Representation theory of Lie algebras

In these notes, we give a brief overview of the (finite dimensional) representation theory of finite dimensional semisimple Lie algebras. We first study the example of sl2(C) and then provide the general (additive) theory, along with an analysis of the representations of sl3(C). In the last section, we have a look at the multiplicative structure of the representation ring, discussing examples for the Lie algebras sl2(C) and sl3(C). The main source for these notes is the book Representation Theory, A First Course by William Fulton and Joe Harris. The name of the game In this section, we present the fundamental terminology and notation used in the sequel. In particular, we introduce representations of Lie algebras (Subsection 1.2) and some basic constructions (Subsection 1.4). 1.1 Lie algebras The theory of Lie algebras is the linear algebraic counterpart of the (rather geometric) theory of Lie groups. 1 The name of the game Clara Löh – [email protected] Definition (1.1). A Lie algebra is a finite dimensional complex vector space g, equipped with a skew-symmetric bilinear map [ · , · ] : g × g −→ g, the so-called Lie bracket, satisfying the Jacobi identity, i.e., for all x, y ∈ g, [ x, [y, z] ] + [ y, [z, x] ] + [ z, [x, y] ] = 0. A homomorphism of Lie algebras is a homomorphism φ : g −→ h of complex vector spaces that is compatible with the Lie brackets, i.e., for all x, y ∈ g we have φ ( [x, y]g ) = [ φ(x), φ(y) ] h. ! Example (1.2). Let V be a finite dimensional complex vector space. Then the trivial Lie bracket [ · , · ] = 0 turns V into a Lie algebra. Lie algebras of this type are called Abelian. " Example (1.3). The main source of Lie algebras are matrix algebras: • Let V be a finite dimensional complex vector space. Then the set of endomorphsisms of V is a Lie algebra, when endowed with the Lie bracket EndV × EndV −→ EndV (A, B) −→ A ◦ B− B ◦ A. We denote this Lie algebra by gl(V). Moreover, we use the notation gln(C) := gl(Cn), and view the elements of gln(C) as matrices rather than endomorphisms. • Let n ∈ N. Then the traceless matrices form a subalgebra of gln(C), denoted by sln(C). " Example (1.4). If G is a Lie group, then the tangent space TeG at the unit element can be endowed with a Lie algebra structure (using the Lie derivative of vector fields or the derivative of the adjoint representation of G [2; Section 8.1]). " Definition (1.5). A Lie algebra is simple, if it contains no non-trivial ideals. Nontrivial Lie algebras that can be decomposed as a direct product of simple Lie algebras are called semisimple. ! Simple Lie algebras can be classified by means of Dynkin diagrams, a purely combinatorial tool [2; Chapter 21]. Example (1.6). For example, the Lie algebras sln(C) are all semisimple (they are even simple Lie algebras) [2; Chapter 21]. "