One more little side trip before we proceed with the
differential geometry: Lie algebras. These are like “regular”
associative algebras in that we take a module (often a vector
space) and define a bilinear operation on it. This much is
covered at the top of the post on algebras. The difference is
that instead of insisting that the operation be associative, we
impose different conditions. Also, instead of writing our
operation like a multiplication (and using the word
“multiplication”), we will write it as and call it the “bracket”
of and . Now, our first condition is that the bracket be
antisymmetric: Secondly, and more importantly, we demand that
the bracket should satisfy the “Jacobi identity”: What this
means is that the operation of “bracketing with ” acts like a
derivation on the Lie algebra; we can apply to the bracket by
first applying it to and bracketing the result with , then
bracketing with the result of applying the operation to , and
adding the two together. This condition is often stated in the