Lie algebra-valued differential form

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 $({\mathfrak {g}}\times M)\otimes \Lambda ^{k}T^{*}M$ , where ${\mathfrak {g}}$ 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 $[\omega\wedge\eta]$ , is given by: for ${\mathfrak {g}}$ -valued p-form $\omega$ and ${\mathfrak {g}}$ -valued q-form $\eta$

[\omega \wedge \eta ](v_{1},\cdots ,v_{{p+q}})={1 \over (p+q)!}\sum _{{\sigma }}\operatorname {sgn}(\sigma )[\omega (v_{{\sigma (1)}},\cdots ,v_{{\sigma (p)}}),\eta (v_{{\sigma (p+1)}},\cdots ,v_{{\sigma (p+q)}})]

where 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 ](v_{1},v_{2})={1 \over 2}([\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 φ is a ${\mathfrak {g}}$ -valued form on a manifold, then f(φ) is an ${\mathfrak {h}}$ -valued form on the same manifold obtained by applying f to the values of φ: $f(\varphi )(v_{1},\dots ,v_{k})=f(\varphi (v_{1},\dots ,v_{k}))$ .

Similarly, if f is a multilinear functional on $\textstyle \prod _{1}^{k}{\mathfrak {g}}$ , then one puts^[1]

f(\varphi _{1},\dots ,\varphi _{k})(v_{1},\dots ,v_{q})={1 \over q!}\sum _{{\sigma }}\operatorname {sgn}(\sigma )f(\varphi _{1}(v_{{\sigma (1)}},\dots ,v_{{\sigma (q_{1})}}),\dots ,\varphi _{k}(v_{{\sigma (q-q_{k}+1)}},\dots ,v_{{\sigma (q)}}))

where q = q₁ + … + q_k and φ_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, φ is a ${\mathfrak {g}}$ -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 ${\mathfrak {g}}$ on V; i.e., f determines the representation

\rho :{\mathfrak {g}}\to V,\rho (x)y=f(x,y)

and, conversely, any representation ρ 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 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 ${\mathfrak {gl}}(V)$ -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,

(\alpha \cdot \phi )(v_{1},\dots ,v_{p+q})={1 \over (p+q)!}\sum _{\sigma }\operatorname {sgn} (\sigma )\alpha (v_{\sigma (1)},\dots ,v_{\sigma (p)})\phi (v_{\sigma (p+1)},\dots ,v_{\sigma (p+q)}).

With this notation, one has for example:

\operatorname {ad} (\alpha )\cdot \phi =[\alpha \wedge \phi ]

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

\rho ([\omega \wedge \omega ])\cdot \varphi =2\rho (\omega )\cdot (\rho (\omega )\cdot \varphi ).

^[2]

Forms with values in an adjoint bundle

Notes

↑ Kobayashi–Nomizu, Ch. XII, § 1.
↑ 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
$(\rho ([\omega \wedge \omega ])\cdot \phi )(v,w)={1 \over 2}(\rho ([\omega \wedge \omega ])(v,w)\phi -\rho ([\omega \wedge \omega ])(w,v)\phi )$
is $\rho (\omega (v))\rho (\omega (w))\phi -\rho (\omega (w))\rho (\omega (v))\phi =2(\rho (\omega )\cdot (\rho (\omega )\cdot \phi ))(v,w).$

References

S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2.

External links

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