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Graded bracket of Lie-algebra-valued differential forms

Definition

Let $M$ be a smooth manifold and $g$ a finite-dimensional real Lie algebra. For homogeneous $α \in Ω^{p} (M, g)$ and $β \in Ω^{q} (M, g)$ , define their graded bracket by

$[α \land β] (X_{1}, \dots, X_{p + q}) : = \frac{1}{p! q!} \sum_{σ \in S_{p + q}} sgn (σ) [α (X_{σ (1)}, \dots, X_{σ (p)}), β (X_{σ (p + 1)}, \dots, X_{σ (p + q)})] .$

Here $Ω^{p} (M, g)$ is the space of smooth alternating $p$ -linear forms with values in $g$ , the inner bracket is the Lie-algebra bracket, and the sum is over all permutations. This is a smooth $g$ -valued $(p + q)$ -form. It obeys graded skew-symmetry $[α \land β] = - (- 1)^{p q} [β \land α]$ , graded Jacobi

$[α \land [β \land γ]] = [[α \land β] \land γ] + (- 1)^{p q} [β \land [α \land γ]],$

and the exterior-derivative rule

$d [α \land β] = [d α \land β] + (- 1)^{p} [α \land d β] .$

In particular, for $g$ -valued 1-forms $A, B$ , $[A \land B] (X, Y) = [A (X), B (Y)] - [A (Y), B (X)]$ , so $\frac{1}{2} [A \land A] (X, Y) = [A (X), A (Y)]$ . This fixes the normalization used by the Cartan curvature formula.

Definition references

Vákár, §1.3.1, graded bracket on Lie-algebra-valued forms

Neeb, Chapter 6 §6.1, curvature and bracket conventions

Finite-dimensional real Lie algebra

Finite-dimensional vector-valued forms

Wedge product of differential forms

Graded-commutative algebra of forms

Graded derivation rule for the exterior derivative

Facts & Assumptions

Given: A smooth manifold $M$ , a finite-dimensional real Lie algebra $g$ , and homogeneous forms $α, β, γ$ of degrees $p, q, r$ .

[F1]

A finite-dimensional real Lie algebra has a bilinear alternating bracket satisfying Jacobi. Finite-dimensional Lie algebra

[F2]

A smooth $g$ -valued form has smooth scalar components in any finite basis, and its exterior derivative is taken componentwise. Finite-dimensional vector-valued forms and their exterior derivative

[F3]

The scalar exterior derivative is a graded derivation of the wedge product. The exterior derivative is a graded derivation

[F4]

Differential forms form an associative graded-commutative algebra. The wedge product of differential forms, Differential forms form a graded commutative algebra

Proof

1.1F1F2F4construct

Choose a finite basis $(e_{a})_{a = 1}^{n}$ of $g$ with dual basis $(e^{a})_{a = 1}^{n}$ , and write $α = \sum_{a} e_{a} α^{a}$ , $β = \sum_{b} e_{b} β^{b}$ , where $α^{a} = e^{a} \circ α$ and $β^{b} = e^{b} \circ β$ are smooth scalar forms. Expanding the alternating permutation sum gives $[α \land β] = \sum_{a, b} [e_{a}, e_{b}] α^{a} \land β^{b}$ ; this finite sum is smooth and alternating, hence is in $Ω^{p + q} (M, g)$ .

2.1F1F4step 1.1algebra

Swapping the two blocks of $p$ and $q$ arguments in the defining permutation sum contributes the sign $(- 1)^{p q}$ ; antisymmetry of the inner Lie bracket contributes the minus sign. Therefore $[α \land β] = - (- 1)^{p q} [β \land α]$ .

2.2F2F3step 1.1

In the finite basis expansion of step 1.1, apply the scalar graded derivation rule [F3] to each $α^{a} \land β^{b}$ and retain the fixed coefficient $[e_{a}, e_{b}]$ . Summing the resulting finite terms gives $d [α \land β] = [d α \land β] + (- 1)^{p} [α \land d β]$ .

2.3F1F4step 1.1

For homogeneous $α, β, γ$ , the same basis expansion writes $[α \land [β \land γ]]$ with coefficient $[e_{a}, [e_{b}, e_{c}]]$ and form factor $α^{a} \land β^{b} \land γ^{c}$ . The right-hand terms have coefficients $[[e_{a}, e_{b}], e_{c}]$ and, after commuting $β^{b}$ past $α^{a}$ using [F4], $[e_{b}, [e_{a}, e_{c}]]$ , with the same form factor. The equality of coefficients is exactly the Lie-algebra Jacobi identity in [F1], so the graded Jacobi formula follows.

3.1F1F2F3F4step 1.1step 2.1step 2.2step 2.3∎

If $g = 0$ , all forms and brackets are zero; if $p = 0$ or $q = 0$ , the same formulas use the empty wedge with sign $+ 1$ . The finite basis in step 1.1 is one finite-dimensional coordinate calculation, not a choice from a family. The definition and identities hold on the empty manifold as well.

Graded bracket of Lie-algebra-valued differential forms

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources