exterior algebra (Rev #2) in nLab

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

The exterior algebra (written ⋀A\bigwedge A, ΛA\Lambda A, or AltAAlt A) on AA is the free skew-commutative algebra on AA (where ‘algebra’ should be interpreted as a monoid object in CC). That is, the functor Alt:C→SComMon(C)Alt: C \to SCom Mon (C) is the left adjoint of the forgetful functor SComMon(C)→CSCom Mon (C) \to C. The elements of degree pp in this graded algebra comprise Alt pAAlt_p A, the ppth exterior power of AA.

In more detail, AltAAlt A is generated by the elements of AA (which comprise Alt 1AAlt_1 A) and these operations:

• the operations (addition, scalar multiplication, etc) of the objects of CC, generalising the operations in AA,
• an associative binary operation ∧\wedge of multiplication (the exterior product or wedge product),

subject to these identities:

• the identities necessary for AltAAlt A to be an object of CC,
• ∧\wedge distributes over these operations,
• x∧x=0x \wedge x = 0 for x∈Ax \in A.

A more general form of the last then follows; if v∈Alt pAv \in Alt_p A and w∈Alt qAw \in Alt_q A (that is if they are homogeneous of degree pp and degree qq), then

• v∧v=0v \wedge v = 0 if pp is odd,
• v∧w=(−1) pqw∧vv \wedge w = (-1)^{pq}\, w \wedge v.

That is, AltAAlt A is a skew-commutative algebra.

Examples

In the following examples, it is somewhat traditional to have an inner product structure, so we mention it; but that structure is not necessary for forming the exterior algebra.

Let AA be R 3\mathbf{R}^3. Then an element of Alt 0AAlt_0 A is a scalar (a real number), an element of Alt 1AAlt_1 A is a vector in the elementary sense, an element of Alt 2AAlt_2 A is a bivector (which we may identify with a pseudovector using the standard inner product on AA), and an element of Alt 3AAlt_3 A is (again using the inner product) a pseudoscalar.

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