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 $(\mathfrak{𝔤}\times M)\otimes {\wedge }^k{T}^{*}M$, where $\mathfrak{𝔤}$ is a Lie algebra, $T^{*}M$ is the cotangent bundle of $M$ and ${\wedge }^k$ denotes the $k^{\text{th}}$ 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 $\mathfrak{𝔤}$-valued $p$-form $\omega$ and a $\mathfrak{𝔤}$-valued $q$-form $\eta$, their wedge product $[\omega \wedge \eta ]$ is given by

$[\omega \wedge \eta ](v_1,\dots ,v_{p+q})=\frac{1}{p!q!}\sum _{\sigma }\mathrm{sgn}(\sigma )[\omega (v_{\sigma (1)},\dots ,v_{\sigma (p)}),\eta (v_{\sigma (p+1)},\dots ,v_{\sigma (p+q)})],$

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

$[\omega \wedge \eta ](v_1,v_2)=[\omega (v_1),\eta (v_2)]-[\omega (v_2),\eta (v_1)].$

The operation $[\omega \wedge \eta ]$ can also be defined as the bilinear operation on $\mathrm{\Omega }(M,\mathfrak{𝔤})$ satisfying

$[(g\otimes \alpha )\wedge (h\otimes \beta )]=[g,h]\otimes (\alpha \wedge \beta )$

for all $g,h\in \mathfrak{𝔤}$ and $\alpha ,\beta \in \mathrm{\Omega }(M,\mathbb{ℝ})$. 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 $\omega \in {\mathrm{\Omega }}^p(M,\mathfrak{𝔤})$ and $\eta \in {\mathrm{\Omega }}^q(M,\mathfrak{𝔤})$ then

$[\omega \wedge \eta ]=\omega \wedge \eta -(-1{)}^{pq}\eta \wedge \omega ,$

where $$\begin{gathered} \omega \wedge \eta , \\ \eta \wedge \omega \in {\mathrm{\Omega }}^{p+q}(M,\mathfrak{𝔤}) \end{gathered}$$ 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 $\phi$ is a $\mathfrak{𝔤}$-valued form on a manifold, then $f(\phi )$ is an $\mathfrak{𝔥}$-valued form on the same manifold obtained by applying $f$ to the values of $\phi$: $f(\phi )(v_1,\dots ,v_k)=f(\phi (v_1,\dots ,v_k))$. Similarly, if $f$ is a multilinear functional on ${\prod }_1^k\mathfrak{𝔤}$, then one puts

$f({\phi }_1,\dots ,{\phi }_k)(v_1,\dots ,v_q)=\frac{1}{q!}\sum _{\sigma }\mathrm{sgn}(\sigma )f({\phi }_1(v_{\sigma (1)},\dots ,v_{\sigma (q_1)}),\dots ,{\phi }_k(v_{\sigma (q-q_k+1)},\dots ,v_{\sigma (q)}))$

where $q=q_1+\dots +q_k$ and ${\phi }_i$ 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 $f(\phi ,\eta )$ when

$f:\mathfrak{𝔤}\times V\to V$

is a multilinear map, $\phi$ is a $\mathfrak{𝔤}$-valued form and $\eta$ 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 $\rho$ 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 $\rho =\mathrm{ad}$, the adjoint representation. (Note the relation between $f$ and $\rho$ above is thus like the relation between a bracket and $\mathrm{ad}$.) In general, if $\alpha$ is a $\mathfrak{𝔤}\mathfrak{𝔩}(V)$-valued $p$-form and $\phi$ is a $V$-valued $q$-form, then one more commonly writes $\alpha \cdot \phi =f(\alpha ,\phi )$ when $f(T,x)=Tx$. Explicitly,

$(\alpha \cdot \varphi )(v_1,\dots ,v_{p+q})=\frac{1}{(p+q)!}\sum _{\sigma }\mathrm{sgn}(\sigma )\alpha (v_{\sigma (1)},\dots ,v_{\sigma (p)})\varphi (v_{\sigma (p+1)},\dots ,v_{\sigma (p+q)}).$

With this notation, one has for example:

$\mathrm{ad}(\alpha )\cdot \varphi =[\alpha \wedge \varphi ]$.

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

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

Forms with values in an adjoint bundle

See also: adjoint bundle

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

${\mathfrak{𝔤}}_P=P{\times }_{\mathrm{Ad}}\mathfrak{𝔤}.$

Any ${\mathfrak{𝔤}}_P$-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

References

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

External links

• Wedge Product of Lie Algebra Valued One-Form
• groupoid of Lie-algebra valued forms at the nLab