# Exterior algebra of a vector bundle

Associated to any vector space $V$ is its exterior algebra $\Lambda(V)$ which has the direct sum decomposition $\Lambda(V) = \bigoplus_{i=0}^n\Lambda^i(V)$ where $n = \dim V$.

My first interaction with the concept of an exterior algebra was in differential geometry where one defines a $k$-form on a smooth manifold $M$ to be an element of $\Gamma(M, \Lambda^k(T^*M))$. Here $\Lambda^k(T^*M)$ is a vector bundle; in particular, $\Lambda^k(T^*M) = \bigsqcup_{m\in M}\Lambda^k(T_m^*M)$. A little bit further into my differential geometry studies, I encountered the concept of an $E$-valued $k$-form, where $E$ is a vector bundle on $M$, which is defined to be an element of $\Gamma(M, \Lambda^k(M)\otimes E)$.

Before seeing the definition of an $E$-valued form (or truly understanding the concept), I was under the impression that the exterior algebra of $E$ would appear in the definition. I now know why it doesn't, but I have come to realise that I have never seen the exterior algebra associated to a vector bundle other than the cotangent bundle. Therefore, I ask the following question:

Are there any situations in which one wants/needs to consider the exterior algebra of a vector bundle other than the cotangent bundle?

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Are you distinguishing considering the entire exterior algebra from considering individual exterior powers? –  Qiaochu Yuan Jul 15 '13 at 20:21

If $E$ is a complex vector bundle of rank $r$, its first Chern class is equal to the first chern class of its top exterior product: $$c_1(E)=c_1(\wedge ^r E)$$

This is extremely useful since the first (and only!) chern class of a line bundle is generally easy to compute.
For example on a compact Riemann surface or on a smooth projective curve, the line bundle $L=\mathcal O(D)$ associated to a divisor $D$ has its first chern class equal to the degree of the divisor: $$c_1(L)=\text {deg} D$$

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Tractor bundles are certain vector bundles that are more suitable than tensor bundles for the construction of invariant differential operators on some (so called parabolic) geometries.

For instance, a conformal structure $c = [g]$ on a smooth manifold $M$ defines a parabolic geometry in this sense (conformal geometry), and there exist so called (standard conformal) tractor bundle which in any choice of a metric $g \in c$ from the conformal class is just the direct sum $$\Bbb T = \Omega^0 \oplus \Omega^1 \oplus \Omega^0$$ but when the other metric $\hat{g} \in c$ is chosen this direct sum decomposition transforms nicely so that the (standard conformal) tractor metric and the (standard conformal) tractor connection are well-defined (invariant) on the tractor bundle.

In "Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature" of T.Branson and A.R. Gover, see e.g. here, exterior powers $$\Bbb T^k = \underbrace{\Bbb T \wedge \dots \wedge \Bbb T}_{\text{k times}}$$ of the tractor bundles were introduced (under the name of $k$-form tractors, see p. 24 there). They have a number of applications in the theory of conformally invariant differential operators.

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In Poisson geometry and related fields (e.g., deformation quantisation à la Kontsevich), one does actually consider the bundle $\wedge TM$ of multivector fields together with a generalisation of the Lie bracket on $\Gamma(TM)$ to $\Gamma(\wedge TM)$ called the Schouten--Nijenhuis bracket. In particular, specifying a Poisson bracket $\{\cdot,\cdot\}_M$ on a manifold $M$ is equivalent to specifying a Poisson bivector, i.e., a section $\eta \in \Gamma(\wedge^2 TM)$ such that $[\eta,\eta]=0$, via $\{f,g\}_M = (df \otimes dg)(\eta)$.