TITLE: Why are superalgebras so important?
QUESTION [20 upvotes]: I know that a superalgebra is a $\mathbb Z/2\mathbb Z$-graded algebra and that it behaves nicely. I know very little physics though, so even though I know that the super- prefix is related to supersymmetry, I don't know what that means; is there a compelling mathematical reason to consider superalgebras?

REPLY [5 votes]: If you are willing to build a formalism of supergeometry then some constructions are easier to state in this language, notably differential forms and the De Rham complex.  The first one is just functions from the odd line ($R^{(0 | 1)}$) to your manifold, and the De Rham differential comes from super-diffeomorphisms acting on that line.
Also, there is a symplectic/orthogonal duality in representation theory that some people (most famously Kontsevich) have advocated as best understood in terms of Lie superalgebras.
Constructions using a formally negative-dimensional object (or calculations that look like they might come from such an object) sometimes can be interpreted in terms of bona fide $Z/2$ graded objects whose superdimension (even dimension minus odd dimension) is the negative dimension in question.
edit: you will get more knowledgeable answers if you post to Mathoverflow, where some of the contributors have written papers on super- or noncommutative geometry.