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.
A Lie algebra-valued differential k-form on a manifold, , is a smooth section of the bundle , where is a Lie algebra, is the cotangent bundle of and Λk denotes the kth exterior power.
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 , is given by: for -valued p-form and -valued q-form
where vi's are tangent vectors. The notation is meant to indicate both operations involved. For example, if and are Lie algebra-valued one forms, then one has
The operation can also be defined as the bilinear operation on satisfying
for all and .
Some authors have used the notation instead of . The notation , which resembles a commutator, is justified by the fact that if the Lie algebra is a matrix algebra then is nothing but the graded commutator of and , i. e. if and then
where are wedge products formed using the matrix multiplication on .
Let be a Lie algebra homomorphism. If φ is a -valued form on a manifold, then f(φ) is an -valued form on the same manifold obtained by applying f to the values of φ: .
Similarly, if f is a multilinear functional on , then one puts
where q = q1 + … + qk and φi are -valued qi-forms. Moreover, given a vector space V, the same formula can be used to define the V-valued form when
is a multilinear map, φ is a -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 on V; i.e., f determines the representation
and, conversely, any representation ρ determines f with the condition (*). For example, if (the bracket of ), then we recover the definition of 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 -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,
With this notation, one has for example:
Example: If ω is a -valued one-form (for example, a connection form), ρ a representation of on a vector space V and φ a V-valued zero-form, then
Forms with values in an adjoint bundle
Let P be a smooth principal bundle with structure group G and . G acts on via adjoint representation and so one can form the associated bundle:
Any -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.
- ^ Kobayashi–Nomizu, Ch. XII, § 1.
- ^ Since , we have that