## Some Natural Identifications

**Point of Post: **In this post we take our discussion last time concerning tensor algebras and their quotients to the case of vector spaces, and show that there are simple identifications that can be made. We also discuss the notion of orientations and volume forms.

**Motivation**

We now are able to take the very formal, very general way of defining tensor powers/exterior powers and show how they relate to much tamer objects–multilinear and alternating forms. The main reason for doing this is, as mentioned before, to help explain the somewhat confusing notation used in some differential geometry textbooks. For example, they merely *denote *the space of alternating -forms on as . But, why? This notation makes no sense, especially considering that this notation has other meaning in mathematics (the exterior power of the dual space of ). They should really only conflate these two if there is some sort of natural isomorphism between these two. Well, to state the obvious, they are naturally isomorphic. But, this is never stated, let alone proven, in differential geometry textbooks. This is the point of this post.

**Natural Identification of with **

We begin with some finite dimensional vector space over a field . We recall that where , the dual space, is . Moreover, we recall that if is a basis for then defined by is the dual basis–obviously a basis for .

Recall also that is the space of all -linear forms –if we just take the forms to be . Our first claim is that there is a natural identification . Indeed, define by saying that is sent to a funtion which acts as

It’s evident not only that this function associated to is a -linear form but also that the association itself is -linear. Thus, we get a map where

It’s easy to see that this is an -map. To prove that this is an injection suppose that is the zero map .

Now, suppose that is another -space and is a given map. We know that we get a map by precomposing any functional with . But, we then get from the construction of the tensor product algebra a map given on simple tensors by

Similarly, we get a map defined by for every -linear form . Note then that for each simple tensor and each vector we have that

and

So that the diagram

Commutes, and thus we may finally conclude the following:

**Theorem: ***Let be a finite dimensional vector space. Then, the map as defined above is a natural isomorphism.*

Similarly, we note that there is a natural isomorphism where is . This is an isomorphism of -algebras when we define the multiplication, auspiciously denoted , in by saying that if and then is given by

This is easy to prove.

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

[6] Conrad, Keith. “Exterior Powers.” *Www.math.uconn.edu/~kconrad*. Web. <http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/extmod.pdf>.

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