Lie algebra–valued differential form

Formal definition

A Lie-algebra-valued differential k-form on a manifold, M, is a smooth section of the bundle (\mathfrak{g} \times M) \otimes \wedge^k T^*M, where \mathfrak{g} is a Lie algebra, T^*M is the cotangent bundle of M and \wedge^k denotes the k^{\text{th}} exterior power.

Wedge product

The wedge product of ordinary, real-valued differential forms is defined using multiplication of real numbers. For a pair of Lie algebra–valued differential forms, the wedge product can be defined similarly, but substituting the bilinear Lie bracket operation, to obtain another Lie algebra–valued form. For a \mathfrak{g}-valued p-form \omega and a \mathfrak{g}-valued q-form \eta, their wedge product [\omega\wedge\eta] is given by :[\omega\wedge\eta](v_1, \dotsc, v_{p+q}) = {1 \over p!q!}\sum_{\sigma} \operatorname{sgn}(\sigma) [\omega(v_{\sigma(1)}, \dotsc, v_{\sigma(p)}), \eta(v_{\sigma(p+1)}, \dotsc, v_{\sigma(p+q)})], where the 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 = [\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 \varphi is a \mathfrak{g}-valued form on a manifold, then f(\varphi) is an \mathfrak{h}-valued form on the same manifold obtained by applying f to the values of \varphi: f(\varphi)(v_1, \dotsc, v_k) = f(\varphi(v_1, \dotsc, v_k)).

Similarly, if f is a multilinear functional on \textstyle \prod_1^k \mathfrak{g}, then one puts :f(\varphi_1, \dotsc, \varphi_k)(v_1, \dotsc, v_q) = {1 \over q!} \sum_{\sigma} \operatorname{sgn}(\sigma) f(\varphi_1(v_{\sigma(1)}, \dotsc, v_{\sigma(q_1)}), \dotsc, \varphi_k(v_{\sigma(q - q_k + 1)}, \dotsc, v_{\sigma(q)})) where q = q_1 + \ldots + q_k and \varphi_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, \varphi is a \mathfrak{g}-valued form and \eta is a V-valued form. Note that, when :f([x, y], z) = f(x, f(y, z)) - f(y, f(x, z)) {,} \qquad () 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 \rho 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 \rho = \operatorname{ad}, the adjoint representation. (Note the relation between f and \rho above is thus like the relation between a bracket and \operatorname{ad}.)

In general, if \alpha is a \mathfrak{gl}(V)-valued p-form and \varphi is a V-valued q-form, then one more commonly writes \alpha \cdot \varphi = f(\alpha, \varphi) when f(T, x) = T x. Explicitly, :(\alpha \cdot \phi)(v_1, \dotsc, v_{p+q}) = {1 \over (p+q)!} \sum_{\sigma} \operatorname{sgn}(\sigma) \alpha(v_{\sigma(1)}, \dotsc, v_{\sigma(p)}) \phi(v_{\sigma(p+1)}, \dotsc, v_{\sigma(p+q)}). With this notation, one has for example: :\operatorname{ad}(\alpha) \cdot \phi = [\alpha \wedge \phi].

Example: If \omega is a \mathfrak{g}-valued one-form (for example, a connection form), \rho a representation of \mathfrak{g} on a vector space V and \varphi a V-valued zero-form, then :\rho([\omega \wedge \omega]) \cdot \varphi = 2 \rho(\omega) \cdot (\rho(\omega) \cdot \varphi).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 \varphi)(v, w) = {1 \over 2} (\rho([\omega \wedge \omega])(v, w) \varphi - \rho([\omega \wedge \omega])(w, v) \phi) is \rho(\omega(v))\rho(\omega(w))\varphi - \rho(\omega(w))\rho(\omega(v))\phi = 2(\rho(\omega) \cdot (\rho(\omega) \cdot \phi))(v, w).

Forms with values in an adjoint bundle

Let P be a smooth principal bundle with structure group G and \mathfrak{g} = \operatorname{Lie}(G). G acts on \mathfrak{g} via adjoint representation and so one can form the associated bundle: :\mathfrak{g}P = P \times{\operatorname{Ad}} \mathfrak{g}. Any \mathfrak{g}_P-valued forms on the base space of P are in a natural one-to-one correspondence with any tensorial forms on P of adjoint type.

Notes

References

References

1. {{Kobayashi-Nomizu Chapter XII, § 1.