Special Linear Groups
Recall the special linear groups:
Claim that the set of all linear operators such that determinant is
Link to originalis a group. Write it as where denotes the dimension of the corresponding square matrix.
e.g.
- The Lie algebra
consists of all real matrices with trace zero. A standard basis is given by with Lie brackets As is so important in physics, people give special names to these basis elements: is called the raising operator, the lowering operator, and the Cartan operator. - The complexification of
is . i.e., . So the above basis also works for . It is also sometimes useful to use Pauli matrices as a basis of it. Suppose is a basis of satisfying Then the Casimir operator of is defined as A notable result is that lies in the center of , for which one can verify that