elvum

A further word of explanation: a "representation" of a Lie algebra is a model for it in terms of matrix multiplication. In more detail: suppose you have a vector space A, with an associative linear multiplication operation on it (such things are called associative algebras). Then you can define a Lie bracket on it, by setting [a,b] = ab - ba. The space of nxn matrices forms an associative algebra under matrix multiplication, so we can form a Lie algebra of nxn matrices (equivalently, we can form a Lie algebra of linear transformations on an n-dimensional vector space V). A representation of a Lie algebra g is then a Lie algebra homomorphism from g to such a Lie algebra of matrices: equivalently, it's a choice of matrix A for each vector a in g so that the Lie bracket is realized by AB - BA. You can learn a lot about Lie algebras by studying their representations, but I can't tell you exactly what, because my Lie algebra lecturer used to schedule all his lectures for 9am on the grounds that undergraduates would be bright and fresh at that time. Consequently, I was unusually vulnerable to attacks of somnolence, and representation theory sends me to sleep at the best of times :-)