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.

A Lie-algebra-valued differential ${\displaystyle k}$-form on a manifold, ${\displaystyle M}$, is a smooth section of the bundle ${\displaystyle ({\mathfrak {g}}\times M)\otimes \wedge ^{k}T^{*}M}$, where ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra, ${\displaystyle T^{*}M}$ is the cotangent bundle of ${\displaystyle M}$ and ${\displaystyle \wedge ^{k}}$ denotes the ${\displaystyle k^{\text{th}}}$ exterior power.

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 ${\displaystyle {\mathfrak {g}}}$-valued ${\displaystyle p}$-form ${\displaystyle \omega }$ and a ${\displaystyle {\mathfrak {g}}}$-valued ${\displaystyle q}$-form ${\displaystyle \eta }$, their wedge product ${\displaystyle [\omega \wedge \eta ]}$ is given by

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

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

${\displaystyle [\omega \wedge \eta ](v_{1},v_{2})=[\omega (v_{1}),\eta (v_{2})]-[\omega (v_{2}),\eta (v_{1})].}$

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

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

for all ${\displaystyle g,h\in {\mathfrak {g}}}$ and ${\displaystyle \alpha ,\beta \in \Omega (M,\mathbb {R} )}$.

Some authors have used the notation ${\displaystyle [\omega ,\eta ]}$ instead of ${\displaystyle [\omega \wedge \eta ]}$. The notation ${\displaystyle [\omega ,\eta ]}$, which resembles a commutator, is justified by the fact that if the Lie algebra ${\displaystyle {\mathfrak {g}}}$ is a matrix algebra then ${\displaystyle [\omega \wedge \eta ]}$ is nothing but the graded commutator of ${\displaystyle \omega }$ and ${\displaystyle \eta }$, i. e. if ${\displaystyle \omega \in \Omega ^{p}(M,{\mathfrak {g}})}$ and ${\displaystyle \eta \in \Omega ^{q}(M,{\mathfrak {g}})}$ then

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

where ${\displaystyle \omega \wedge \eta ,\ \eta \wedge \omega \in \Omega ^{p+q}(M,{\mathfrak {g}})}$ are wedge products formed using the matrix multiplication on ${\displaystyle {\mathfrak {g}}}$.

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

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

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

where ${\displaystyle q=q_{1}+\ldots +q_{k}}$ and ${\displaystyle \varphi _{i}}$ are ${\displaystyle {\mathfrak {g}}}$-valued ${\displaystyle q_{i}}$-forms. Moreover, given a vector space ${\displaystyle V}$, the same formula can be used to define the ${\displaystyle V}$-valued form ${\displaystyle f(\varphi ,\eta )}$ when

${\displaystyle f:{\mathfrak {g}}\times V\to V}$

is a multilinear map, ${\displaystyle \varphi }$ is a ${\displaystyle {\mathfrak {g}}}$-valued form and ${\displaystyle \eta }$ is a ${\displaystyle V}$-valued form. Note that, when

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

giving ${\displaystyle f}$ amounts to giving an action of ${\displaystyle {\mathfrak {g}}}$ on ${\displaystyle V}$; i.e., ${\displaystyle f}$ determines the representation

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

and, conversely, any representation ${\displaystyle \rho }$ determines ${\displaystyle f}$ with the condition ${\displaystyle (*)}$. For example, if ${\displaystyle f(x,y)=[x,y]}$ (the bracket of ${\displaystyle {\mathfrak {g}}}$), then we recover the definition of ${\displaystyle [\cdot \wedge \cdot ]}$ given above, with ${\displaystyle \rho =\operatorname {ad} }$, the adjoint representation. (Note the relation between ${\displaystyle f}$ and ${\displaystyle \rho }$ above is thus like the relation between a bracket and ${\displaystyle \operatorname {ad} }$.)

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

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

With this notation, one has for example:

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

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

${\displaystyle \rho ([\omega \wedge \omega ])\cdot \varphi =2\rho (\omega )\cdot (\rho (\omega )\cdot \varphi ).}$^[2]

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

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

Any ${\displaystyle {\mathfrak {g}}_{P}}$-valued forms on the base space of ${\displaystyle P}$ are in a natural one-to-one correspondence with any tensorial forms on ${\displaystyle P}$ of adjoint type.

• Maurer–Cartan form
• Adjoint bundle

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

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