Lie Algebra Representations

Representation of Lie Algebras

Let be a Lie algebra over a field . A representation of on a vector space is a Lie algebra homomorphism where is the Lie algebra of all linear endomorphisms on .

Weights and Maximal Vectors

Weight

Let be a representation of a Lie algebra over a field . Then a vector is said to have weight if

Classification of Representations

Proposition

The Lie algebra has precisely one irreducible representation of dimension for every .

Proof Let us use the standard basis of : Suppose is an irreducible representation. The commutation relation above immediately implies that is diagonalisable, so by theorem, is always diagonalizable. Hence is a direct sum of weight spaces: Note that the commutation relations imply that raises weights by 2, while lowers weights by 2. is finite dimensional, so only finitely many can be nontrivial. Let be the highest weight such that but . Let be the highest weight vector, and . Then we observe the following:

Now let be the smallest integer such that but . Consider Then is a subrepresentation of . Since is irreducible, . Thus, forms a basis of , and .

e.g. Here are some concrete examples:

• . The trivial representation with being the zero map.
• . The standard representation on given by inclusion .
• . This is isomorphic to the adjoint representation .

Weyl's Theorem

Let be a semisimple Lie algebra over , with a finite dimensional representation . Then is expressable as a direct sum of irreducible representations . That is, the category of representations of semisimple Lie algebras is a semisimple category.