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 $(\mathfrak \times M) \otimes \wedge^k T^*M$, where $\mathfrak$ is a Lie algebra, $T^*M$ is the cotangent bundle of $M$ and $\wedge^k$ denotes the $k^$ 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. For a $\mathfrak$-valued $p$-form $\omega$ and a $\mathfrak$-valued $q$-form $\eta$, their wedge product omega\wedge\eta/math> is given by
: omega\wedge\etav_1, \dotsc, v_) = \sum_ \operatorname(\sigma) omega(v_, \dotsc, v_), \eta(v_, \dotsc, v_)
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\etav_1,v_2) = ( omega(v_1), \eta(v_2)- omega(v_2),\eta(v_1).

The operation omega\wedge\eta/math> can also be defined as the bilinear operation on $\Omega(M, \mathfrak)$ satisfying
: g \otimes \alpha) \wedge (h \otimes \beta)= , h\otimes (\alpha \wedge \beta)
for all $g, h \in \mathfrak$ and $\alpha, \beta \in \Omega(M, \mathbb R)$.

Some authors have used the notation omega, \eta/math> instead of omega\wedge\eta/math>. The notation omega, \eta/math>, which resembles a commutator, is justified by the fact that if the Lie algebra $\mathfrak g$ is a matrix algebra then omega\wedge\eta/math> 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)^\eta\wedge\omega,
where $\omega \wedge \eta,\ \eta \wedge \omega \in \Omega^(M, \mathfrak g)$ are wedge products formed using the matrix multiplication on $\mathfrak g$.

Operations

Let $f : \mathfrak \to \mathfrak$ be a Lie algebra homomorphism. If $\varphi$ is a $\mathfrak$-valued form on a manifold, then $f(\varphi)$ is an $\mathfrak$-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 $\textstyle \prod_1^k \mathfrak$, then one puts
:$f(\varphi_1, \dotsc, \varphi_k)(v_1, \dotsc, v_q) = \sum_ \operatorname(\sigma) f(\varphi_1(v_, \dotsc, v_), \dotsc, \varphi_k(v_, \dotsc, v_))$
where $q = q_1 + \ldots + q_k$ and $\varphi_i$ are $\mathfrak$-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
:$f: \mathfrak \times V \to V$
is a multilinear map, $\varphi$ is a $\mathfrak$-valued form and $\eta$ is a $V$-valued form. Note that, when
:$f($, y z) = f(x, f(y, z)) - f(y, f(x, z)) \qquad (*)
giving $f$ amounts to giving an action of $\mathfrak$ on $V$; i.e., $f$ determines the representation
:$\rho: \mathfrak \to V, \rho(x)y = f(x, y)$
and, conversely, any representation $\rho$ determines $f$ with the condition $(*)$. For example, if $f(x, y) =$, y/math> (the bracket of $\mathfrak$), then we recover the definition of cdot \wedge \cdot/math> given above, with $\rho = \operatorname$, the adjoint representation. (Note the relation between $f$ and $\rho$ above is thus like the relation between a bracket and $\operatorname$.)

In general, if $\alpha$ is a $\mathfrak(V)$-valued $p$-form and $\varphi$ is a $V$-valued $q$-form, then one more commonly writes $\alpha \cdot \varphi = f(\alpha, \varphi)$ when $f(T, x) = T x$. Explicitly,
:$(\alpha \cdot \phi)(v_1, \dotsc, v_) = \sum_ \operatorname(\sigma) \alpha(v_, \dotsc, v_) \phi(v_, \dotsc, v_).$
With this notation, one has for example:
:\operatorname(\alpha) \cdot \phi = alpha \wedge \phi/math>.

Example: If $\omega$ is a $\mathfrak$-valued one-form (for example, a connection form), $\rho$ a representation of $\mathfrak$ 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).Since \rho( omega \wedge \omega(v, w) = \rho( omega \wedge \omegav, w)) = \rho( omega(v), \omega(w) = \rho(\omega(v))\rho(\omega(w)) - \rho(\omega(w))\rho(\omega(v)), we have that
:(\rho( omega \wedge \omega \cdot \varphi)(v, w) = (\rho( omega \wedge \omega(v, w) \varphi - \rho( omega \wedge \omega(w, v) \phi)
is $\rho(\omega(v))\rho(\omega(w))\varphi - \rho(\omega(w))\rho(\omega(v))\phi = 2(\rho(\omega) \cdot (\rho(\omega) \cdot \phi))(v, w).$

Forms with values in an adjoint bundle

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

Notes

References

*

External links

Wedge Product of Lie Algebra Valued One-Form
* {{nlab, id=groupoid+of+Lie-algebra+valued+forms, title=groupoid of Lie-algebra valued forms

Differential forms
Lie algebras