Lie algebra–valued differential form explained

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.

Formal definition

A Lie-algebra-valued differential

k -form on a manifold, M , is a smooth section of the bundle ( ak{g} x M) ⊗ \wedge ^{k} T ^{*M} , where ak{g} is a Lie algebra, T ^{*M} is the cotangent bundle of M and \wedge ^{k} denotes the k ^{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

ak{g} -valued p -form \omega and a ak{g} -valued q -form η , their wedge product [\omega\wedge η ] is given by [\omega\wedge η ](v _{1,} ... c, v _{p+q} ) = { 1 \over p!q!}\sum _{\sigma} \operatorname{sgn}(\sigma) [\omega(v _{\sigma(1)} , ... c, v _{\sigma(p)} ), η (v _{\sigma(p+1)} , ... c, 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 η are Lie-algebra-valued one forms, then one has[\omega\wedge η ](v _{1,v} _{2)} = [\omega(v _{1),} η (v _{2)]} - [\omega(v _{2),η(v} _{1)].}

The operation

[\omega\wedge η ] can also be defined as the bilinear operation on \Omega(M, ak{g}) satisfying[(g ⊗ \alpha) \wedge (h ⊗ \beta)] = [g, h] ⊗ (\alpha \wedge \beta) for all g, h \in ak{g} and \alpha, \beta \in \Omega(M, R) .

Some authors have used the notation

[\omega, η ] instead of [\omega\wedge η ] . The notation [\omega, η ] , which resembles a commutator, is justified by the fact that if the Lie algebra ak g is a matrix algebra then [\omega\wedge η ] is nothing but the graded commutator of \omega and η , i. e. if \omega \in \Omega ^{p(M,} ak g) and η \in \Omega ^{q(M,} ak g) then[\omega\wedge η ] = \omega\wedge η - ( -1 ) ^{pq} η \wedge\omega, where \omega \wedge η , η \wedge \omega \in \Omega ^{p+q} (M, ak g) are wedge products formed using the matrix multiplication on ak g .

Operations

Let

f : ak{g} \to ak{h} be a Lie algebra homomorphism. If \varphi is a ak{g} -valued form on a manifold, then f(\varphi) is an ak{h} -valued form on the same manifold obtained by applying f to the values of \varphi : f(\varphi)(v _{1,} ... c, v _{k)} = f(\varphi(v _{1,} ... c, v _{k))} .

Similarly, if

f is a multilinear functional on style \prod ^{k} _{1} ak{g} , then one puts^[1] f(\varphi _{1,} ... c, \varphi _{k)(v} _{1,} ... c, v _{q)} = { 1 \over q!} \sum _{\sigma} \operatorname{sgn}(\sigma) f(\varphi _{1(v} _{\sigma(1)} , ... c, v _{\sigma(q _{1)}} ), ... c, \varphi _{k(v} ^{\sigma(q - q _{k} +1 )} , ... c, v _{\sigma(q)} )) where q = q _{1} + \ldots + q _{k} and \varphi _{i} are ak{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, η ) whenf: ak{g} x V \to V is a multilinear map, \varphi is a ak{g} -valued form and η is a V -valued form. Note that, whenf([x, y], z) = f(x, f(y, z)) - f(y, f(x, z)) {,} (*) giving f amounts to giving an action of ak{g} on V ; i.e., f determines the representation\rho: ak{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 ak{g} ), then we recover the definition of [ ⋅ \wedge ⋅ ] 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 ak{gl}(V) -valued p -form and \varphi is a V -valued q -form, then one more commonly writes \alpha ⋅ \varphi = f(\alpha, \varphi) when f(T, x) = T x . Explicitly,(\alpha ⋅ \phi)(v _{1,} ... c, v _{p+q} ) = { 1 \over (p + q)!} \sum _{\sigma} \operatorname{sgn}(\sigma) \alpha(v _{\sigma(1)} , ... c, v _{\sigma(p)} ) \phi(v _{\sigma(p+1)} , ... c, v _{\sigma(p+q)} ). With this notation, one has for example:\operatorname{ad}(\alpha) ⋅ \phi = [\alpha \wedge \phi] .

Example: If

\omega is a ak{g} -valued one-form (for example, a connection form), \rho a representation of ak{g} on a vector space V and \varphi a V -valued zero-form, then\rho([\omega \wedge \omega]) ⋅ \varphi = 2 \rho(\omega) ⋅ (\rho(\omega) ⋅ \varphi). ^[2]

Forms with values in an adjoint bundle

See also: adjoint bundle. Let

P be a smooth principal bundle with structure group G and ak{g} = \operatorname{Lie}(G) . G acts on ak{g} via adjoint representation and so one can form the associated bundle:ak{g} _{P} = P x _{\operatorname{Ad} } \mathfrak.Any ak{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.

See also

• Maurer–Cartan form
• Adjoint bundle

External links

• Wedge Product of Lie Algebra Valued One-Form

Notes and References

1. Chapter XII, § 1.}}
2. 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]) ⋅ \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) ⋅ (\rho(\omega) ⋅ \phi))(v, w).