nLab
exterior algebra

Idea

The exterior algebra of some object $V$ in a context with a tensor product is the free graded-commutative tensor algebra over $V$.

Definition

Let $A$ be an abelian group (or vector space, velc; any object of a symmetric monoidal abelian category $C$ should do fine, and probably something more general than that).

The exterior algebra (written $\bigwedge A$, $\Lambda A$, or $\mathrm{Alt}A$) on $A$ is the free skew-commutative algebra on $A$ (where ‘algebra’ should be interpreted as a monoid object in $C$). That is, the functor $\mathrm{Alt}:C\to \mathrm{SCom}\mathrm{Mon}(C)$ is the left adjoint of the forgetful functor $\mathrm{SCom}\mathrm{Mon}(C)\to C$. The elements of degree $p$ in this graded algebra comprise ${\mathrm{Alt}}_{p}A$, the $p$th exterior power (or $p$th alternating power) of $A$.

In more detail, $\mathrm{Alt}A$ is generated by the elements of $A$ (which comprise ${\mathrm{Alt}}_{1}A$) and these operations:

• the operations (addition, scalar multiplication, etc) of the objects of $C$, generalising the operations in $A$,
• an associative binary operation $\wedge$ of multiplication (the exterior product or wedge product),

subject to these identities:

• the identities necessary for $\mathrm{Alt}A$ to be an object of $C$,
• $\wedge$ distributes over these operations,
• $x\wedge x=0$ for $x\in A$.

A more general form of the last then follows; if $v\in {\mathrm{Alt}}_{p}A$ and $w\in {\mathrm{Alt}}_{q}A$ (that is if they are homogeneous of degree $p$ and degree $q$), then

• $v\wedge v=0$ if $p$ is odd,
• $v\wedge w=(-1{)}^{pq}w\wedge v$.

That is, $\mathrm{Alt}A$ is a skew-commutative algebra.

Examples

over a vector space: Grassmann algebra

For $A=V$ a vector space or super vector space, the exterior algebra $\bigwedge A$ is often called the Grassmann algebra over $V$

The Grassmann algebra $\Lambda V$ or ${\wedge }^{•}V$ is the free graded commutative? superalgebra on $V$.

Explicitly, for an ordinary vector space this is the quotient of the tensor algebra? $TV$ by the ideal generated by elements of the form $v\otimes w+w\otimes v$. For $V$ a graded vector space these elements are $v\otimes w+(-1{)}^{\mathrm{deg}v\cdot \mathrm{deg}w}w\otimes v$.

The product in this algebra is denoted with a wedge, and called the wedge product. In particular, if $v,w\in V$ for $V$ an ordinary vector space we have in $\Lambda V$ the relation

$$v\wedge w=-w\wedge v.$$v \wedge w = - w \wedge v \,.

For $V$ a graded vector space this is

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

in the presence of 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 $A$ be $R^3$. Then an element of ${\mathrm{Alt}}_{0}A$ is a scalar (a real number), an element of ${\mathrm{Alt}}_{1}A$ is a vector in the elementary sense, an element of ${\mathrm{Alt}}_{2}A$ is a bivector? (which we may identify with a pseudovector? using the standard inner product on $A$), and an element of ${\mathrm{Alt}}_{3}A$ is (again using the inner product) a pseudoscalar?.

More generally, let $A$ be $R^n$, or indeed any real inner product space. Then an element of ${\mathrm{Alt}}_{p}A$ 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 $A$ be the cotangent bundle of $X$. (This is not really an object of any abelian category, but it is a vector bundle over $X$, and we can apply $\mathrm{Alt}$ fibre-wise.) Then a differential form on $X$ is a section of the vector bundle $\mathrm{Alt}A$. If $X$ is a (semi)-Riemannian manifold, then we can identify $p$-forms with $(n-p)$-forms using the Hodge star?

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-∞ 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.