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 exterior powers 
.
Alternating differential k-forms on
instead belong to 
. The product on these
algebras is the wedge product. 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 spanned
by simple 
-tensors with two equal factors.
In particular, 
is spanned by 
for 
. This definition applies in every field
characteristic. The symbol 
denotes the vector
space tensor product. The equivalence class ![[x_1 tensor ... tensor x_p]](/images/equations/ExteriorAlgebra/Inline23.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 
.
When the scalar field has field characteristic zero, the alternating products can be realized as a subspace of the tensor products. Define the linear transformation
 |
(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 tensor 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 
over a field of field
characteristic different from 2, then 
is a decomposition
of 
into two representations.
