nLab
exterior algebra

Idea

The exterior algebra $\Lambda V$ of a vector space is the free graded-commutative algebra? over $V$, where the elements of $V$ are taken to be of degree $1$. (That is, the forgetful functor takes a graded-commutative algebra to its vector space of degree-$1$ 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 $V$ is a vector space over a field $K$. Then the exterior algebra $\Lambda V$ is generated by the elements of $V$ 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 $\Lambda V$ to be an associative algebra
• the identity $v\wedge v=0$ for all $v\in V$.

It then follows that $\Lambda V$ is a graded algebra where ${\Lambda }^{p}V$ is spanned by $p$-fold wedge products, that is, elements of the form

$$v_1\wedge \cdots \wedge v_p$$v_1 \wedge \cdots \wedge v_p

where $v_1,\dots ,v_p\in V$. It also follows that $\Lambda V$ is graded commutative?: that is, if $\omega \in {\Lambda }^{p}V$ and $\nu \in {\Lambda }^{q}V$, then

• $\omega \wedge \nu =(-1{)}^{pq}\nu \wedge \omega$.

If $K$ is a field not of characteristic $2$, we may replace the relations

(1)$$v\wedge v=0$$v \wedge v = 0

by the relations

(2)$$v\wedge w=-w\wedge v$$v \wedge w = - w \wedge v

for all $v,w\in V$. If we can divide by $2$, 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 $\bigwedge V$, ${\bigwedge }^{•}V$, or $\mathrm{Alt}V$.

In general

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

$$TV=\underset{n\ge 0}{⨁}{V}^{\otimes n}$$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, $TV$ becomes a monoid object in $C$, with multiplication given by the obvious maps

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

This monoid object is called the tensor algebra of $V$.

The symmetric group $S_n$ acts on $V^{\otimes n}$, and if $C$ is a linear category over a field of characteristic zero, then we can form the antisymmetrization map

$$p_A:V^{\otimes n}\to V^{\otimes n}$$p_A : V^{\otimes n} \to V^{\otimes n}

given by

$$p_A=\frac{1}{n!}\sum _{\sigma \in S_n}\mathrm{sgn}(\sigma )\sigma$$p_A = \frac{1}{n!} \sum_{\sigma \in S_n} sgn(\sigma) \, \sigma

This is an idempotent, so if idempotents split in $C$ we can form its cokernel, called the $n$th antisymmetric tensor power or alternating power ${\Lambda }^{n}V$. The coproduct

$$\Lambda V=\underset{n\ge 0}{⨁}{\Lambda }^{n}V$$\Lambda V = \bigoplus_{n \ge 0} \Lambda^n V

becomes a monoid object called the exterior algebra of $V$.

If $C$ 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 ${\Lambda }^{n}V$.

Examples

Over a super vector space

For $V$ a super vector space, the exterior algebra $\Lambda V$ is often called the Grassmann algebra over $V$. This is the $\Lambda V$ or ${\wedge }^{•}V$ is the free graded commutative? superalgebra on $V$.

Explicitly, this is the quotient of the tensor algebra $TV$ by the ideal generated by elements of the form

$$v\otimes w+(-1{)}^{\mathrm{deg}v\cdot \mathrm{deg}w}w\otimes v.$$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

$$v\wedge w=-(-1{)}^{\mathrm{deg}v\cdot \mathrm{deg}w}w\wedge v.$$v \wedge w = - (-1)^{deg v \cdot deg w} w \wedge v \,.

With an inner product

If $V$ is equipped with a bilinear form then there is also the Clifford algebra on $V$. 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 $V$ be $R^3$ equipped with its standard inner product. Then an element of ${\Lambda }^{0}V$ is a scalar (a real number), an element of ${\Lambda }^{1}V$ may be identified with a vector in the elementary sense, an element of ${\Lambda }^{2}V$ may be identified with a bivector or pseudovector?, and an element of ${\Lambda }^{2}V$ may be identified a pseudoscalar?.

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

On a manifold (or generalized smooth space) $X$, let $T^{*}X$ be the cotangent bundle of $X$. Then we may define $\Lambda T^{*}X$ using the abstract nonsense describe earlier, taking $C$ to be the category of vector bundles over $X$. Then a differential form on $X$ is a section of the vector bundle $\Lambda T^{*}X$. If $X$ is an oriented (semi)-Riemannian manifold, then we can identify $p$-forms with $(n-p)$-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 $L_{\mathrm{\infty }}$-algebras and Lie ∞-algebroids.

Differential forms / deRham complex

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

The deRham complex of $X$ is the exterior algebra

$${\Omega }^{•}(X)=\underset{C^{\mathrm{\infty }}(X)}{\bigwedge }\Gamma (T^{*}X).$$\Omega^\bullet(X) = \bigwedge_{C^\infty(X)} \Gamma(T^* X) \,.

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