Exterior Algebra

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 k 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 V is constructed by forming monomials u, v ^ w, x ^ y ^ z, etc., where u, v, w, x, y, and z are vectors in V 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,

 Lambda^pV= tensor ^pV/W_p,

where W_p is the subspace of p-tensors generated by transpositions such as W_2=<x tensor y+y tensor x> and  tensor denotes the vector space tensor product. The equivalence class [x_1 tensor ... tensor x_p] is denoted x_1 ^ ... ^ x_p. For instance,

 x ^ y+y ^ x=0,

since the representatives add to an element of W_2. Consequently, x ^ y=-y ^ x. Sometimes Lambda^pV is called the pth exterior power of V, and may also be denoted by Alt^pV.

The alternating products are a subspace of the tensor products. Define the linear map

 Alt: tensor ^pV-> tensor ^pV


 Alt(v_(i_1) tensor ... tensor v_(i_p))=1/(p!)sum_(sigma)pi(sigma)v_(i_(sigma(1))) tensor ... tensor v_(i_(sigma(p))),

where sigma ranges over all permutations of {1,...,p}, and pi(sigma) is the signature of the permutation, given by the permutation symbol. Then Lambda^pV is the image of Alt, as W_p is its null space. The constant factor 1/p! , which is sometimes not used, makes Alt into a projection operator.

For example, if V has the vector basis {e_1,e_2,e_3,e_4}, then

Lambda^2V=<e_1 ^ e_2,e_1 ^ e_3,e_1 ^ e_4,e_2 ^ e_3,e_2 ^ e_4,e_3 ^ e_4>
Lambda^3V=<e_1 ^ e_2 ^ e_3,e_1 ^ e_2 ^ e_4,e_1 ^ e_3 ^ e_4,e_2 ^ e_3 ^ e_4>
Lambda^4V=<e_1 ^ e_2 ^ e_3 ^ e_4>,

and Lambda^kV={0} where k>dimV and <v,w> is the vector space spanned by v and w. For a general vector space V of dimension n, the space Lambda^pV has dimension (n; p).

The space Lambda^*= direct sum _pLambda^pV becomes an algebra with the wedge product, defined using the function Alt. Also, if T:V->W is a linear transformation, then the map T_(*,p):Lambda^pV->Lambda^pW sends v_1 ^ ... ^ v_p to T(v_1) ^ ... ^ T(v_p). If n=dimV and T(v)=Av where A is a square matrix, then T_(*,n)(e_1 ^ ... ^ e_n)=(detA)e_1 ^ ... ^ e_n.

The alternating algebra, also called the exterior algebra, Lambda^*V is a 2^n dimensional algebra. In the Wolfram Language, an element of the alternating algebra can be represented by an n-nested binary list. For example, {{{1, 2}, {0, 0}}, {{3, 0}, {4, 5}}}represents e_1 ^ e_2 ^ e_3+2e_1 ^ e_3+3e_2 ^ e_3+4e_3+5.

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 V to have the structure of a module. So exterior algebras come up in representation theory. For example, if V is a group representation of a group G, then Sym_2V direct sum Lambda^2V is a decomposition of V tensor V into two representations.

See also

Differential k-Form, Form Envelope, Group Representation, Symmetric Group, Tensor Product, Vector Space, Wedge Product

Portions of this entry contributed by Todd Rowland

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Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

Referenced on Wolfram|Alpha

Exterior Algebra

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Exterior Algebra." From MathWorld--A Wolfram Web Resource.

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