SL2modules

sl2modules refers to the representations of the Lie algebra sl2, the algebra of 2×2 traceless matrices over a field (often the complex numbers). A sl2-module V is a vector space equipped with a Lie algebra action ρ: sl2 → End(V) that satisfies ρ([x,y]) = ρ(x)ρ(y) − ρ(y)ρ(x). With the standard basis of sl2 given by h, e, f and relations [h,e]=2e, [h,f]=-2f, [e,f]=h, the action encodes how the three generators transport vectors within V.

Weight space decomposition is central for finite-dimensional modules: h acts diagonally, so V decomposes into weight spaces Vλ = {v in V | h·v = λv}. The operators e and f raise and lower weights by 2, respectively. Finite-dimensional sl2-modules thus have a structured ladder of weights with symmetric endpoints.

Finite-dimensional irreducibles over a field of characteristic zero are classified by a nonnegative integer n. There is a unique irreducible module L(n) of dimension n+1, with a highest weight vector v such that e·v = 0 and h·v = n v. The weights of L(n) are n, n−2, …, −n, and the whole module is generated by repeatedly applying f to the highest weight vector. Each L(n) is isomorphic to the nth symmetric power of the standard 2-dimensional representation.

In addition to finite-dimensional theory, there are Verma modules M(n) generated by a highest weight n; these have unique simple quotients L(n). Over C, finite-dimensional sl2-modules are completely reducible, and tensor products decompose via the Clebsch–Gordan formula: V(m) ⊗ V(n) ≅ ⊕_{k=0}^{min(m,n)} V(m+n−2k).

sl2-modules form a foundational part of representation theory and illuminate the structure of more complex Lie algebras and algebraic groups, with many concepts extending to higher rank cases.