sl2module

sl2module refers to a module (representation) of the Lie algebra sl2, the algebra of 2x2 complex traceless matrices. Over fields of characteristic zero, sl2 is commonly presented with generators e, f, h and relations [h,e] = 2e, [h,f] = -2f, [e,f] = h. An sl2-module V is a vector space equipped with a Lie algebra homomorphism sl2 → End(V), giving an action of sl2 on V.

For finite-dimensional sl2-modules, the action of h is diagonalizable and V decomposes into weight spaces V_μ = {v ∈ V : h·v = μv}. All weights μ are integers, and there exists a highest weight n ≥ 0 with e·v = 0 for some weight vector v, from which the entire module is generated by repeated application of f. Each finite-dimensional irreducible sl2-module is unique up to isomorphism and is denoted V(n), having dimension n+1 and weights n, n−2, …, −n. Special cases include V(0), the trivial 1-dimensional module, V(1), the standard 2-dimensional representation, and V(2), the 3-dimensional adjoint representation.

Tensor products of finite-dimensional modules follow the Clebsch–Gordan rule: V(m) ⊗ V(n) ≅ ⊕_{k=0}^{min(m,n)} V(m+n−2k). This provides a complete decomposition into irreducibles.

Beyond finite dimensions, sl2-modules encompass infinite-dimensional objects such as Verma modules M(λ) with highest weight λ, and their simple quotients L(λ). If λ ∈ Z_{\ge 0}, L(λ) ≅ V(λ); for other λ, L(λ) is infinite-dimensional. These structures are central to the broader theory of representations of semisimple Lie algebras and have applications in mathematical physics and related areas.