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

$[\omega\wedge\eta](v_1, \cdots, v_{p+q}) = {1 \over (p + q)!}\sum_{\sigma} \operatorname{sgn}(\sigma) [\omega(v_{\sigma(1)}, \cdots, v_{\sigma(p)}), \eta(v_{\sigma(p+1)}, \cdots, v_{\sigma(p+q)})]$

where 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

$[\omega\wedge\eta](v_1,v_2) = {1 \over 2} ([\omega(v_1),\eta(v_2)] - [\omega(v_2),\eta(v_1)]).$

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 φ is a $\mathfrak{g}$-valued form on a manifold, then f(φ) is an $\mathfrak{h}$-valued form on the same manifold obtained by applying f to the values of φ: $f(\varphi)(v_1, \dots, v_k) = f(\varphi(v_1, \dots, v_k))$.

Similarly, if f is a multilinear functional on $\textstyle \prod_1^k \mathfrak{g}$, then one puts^[1]

$f(\varphi_1, \dots, \varphi_k)(v_1, \dots, v_q) = {1 \over q!} \sum_{\sigma} \operatorname{sgn}(\sigma) f(\varphi_1(v_{\sigma(1)}, \dots, v_{\sigma(q_1)}), \dots, \varphi_k(v_{\sigma(q - q_k + 1)}, \dots, v_{\sigma(q)}))$

where q = q₁ + … + q_k and φ_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, φ is a $\mathfrak{g}$-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 $\mathfrak{g}$ on V; i.e., f determines the representation

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

and, conversely, any representation ρ 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 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 $\mathfrak{gl}(V)$-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,

$(\alpha \cdot \phi)(v_1, \dots, v_{p+q}) = {1 \over (p+q)!} \sum_{\sigma} \operatorname{sgn}(\sigma) \alpha(v_{\sigma(1)}, \dots, v_{\sigma(p)}) \phi(v_{\sigma(p+1)}, \dots, v_{\sigma(p+q)}).$

With this notation, one has for example:

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

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

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

Forms with values in an adjoint bundle

Let P be a smooth principal bundle with structure group G and $\mathfrak{g} = \operatorname{Lie}(G)$. 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

Notes

1. Kobayashi–Nomizu, Ch. XII, § 1.
2. Since $\rho([\omega \wedge \omega])(v, w) = \rho([\omega \wedge \omega](v, w)) = \rho([\omega(v), \omega(w)]) = \rho(\omega(v))\rho(\omega(w)) - \rho(\omega(w))\rho(\omega(v))$, we have that

   $(\rho([\omega \wedge \omega]) \cdot \phi)(v, w) = {1 \over 2} (\rho([\omega \wedge \omega])(v, w) \phi - \rho([\omega \wedge \omega])(w, v) \phi)$

   is $\rho(\omega(v))\rho(\omega(w))\phi - \rho(\omega(w))\rho(\omega(v))\phi = 2(\rho(\omega) \cdot (\rho(\omega) \cdot \phi))(v, w).$

References

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

External links

• http://mathoverflow.net/questions/123632/wedge-product-of-lie-algebra-valued-one-form
• http://ncatlab.org:8080/nlab/show/groupoid+of+Lie-algebra+valued+forms