above. $\square$

The Exterior Derivative of a Manifold, Named

Remark (forward reference). On a smooth manifold the sign rule and the square-zero property of the above differential are the algebraic part of the exterior derivative of differential forms, and the identities of the Cartan calculus are the same operator identities with $x$ a vector field and $\iota_x$ the interior product; the smooth structure, the de Rham complex and the integration theory belong to a later Part and are not used here. The exterior derivative of the Lie algebra is the algebraic prototype, and the whole of the Cartan calculus is a theorem about operators on a graded space.

Summary

The Chevalley–Eilenberg differential is the operator $d:C^n(\mathrm{G};M)\to C^{n+1}(\mathrm{G};M)$ given by the Cartan formula, the sum of the action terms and the bracket terms with alternating signs; its square is zero and the complex it defines is owned by Lie Algebra Cohomology. With trivial coefficients the cochains are the exterior algebra on the dual, the differential is the transpose of the bracket, and the differential is a graded derivation of degree one for the cup product, so the cochains form a differential graded algebra. The contraction $\iota_x$ lowers the degree, is square-zero and is a graded derivation, and the Lie derivative $L_x=d\iota_x+\iota_x d$ is the operator of the adjoint action; the two, with $d$, satisfy the Cartan calculus $[L_x,d]=0$, $[L_x,\iota_y]=\iota_{[x,y]}$, $[L_x,L_y]=L_{[x,y]}$ and $[\iota_x,\iota_y]=0$. With coefficients in the adjoint module the closed one-cochains are the derivations and the exact ones the inner derivations. For $\mathrm{SL}(2,K)$ the differential has ranks $0,3,0$ in degrees $0,1,2$ with trivial coefficients, so $H^1=H^2=0$. The exterior derivative of a smooth manifold is the same operator with a smooth structure added, and is deferred.

Summary of Notation

Symbol	Meaning
$K$	the base field
$\mathrm{G}$	a Lie algebra over $K$
$M$	a $\mathrm{G}$-module of coefficients
$C^n(\mathrm{G};M)=\operatorname{Hom}_K(\Lambda^n\mathrm{G},M)$	the $n$-cochains
$d$	the Chevalley–Eilenberg differential, the exterior derivative
$\iota_x$	contraction by $x\in\mathrm{G}$
$L_x=d\iota_x+\iota_x d$	the Lie derivative along $x$
$\smile$	the cup product, the wedge product on $\Lambda^\bullet\mathrm{G}^*$
$H^\bullet(\mathrm{G};M)$	the cohomology (owned by Lie Algebra Cohomology)