Exterior algebra is the algebra of the wedge product, also called an alternating algebra or Grassmann algebra. The study of exterior algebra is also called Ausdehnungslehre or extensions calculus. Exterior algebras are graded algebras.
In particular, the exterior algebra of a vector space is the direct sum over 
in the natural numbers of the vector
spaces of alternating differential k-forms
on that vector space. The product on this algebra
is then the wedge product of forms. The exterior
algebra for a vector space 
is constructed by forming monomials 
, 
, 
, etc., where 
, 
, 
, 
, 
, and 
are vectors in 
and 
is wedge product. The sums
formed from linear combinations of the monomials
are the elements of an exterior algebra.
The exterior algebra of a vector space can also be described as a quotient vector space,
 |
(1)
|
where 
is the subspace of 
-tensors
generated by transpositions such as 
and 
denotes the vector
space tensor product. The equivalence class ![[x_1 tensor ... tensor x_p]](/images/equations/ExteriorAlgebra/Inline18.svg)
is denoted
. For instance,
 |
(2)
|
since the representatives add to an element of 
. Consequently, 
. Sometimes 
is called the 
th exterior power of 
, and may also be denoted by 
.
The alternating products are a subspace of the tensor products. Define the linear map
 |
(3)
|
by
 |
(4)
|
where 
ranges over all permutations of 
, and 
is the signature of the permutation,
given by the permutation symbol. Then 
is the image of Alt, as 
is its null
space. The constant factor 
, which is sometimes not used, makes Alt into a projection
operator.
For example, if 
has the vector basis 
, then
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
and 
where 
and 
is the vector space spanned by 
and 
. For a general vector space 
of dimension 
, the space 
has dimension 
.
The space 
becomes an algebra with the wedge
product, defined using the function Alt. Also, if 
is a linear transformation,
then the map 
sends 
to 
.
If 
and 
where 
is a square matrix, then 
.
The alternating algebra, also called the exterior algebra, 
is a 
dimensional algebra. In the
Wolfram Language, an element of the
alternating algebra can be represented by an 
-nested binary list. For example, 
1, 2
, 
0, 0
,
3, 0
, 
4, 5
represents 
.
The rank of an alternating form has a couple different definitions. The rank of a form, used in studying integral manifolds of differential ideals, is the dimension of its form envelope. Another definition is its rank as a tensor.
The differential k-forms in modern geometry are an exterior algebra, and play a role in multivariable calculus. In general,
it is only necessary for 
to have the structure of a module.
So exterior algebras come up in representation
theory. For example, if 
is a group representation
of a group 
,
then 
is a decomposition of 
into two representations.
