exterior algebra

⋀ V : = k = 0 ⨁ \infty \land^{k} V = k = 0 ⨁ d i m V \land^{k} V

Where $\land^{k}$ is the exterior product.

This is an example of an graded algebra.

”multiplication” in $\land V$

We can multiply vectors in $\land V$ by

(v_{1} \land \dots \land v_{k}) \cdot (v_{k + 1} \land \dots \land v_{k + l}) = v_{1} \land \dots \land v_{k + l}

Where $\cdot$ is usually denoted by $\land$ .