$a$ and $b$ both of which square to zero.

Having explained what a representation of $V_{4}$ is, the following two questions have to be answered.

1.  Why do we care about the representations of $V_{4}$?

2.  What is unique about our approach?

In answer to the first question, first note that groups act naturally on various algebraic objects including vector spaces, rings, algebraic varieties and topological spaces. These actions tend to be quite complex in general. Therefore it is important to find simple pieces of this action and find ways to glue these pieces together to reconstruct the original action. Often this is related to other invariants of the group or our given representation like cohomology groups and support varieties. Amazingly, this goal in modular representation theory turns out to be exceedingly difficult. It turns out that besides the cyclic groups whose representations are very easily understood, Klein four group is one of the very few (other groups are the dihedral groups) interesting yet non-trivial examples for which representation theorists are able to completely classify all the finite dimensional modular representations. There is a lot to be learned by studying the representation theory of this one group and it goes 