The exterior algebra ΛV\Lambda V of a vector space is the free graded-commutative algebra over VV, where the elements of VV are taken to be of degree 11. (That is, the forgetful functor takes a graded-commutative algebra to its vector space of degree-11 elements.)

This construction generalizes to group representations, chain complexes, vector bundles, coherent sheaves, and indeed objects in any symmetric monoidal linear categories with enough colimits, where the tensor product distributes over those colimits (as in a 2-rig).

Explicit definition

We begin with the construction for vector spaces and then sketch how to generalize it.

For vector spaces

Suppose VV is a vector space over a field KK. Then the exterior algebra ΛV\Lambda V is generated by the elements of VV using these operations:

addition and scalar multiplication
an associative binary operation ∧\wedge called the exterior product or wedge product,

subject to these identities:

the identities necessary for ΛV\Lambda V to be an associative algebra
the identity v∧v=0v \wedge v = 0 for all v∈Vv \in V.

It then follows that ΛV\Lambda V is a graded algebra where Λ pV\Lambda^p V is spanned by pp-fold wedge products, that is, elements of the form

v 1∧⋯∧v pv_1 \wedge \cdots \wedge v_p

where v 1,…,v p∈Vv_1, \dots, v_p \in V. It also follows that ΛV\Lambda V is graded commutative: that is, if ω∈Λ pV\omega \in \Lambda^p V and ν∈Λ qV\nu \in \Lambda^q V, then

ω∧ν=(−1) pqν∧ω\omega \wedge \nu = (-1)^{p q}\, \nu \wedge \omega.

If KK is a field not of characteristic 22, we may replace the relations

(1)v∧v=0 v \wedge v = 0

by the relations

(2)v∧w=−w∧v v \wedge w = - w \wedge v

for all v,w∈Vv, w \in V (Grassmann 1844, §37, §55). If we can divide by 22, then the relations (2) imply (1), while the converse holds in any characteristic.

The exterior algebra of a vector space is also called the Grassmann algebra or alternating algebra. It is also denoted ⋀V\bigwedge V, ⋀ •V\bigwedge^\bullet V, or AltVAlt V.

In general

More generally, suppose CC is any symmetric monoidal category and V∈CV \in C is any object. Then we can form the tensor powers V ⊗nV^{\otimes n}. If CC has countable coproducts we can form the coproduct

TV=⨁ n≥0V ⊗n T V = \bigoplus_{n \ge 0} V^{\otimes n}

(which we write here as a direct sum), and if the tensor product distributes over these coproducts, TVT V becomes a monoid object in CC, with multiplication given by the obvious maps

V ⊗p⊗V ⊗q→V ⊗(p+q) V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes (p+q)}

This monoid object is called the tensor algebra of VV.

The symmetric group S nS_n acts on V ⊗nV^{\otimes n}, and if CC is a symmetric monoidal 𝕂\mathbb{K}-linear category over a field 𝕂\mathbb{K} of characteristic zero, then we can form the antisymmetrization map

p A:V ⊗n→V ⊗n p_A : V^{\otimes n} \to V^{\otimes n}

given by

p A=1n!∑ σ∈S nsgn(σ)σ p_A = \frac{1}{n!} \sum_{\sigma \in S_n} sgn(\sigma) \, \sigma

The cokernel of 1−p A1 - p_A is called the nnth antisymmetric tensor power or alternating power Λ nV\Lambda^n V. The coproduct

ΛV=⨁ n≥0Λ nV \Lambda V = \bigoplus_{n \ge 0} \Lambda^n V

becomes a monoid object called the exterior algebra of VV.

If CC is a linear category over a field of positive characteristic (or more generally, over a commutative ring in which not every positive integer is invertible, that is which is not itself an algebra over the rational numbers), then we need a different construction of Λ nV\Lambda^n V; we define … (please complete this!).

Examples

Over a super vector space

For VV a super vector space, the exterior algebra ΛV\Lambda V is often called the Grassmann algebra over VV. This ΛV\Lambda V or ∧ •V\wedge^\bullet V is the free graded commutative superalgebra on VV.

Explicitly, this is the quotient of the tensor algebra TVT V by the ideal generated by elements of the form

v⊗w+(−1) degv⋅degww⊗v.v \otimes w + (-1)^{deg v \cdot deg w } w \otimes v \, .

The product in this algebra is denoted with a wedge, and called the wedge product. It obeys the relation

v∧w=−(−1) degv⋅degww∧v. v \wedge w = - (-1)^{deg v \cdot deg w} w \wedge v \,.

With an inner product

If VV is equipped with a bilinear form then there is also the Clifford algebra on VV. This reduces to the Grassmann algebra for vanishing bilinear form.

But sometimes it is useful to consider the Grassmann algebra even in the presence of a non-degenerate bilinear form, in which case the inner product still serves to induce identifications between elements of the Grassmann algebra in different degree.

Let VV be R 3\mathbf{R}^3 equipped with its standard inner product. Then an element of Λ 0V\Lambda^0 V is a scalar (a real number), an element of Λ 1V\Lambda^1 V may be identified with a vector in the elementary sense, an element of Λ 2V\Lambda^2 V may be identified with a bivector or pseudovector, and an element of Λ 3V\Lambda^3 V may be identified a pseudoscalar.

More generally, let VV be R n\mathbf{R}^n, or indeed any real inner product space. Then an element of Λ pV\Lambda^p V is a pp-vector as studied in geometric algebra. Using the inner product, we can identify pp-vectors with (n−p)(n-p)-pseudovectors.

On a manifold (or generalized smooth space) XX, let T *XT^*X be the cotangent bundle of XX. Then we may define ΛT *X\Lambda T^*X using the abstract nonsense describe earlier, taking CC to be the category of vector bundles over XX. Then a differential form on XX is a section of the vector bundle ΛT *X\Lambda T^*X . If XX is an oriented (semi)-Riemannian manifold, then we can identify pp-forms with (n−p)(n-p)-forms using the Hodge star operator.

Semi-free dg-algebras

A semi-free dg-algebra is a dg-algebra whose underlying graded commutative algebra is free, i.e. is an exterior algebra. Examples include in particular Chevalley-Eilenberg algebras of Lie algebras, of L ∞L_\infty-algebras and Lie ∞-algebroids.

Differential forms / deRham complex

For XX a manifold consider the category of modules over its ring of smooth functions C ∞(X)C^\infty(X). One such module is Ω 1(X)=Γ(T *X)\Omega^1(X) = \Gamma(T^* X), the space of smooth sections of the cotangent bundle of XX.

The deRham complex of XX is the exterior algebra

Ω •(X)=⋀ C ∞(X)Γ(T *X). \Omega^\bullet(X) = \bigwedge_{C^\infty(X)} \Gamma(T^* X) \,.

This is really a special case of the previous class of examples, as Ω •(X)\Omega^\bullet(X) equipped with the deRham differential is the Chevalley-Eilenberg algebra of the tangent Lie algebroid.

References

The concept originates in

Hermann Grassmann, Ausdehnungslehre, 1844

where the graded-commutativity of the exterior product appears in §37, §55.

In monographs:

Felix A. Berezin (edited by Alexandre A. Kirillov): Grassmann algebra, Ch. 1 in: Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6_2]
Igor R. Shafarevich, Alexey O. Remizov: §10 in: Linear Algebra and Geometry (2012) [doi:10.1007/978-3-642-30994-6, MAA-review]

For the case of modules over a commutative ring, see

Stacks Project, Tag 00DM.
Bourbaki, Algèbre, chap. III, § 7.

Discussion of Grassmann algebras internal to any symmetric monoidal category is on p. 165 of

Pierre Deligne, Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.