Exterior Algebras

Since V2.15 (December 2008), Magma has a special type for exterior algebras. Such an algebra is skew-commutative and is a quotient of the free algebra \(K\langle x_1, \ldots, x_n\rangle\) by the relations \(x_i^2 = 0\) and \(x_i x_j = -x_j x_i\) for \(1 \leq i,j \leq n\), \(i \not= j\). Because of these relations, elements of the algebra can be written in terms of commutative monomials in the variables (via a collection algorithm), and the associated algorithms are much more efficient than for the general noncommutative case. Also, a Gröbner basis of an ideal of an exterior algebra is always finite (in fact, the whole exterior algebra has dimension \(2^n\) as a \(K\)-vector space). Exterior algebras may be constructed with the ExteriorAlgebra function below, and all operations applicable to general FP algebras are also applicable to them (so will not be duplicated here). Furthermore, modules over exterior algebras are also allowed: see Chapter Modules over Multivariate Rings for details.