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

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

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 $\Omega (M,{\mathfrak {g}})$ 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 $\varphi$ is a ${\mathfrak {g}}$-valued form on a manifold, then $f(\varphi )$ is an ${\mathfrak {h}}$-valued form on the same manifold obtained by applying $f$ to the values of $\varphi$: $f(\varphi )(v_{1},\dotsc ,v_{k})=f(\varphi (v_{1},\dotsc ,v_{k}))$.

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

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

where $q=q_{1}+\ldots +q_{k}$ and $\varphi _{i}$ 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, $\varphi$ is a ${\mathfrak {g}}$-valued form and $\eta$ is a $V$-valued form. Note that, when

$f([x,y],z)=f(x,f(y,z))-f(y,f(x,z)){,}\qquad (*)$

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 $\rho =\operatorname {ad}$, the adjoint representation. (Note the relation between $f$ and $\rho$ above is thus like the relation between a bracket and $\operatorname {ad}$.)

In general, if $\alpha$ is a ${\mathfrak {gl}}(V)$-valued $p$-form and $\varphi$ is a $V$-valued $q$-form, then one more commonly writes $\alpha \cdot \varphi =f(\alpha ,\varphi )$ when $f(T,x)=Tx$. Explicitly,

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

With this notation, one has for example:

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

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 $\varphi$ a $V$-valued zero-form, then

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

Forms with values in an adjoint bundle

See also: adjoint bundle

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

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

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