Lie algebra-valued differential form

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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[edit]

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.

Wedge product[edit]

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 .

Operations[edit]

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[1]

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

[2]

Forms with values in an adjoint bundle[edit]

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.

See also[edit]

Notes[edit]

  1. ^ Kobayashi–Nomizu, Ch. XII, § 1.
  2. ^ Since , we have that
    is

References[edit]

External links[edit]