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.
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
-valued
-form
and a
-valued
-form
, their wedge product
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)})],}](//wikimedia.org/api/rest_v1/media/math/render/svg/0675c1789f445cc4026642657538b3c008579a57)
where the
's are tangent vectors. The notation is meant to indicate both operations involved. For example, if
and
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})].}](//wikimedia.org/api/rest_v1/media/math/render/svg/2e3d3ab037d88cfdcb4c0e3374fcdf6631d5f253)
The operation
can also be defined as the bilinear operation on
satisfying
![{\displaystyle [(g\otimes \alpha )\wedge (h\otimes \beta )]=[g,h]\otimes (\alpha \wedge \beta )}](//wikimedia.org/api/rest_v1/media/math/render/svg/b75b2bb82a263398134c17b2756affc04d17e71e)
for all
and
.
Some authors have used the notation
instead of
. The notation
, which resembles a commutator, is justified by the fact that if the Lie algebra
is a matrix algebra then
is nothing but the graded commutator of
and
, i. e. if
and
then
![{\displaystyle [\omega \wedge \eta ]=\omega \wedge \eta -(-1)^{pq}\eta \wedge \omega ,}](//wikimedia.org/api/rest_v1/media/math/render/svg/127ab18af21042a199e74c283c64d6c5e6e5a5b1)
where
are wedge products formed using the matrix multiplication on
.
Let
be a Lie algebra homomorphism. If
is a
-valued form on a manifold, then
is an
-valued form on the same manifold obtained by applying
to the values of
:
.
Similarly, if
is a multilinear functional on
, 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)}))}](//wikimedia.org/api/rest_v1/media/math/render/svg/16d3490ed807d197ca087d9f80ee260e6db52686)
where
and
are
-valued
-forms. Moreover, given a vector space
, the same formula can be used to define the
-valued form
when
![{\displaystyle f:{\mathfrak {g}}\times V\to V}](//wikimedia.org/api/rest_v1/media/math/render/svg/34fa881f8f18fad6c5a1263c6d4f33bc9fcd2a0b)
is a multilinear map,
is a
-valued form and
is a
-valued form. Note that, when
![{\displaystyle f([x,y],z)=f(x,f(y,z))-f(y,f(x,z)){,}\qquad (*)}](//wikimedia.org/api/rest_v1/media/math/render/svg/b01025dc3b07a0ac16ed40740779ce057c8629a4)
giving
amounts to giving an action of
on
; i.e.,
determines the representation
- :{\mathfrak {g}}\to V,\rho (x)y=f(x,y)}
![{\displaystyle \rho :{\mathfrak {g}}\to V,\rho (x)y=f(x,y)}](//wikimedia.org/api/rest_v1/media/math/render/svg/31044e86b299f47d0671bc0300abb6f104a83794)
and, conversely, any representation
determines
with the condition
. For example, if
(the bracket of
), then we recover the definition of
given above, with
, the adjoint representation. (Note the relation between
and
above is thus like the relation between a bracket and
.)
In general, if
is a
-valued
-form and
is a
-valued
-form, then one more commonly writes
when
. 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)}).}](//wikimedia.org/api/rest_v1/media/math/render/svg/9ff88cb1ec2172b925752c7804c4841c6f577c12)
With this notation, one has for example:
.
Example: If
is a
-valued one-form (for example, a connection form),
a representation of
on a vector space
and
a
-valued zero-form, then
[2]
Since
, we have that
![{\displaystyle (\rho ([\omega \wedge \omega ])\cdot \varphi )(v,w)={1 \over 2}(\rho ([\omega \wedge \omega ])(v,w)\varphi -\rho ([\omega \wedge \omega ])(w,v)\phi )}](//wikimedia.org/api/rest_v1/media/math/render/svg/c64e603bb2fe3b2cab8a87ae9bb539456f053dd4)
is ![{\displaystyle \rho (\omega (v))\rho (\omega (w))\varphi -\rho (\omega (w))\rho (\omega (v))\phi =2(\rho (\omega )\cdot (\rho (\omega )\cdot \phi ))(v,w).}](//wikimedia.org/api/rest_v1/media/math/render/svg/871eb7848fd39eb9eadb0afae441317e67d25daa)