representation theory of
The special linear Lie algebra of matricies, denoted by , is defined to be the span (over ) of the matricies
The representation theory of is a very important tool for understanding the structure theory and representation theory of other Lie algebras (semi-simple finite dimensional Lie algebras, as well as infinite dimensional Kac-Moody Lie algebras).
The finite dimensional, irreducible, representations of are in bijection with the non-negative integers as follows. Let , be a -vector space spanned by vectors . The following action of on define the unique (up to isomorphism) irreducible representation of of dimension (or of highest weight ):
The main points are that the one dimensional spaces are eigenspaces for with eigenvalue , the operator corresponding to kills and otherwise sends , while kills and otherwise sends . The operator corresponding to is often called a raising operator since it raises the eigenvalue for , and that of is called a lowering operator since it lowers the eigenvalue for .
is a simple Lie algebra, thus by Weyl’s Theorem all finite dimensional representations for are completely reducible. So any finite dimensional representation of splits into a direct sum of irreducible representations for various non-negative integers as described above.
|Title||representation theory of|
|Date of creation||2013-03-22 15:30:29|
|Last modified on||2013-03-22 15:30:29|
|Last modified by||benjaminfjones (879)|
|Defines||special linear Lie algebra of 2x2 matricies|