Mixed Exterior Algebra

Abstract

A skew-symmetric map of type (p, q) from E*, E into a vector space H is a (p + q)-linear mapping

$$\psi :\underbrace {E* \times \ldots \times E*}_p \times \underbrace {E \times \ldots \times E}_q \to H$$

which satisfies

$$\psi (x_{\sigma (1)}^*, \ldots ,x_{\sigma (p)}^*;{x_{\tau (q)}}) = {\varepsilon _\sigma }{\varepsilon _\tau }\psi (x_1^*, \ldots ,x_p^*;{x_1}, \ldots ,{x_q})$$

for all permutations σ ∊ S _p and ? ∊ S _q .

Throughout this chapter E*, E will denote a pair of dual vector spaces over a field of characteristic zero.