Lie algebra-valued differential form

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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, ${\displaystyle M}$, is a smooth section of the bundle ${\displaystyle ({\mathfrak {g}}\times M)\otimes \Lambda ^{k}T^{*}M}$, where ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra, ${\displaystyle T^{*}M}$ is the cotangent bundle of ${\displaystyle M}$ and Λk denotes the kth 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 ${\displaystyle [\omega \wedge \eta ]}$, is given by: for ${\displaystyle {\mathfrak {g}}}$-valued p-form ${\displaystyle \omega }$ and ${\displaystyle {\mathfrak {g}}}$-valued q-form ${\displaystyle \eta }$

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

where vi'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

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

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 ${\displaystyle {\mathfrak {g}}}$.

Operations

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

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

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

where q = q1 + … + qk and φi are ${\displaystyle {\mathfrak {g}}}$-valued qi-forms. Moreover, given a vector space V, the same formula can be used to define the V-valued form ${\displaystyle f(\varphi ,\eta )}$ when

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

is a multilinear map, φ is a ${\displaystyle {\mathfrak {g}}}$-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 ${\displaystyle {\mathfrak {g}}}$ on V; i.e., f determines the representation

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

and, conversely, any representation ρ determines f with the condition (*). 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 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 ${\displaystyle {\mathfrak {gl}}(V)}$-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,

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

With this notation, one has for example:

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

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

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

Forms with values in an adjoint bundle

Let P be a smooth principal bundle with structure group G and ${\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)}$. 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 P are in a natural one-to-one correspondence with any tensorial forms on P of adjoint type.

2. ^ Since ${\displaystyle \rho ([\omega \wedge \omega ])(v,w)=\rho ([\omega \wedge \omega ](v,w))=\rho ([\omega (v),\omega (w)])=\rho (\omega (v))\rho (\omega (w))-\rho (\omega (w))\rho (\omega (v))}$, we have that
${\displaystyle (\rho ([\omega \wedge \omega ])\cdot \phi )(v,w)={1 \over 2}(\rho ([\omega \wedge \omega ])(v,w)\phi -\rho ([\omega \wedge \omega ])(w,v)\phi )}$
is ${\displaystyle \rho (\omega (v))\rho (\omega (w))\phi -\rho (\omega (w))\rho (\omega (v))\phi =2(\rho (\omega )\cdot (\rho (\omega )\cdot \phi ))(v,w).}$