3.1. Algebras¶

3.1.1. Grassman algebra¶

The Grassman algebra is the exterior algebra of \(p\)-forms. The exterior algebra may be expressed as the direct sum

\[\Lambda V := \bigoplus_{p=0}^n \Lambda^p V,\]

with dimensionality

\[\text{dim}\, \Lambda V = 2^n.\]

This is relatively straight forward to show.

Proof: We begin by noting a result for the dimension of direct sums, along with eq. (2.9),

\[\begin{split}\text{dim}\, \Lambda V &= \sum_{p=0}^n \text{dim}\, \Lambda^p V. \\ &= \sum_{p=0}^n {n \choose p},\end{split}\]

which is just the binomial theorem,

\[\left( 1 + x \right)^n = \sum_{k=0}^n {n \choose k} x^k,\]

for the case of \(x = 1\). Therefore:

\[\text{dim}\, \Lambda V = 2^n. \ \ \square\]