nLab
exterior algebra

Contents

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 A, ΛA, or AltA) 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 Alt:CSComMon(C) is the left adjoint of the forgetful functor SComMon(C)C. The elements of degree p in this graded algebra comprise Alt pA, the pth exterior power (or pth alternating power) of A.

In more detail, AltA is generated by the elements of A (which comprise Alt 1A) and these operations:

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

subject to these identities:

  • the identities necessary for AltA to be an object of C,
  • distributes over these operations,
  • xx=0 for xA.

A more general form of the last then follows; if vAlt pA and wAlt qA (that is if they are homogeneous of degree p and degree q), then

  • vv=0 if p is odd,
  • vw=(1) pqwv.

That is, AltA 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 A is often called the Grassmann algebra over V

The Grassmann algebra ΛV or 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 vw+wv. For V a graded vector space these elements are vw+(1) degvdegwwv.

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

vw=wv.v \wedge w = - w \wedge v \,.

For V a graded vector space this is

vw=(1) degvdegwwv.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 Alt 0A is a scalar (a real number), an element of Alt 1A is a vector in the elementary sense, an element of Alt 2A is a bivector? (which we may identify with a pseudovector? using the standard inner product on A), and an element of Alt 3A 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 Alt pA is a p-vector as studied in geometric algebra?. Using the inner product, we can identify p-vectors with (np)-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 Alt fibre-wise.) Then a differential form on X is a section of the vector bundle AltA. If X is a (semi)-Riemannian manifold, then we can identify p-forms with (np)-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 (X). One such module is Ω 1(X)=Γ(T *X), the space of smooth sections of the cotangent bundle of X.

The deRham complex of X 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) equipped with the deRham differential is the Chevalley-Eilenberg algebra of the tangent Lie algebroid.