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 ( g × M ) ⊗ Λ k T ∗ M , where g is a Lie algebra, T ∗ M is the cotangent bundle of M and Λ^k denotes the k^th exterior power.

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 [ ω ∧ η ] , is given by: for g -valued p-form ω and g -valued q-form η

[ ω ∧ η ] ( v 1 , ⋯ , v p + q ) = 1 ( p + q ) ! ∑ σ sgn ⁡ ( σ ) [ ω ( v σ ( 1 ) , ⋯ , v σ ( p ) ) , η ( v σ ( p + 1 ) , ⋯ , v σ ( p + q ) ) ]

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

[ ω ∧ η ] ( v 1 , v 2 ) = 1 2 ( [ ω ( v 1 ) , η ( v 2 ) ] − [ ω ( v 2 ) , η ( v 1 ) ] ) .

The operation [ ω ∧ η ] can also be defined as the bilinear operation on Ω ( M , g ) satisfying

[ ( g ⊗ α ) ∧ ( h ⊗ β ) ] = [ g , h ] ⊗ ( α ∧ β )

for all g , h ∈ g and α , β ∈ Ω ( M , R ) .

Some authors have used the notation [ ω , η ] instead of [ ω ∧ η ] . The notation [ ω , η ] , which resembles a commutator, is justified by the fact that if the Lie algebra g is a matrix algebra then [ ω ∧ η ] is nothing but the graded commutator of ω and η , i. e. if ω ∈ Ω p ( M , g ) and η ∈ Ω q ( M , g ) then

[ ω ∧ η ] = ω ∧ η − ( − 1 ) p q η ∧ ω ,

where ω ∧ η ,   η ∧ ω ∈ Ω p + q ( M , g ) are wedge products formed using the matrix multiplication on g .

Operations

Let f : g → h be a Lie algebra homomorphism. If φ is a g -valued form on a manifold, then f(φ) is an h -valued form on the same manifold obtained by applying f to the values of φ: f ( φ ) ( v 1 , … , v k ) = f ( φ ( v 1 , … , v k ) ) .

Similarly, if f is a multilinear functional on ∏ 1 k g , then one puts

f ( φ 1 , … , φ k ) ( v 1 , … , v q ) = 1 q ! ∑ σ sgn ⁡ ( σ ) f ( φ 1 ( v σ ( 1 ) , … , v σ ( q 1 ) ) , … , φ k ( v σ ( q − q k + 1 ) , … , v σ ( q ) ) )

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

f : g × V → V

is a multilinear map, φ is a 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 g on V; i.e., f determines the representation

ρ : g → V , ρ ( 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 g ), 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 g l ( 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,

( α ⋅ ϕ ) ( v 1 , … , v p + q ) = 1 ( p + q ) ! ∑ σ sgn ⁡ ( σ ) α ( v σ ( 1 ) , … , v σ ( p ) ) ϕ ( v σ ( p + 1 ) , … , v σ ( p + q ) ) .

With this notation, one has for example:

ad ⁡ ( α ) ⋅ ϕ = [ α ∧ ϕ ] .

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

ρ ( [ ω ∧ ω ] ) ⋅ φ = 2 ρ ( ω ) ⋅ ( ρ ( ω ) ⋅ φ ) .

Forms with values in an adjoint bundle

Let P be a smooth principal bundle with structure group G and g = Lie ⁡ ( G ) . G acts on g via adjoint representation and so one can form the associated bundle:

g P = P × Ad g .

Any 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.