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 kforms 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 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, 5represents .
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 kforms 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.