Exterior Algebra

The Algebra of the Exterior Product, also called an Alternating Algebra or Grassmann Algebra. The study of exterior algebra is also called Ausdehnungslehre and 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 $k$-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 $u$, $v \wedge w$, $x\wedge y\wedge z$, etc., where $u$, $v$, $w$, $x$, $y$, and $z$ are vectors in $V$ and $\wedge$ is asymmetric multiplication. The sums formed from linear combinations of the Monomials are the elements of an exterior algebra.

References

Forder, H. G. The Calculus of Extension. Cambridge, England: Cambridge University Press, 1941.

Lounesto, P. ``Counterexamples to Theorems Published and Proved in Recent Literature on Clifford Algebras, Spinors, Spin Groups, and the Exterior Algebra.'' http://www.hit.fi/~lounesto/counterexamples.htm.