Lie algebra-valued differential form

In differential geometry, a Lie algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

Wedge product

Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation, denoted by $[\omega\wedge\eta]$ , is given by: for $\mathfrak{g}$ -valued p-form $\omega$ and $\mathfrak{g}$ -valued q-form $\eta$

[\omega\wedge\eta](v_1, \cdots, v_{p+q}) = {1 \over (p + q)!}\sum_{\sigma} \operatorname{sgn}(\sigma) [\omega(v_{\sigma(1)}, \cdots, v_{\sigma(p)}), \eta(v_{\sigma(p+1)}, \cdots, v_{\sigma(p+q)})]

where v_i's are tangent vectors. The notation is meant to indicate both operations involved. For example, if $\omega$ and $\eta$ are Lie algebra-valued one forms, then one has

[\omega\wedge\eta](v_1,v_2) = {1 \over 2} ([\omega(v_1),\eta(v_2)] - [\omega(v_2),\eta(v_1)]).

The operation $[\omega\wedge\eta]$ can also be defined as the bilinear operation on $\Omega(M, \mathfrak g)$ satisfying

[(g \otimes \alpha) \wedge (h \otimes \beta)] = [g, h] \otimes (\alpha \wedge \beta)

for all $g, h \in \mathfrak g$ and $\alpha, \beta \in \Omega(M, \mathbb R)$ .

Some authors have used the notation $[\omega, \eta]$ instead of $[\omega\wedge\eta]$ . The notation $[\omega, \eta]$ , which resembles a commutator, is justified by the fact that if the Lie algebra $\mathfrak g$ is a matrix algebra then $[\omega\wedge\eta]$ is nothing but the graded commutator of $\omega$ and $\eta$ , i. e. if $\omega \in \Omega^p(M, \mathfrak g)$ and $\eta \in \Omega^q(M, \mathfrak g)$ then

[\omega\wedge\eta] = \omega\wedge\eta - (-1)^{pq}\eta\wedge\omega,

where $\omega \wedge \eta,\ \eta \wedge \omega \in \Omega^{p+q}(M, \mathfrak g)$ are wedge products formed using the matrix multiplication on $\mathfrak g$ .

Operations

Let $f: \mathfrak{g} \to \mathfrak{h}$ be a Lie algebra homomorphism. If φ is a $\mathfrak{g}$ -valued form on a manifold, then f(φ) is an $\mathfrak{h}$ -valued form on the same manifold obtained by applying f to the values of φ: $f(\varphi)(v_1, \dots, v_k) = f(\varphi(v_1, \dots, v_k))$ .

Similarly, if f is a multilinear functional on $\textstyle \prod_1^k \mathfrak{g}$ , then one puts^[1]

f(\varphi_1, \dots, \varphi_k)(v_1, \dots, v_q) = {1 \over q!} \sum_{\sigma} \operatorname{sgn}(\sigma) f(\varphi_1(v_{\sigma(1)}, \dots, v_{\sigma(q_1)}), \dots, \varphi_k(v_{\sigma(q - q_k + 1)}, \dots, v_{\sigma(q)}))

where q = q₁ + … + q_k and φ_i are $\mathfrak{g}$ -valued q_i-forms. Moreover, given a vector space V, the same formula can be used to define the V-valued form $f(\varphi, \eta)$ when

f: \mathfrak{g} \times V \to V

is a multilinear map, φ is a $\mathfrak{g}$ -valued form and η is a V-valued form. Note that, when

(*) f([x, y], z) = f(x, f(y, z)) - f(y, f(x, z)),

giving f amounts to giving an action of $\mathfrak{g}$ on V; i.e., f determines the representation

\rho: \mathfrak{g} \to V, \rho(x)y = f(x, y)

and, conversely, any representation ρ determines f with the condition (*). For example, if $f(x, y) = [x, y]$ (the bracket of $\mathfrak{g}$ ), then we recover the definition of $[\cdot \wedge \cdot]$ given above, with ρ = ad, the adjoint representation. (Note the relation between f and ρ above is thus like the relation between a bracket and ad.)

In general, if α is a $\mathfrak{gl}(V)$ -valued p-form and φ is a V-valued q-form, then one more commonly writes α⋅φ = f(α, φ) when f(T, x) = Tx. Explicitly,

(\alpha \cdot \phi)(v_1, \dots, v_{p+q}) = {1 \over (p+q)!} \sum_{\sigma} \operatorname{sgn}(\sigma) \alpha(v_{\sigma(1)}, \dots, v_{\sigma(p)}) \phi(v_{\sigma(p+1)}, \dots, v_{\sigma(p+q)}).

With this notation, one has for example:

\operatorname{ad}(\alpha) \cdot \phi = [\alpha \wedge \phi]

Example: If ω is a $\mathfrak{g}$ -valued one-form (for example, a connection form), ρ a representation of $\mathfrak{g}$ on a vector space V and φ a V-valued zero-form, then

\rho([\omega \wedge \omega]) \cdot \varphi = 2 \rho(\omega) \cdot (\rho(\omega) \cdot \varphi).

^[2]

Forms with values in an adjoint bundle

Notes

↑ Kobayashi–Nomizu, Ch. XII, § 1.
↑ Since $\rho([\omega \wedge \omega])(v, w) = \rho([\omega \wedge \omega](v, w)) = \rho([\omega(v), \omega(w)]) = \rho(\omega(v))\rho(\omega(w)) - \rho(\omega(w))\rho(\omega(v))$ , we have that
$(\rho([\omega \wedge \omega]) \cdot \phi)(v, w) = {1 \over 2} (\rho([\omega \wedge \omega])(v, w) \phi - \rho([\omega \wedge \omega])(w, v) \phi)$
is $\rho(\omega(v))\rho(\omega(w))\phi - \rho(\omega(w))\rho(\omega(v))\phi = 2(\rho(\omega) \cdot (\rho(\omega) \cdot \phi))(v, w).$

References

S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2.

External links

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