# Exterior Algebras

This post will be very similar to last post, where I talked about symmetric algebras. Indeed, the exterior algebra is another universal construction obtained as the quotient of the tensor algebra. Among other applications, this construction is of fundamental importance in differential geometry where it is used to define differential forms, which are in turn used to generalize the machinery of calculus on to calculus on smooth manifolds.

**Definition:** The exterior algebra of an -module is the quotient of the tensor algebra by the ideal generated by the elements of the form for all The exterior algebra is then denoted and the image of in is denoted by

The multiplication in the exterior algebra,

is called the wedge product or the exterior product. Denote by the set of k-tensors in so is the set of k-tensors for which for some The k’th exterior power is the quotient

By this definition, we have in whenever for some Then we have

so the multiplication is anticommutative on the image of simple tensors in i.e.:

From these considerations, we can describe the elements of the k’th exterior power as the set of k-tensors where all elements having two equal entries have been identified with More explicitly, is equal to modulo the submodule generated by all elements of the form

Indeed, the elements of $latex are finite sums of k-tensors having two equal adjacent entries, so every element of is a sum of elements of that form. Conversely, since the wedge product is anticommutative on simple tensors, we can rearrange (modulo a minus sign) any element with equal entries and for these entries to be adjacent.

The exterior algebra is also universal with respect to alternating multilinear maps. Recall that an alternating map is a map such that whenever for some Note that this is equivalent to

This shows that in some sense the exterior algebra is complementary to the symmetric algebra.

Theorem (Universal property for Alternating Multilinear Maps):Let be two -modules for a commutative ring and an alternating multilinear map. Then there is a unique -module homomorphism such that

**Proof: **The proof is completely analogous to the case of the exterior algebra: you take the induced homomorphism

which is then seen to be alternating. Thus, it’s kernel is in and the map

is well defined.

QED

**Reference:** D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.