TITLE: Main use of tensor, symmetric and exterior algebras outside differential geometry?
QUESTION [7 upvotes]: So I've seen these defined when constructing differential forms and in the construction of integration of manifolds. However, these seem to be a standard subject in most graduate algebra books, yet, I've never seen them applied anywhere else than in differential geometry.
My question is: What use do these have as abstract objects? What information do they convey?

REPLY [2 votes]: Symmetric algebras are polynomial algebras. Polynomials appear all over the place. Exterior algebras also appear all over the place. They can be thought of as graded polynomial algebras.
One simple example is that if $\mathfrak g$ is a Lie algebra, then 
$$\cdots\to\Lambda^{k+1} \mathfrak g \to \Lambda^{k} \mathfrak g\to\cdots$$
is a chain complex, where the boundary operator is defined by
$$\partial(x_1\wedge\cdots\wedge x_k)=\sum_{i<j}(-1)^{i+j+1}[x_i,x_j]\wedge x_1\wedge\cdots\wedge \hat{x_i}\wedge\cdots\wedge \hat{x_j}\wedge\cdots\wedge x_k.$$
Here the "hat" notation means to remove those terms from the long wedge. Lie algebra homology is $H_*(\mathfrak g)$ is defined as the the homology of this complex. (This is called the Chevalley-Eilenberg complex.) 
The fact that $[\cdot,\cdot]$ satisfies the Jacobi identity is equivalent to $\partial^2=0,$ and this the key to defining a more flexible gadget callet an $L_{\infty}$-algebra, which is any coderivation on $\Lambda\mathfrak g$ satisfying $d^2=0$.