Enveloping algebras and Casimir elements

Construct U(g) from the tensor algebra by imposing x⊗y−y⊗x=[x,y]. Any Lie-algebra representation extends uniquely to an associative representation of U(g). After choosing an ordered basis of g, PBW says its ordered monomials form a vector-space basis of U(g). Thus g embeds in U(g), and the associated graded algebra for the degree filtration is the symmetric algebra S(g). The associated graded statement does not say U(g) itself is commutative; lower-degree bracket terms survive before taking that graded limit.

A finite-dimensional matrix representation usually has a large kernel as a U(g)-module, even if its restriction to g is faithful. For example, sufficiently high powers of a raising operator can vanish on a finite highest-weight module, while the corresponding ordered monomial remains nonzero in U(g). These are representation-specific annihilator relations. Confusing them with the defining enveloping relations would incorrectly collapse all Verma modules to the dimensions of one chosen matrix example.

The tensor Σxᵢ⊗xⁱ is independent of the basis and invariant under the adjoint action because it corresponds to the identity through the bilinear form. Multiplication into U(g) then produces ΩB, which commutes with every generator and hence is central. Scaling B by a nonzero constant c scales its dual basis and ΩB by 1/c. Centrality survives, but numerical eigenvalues change. A phrase such as the Casimir therefore needs a form convention, especially when comparing a Killing-form calculation with a physics normalization.

Choose an invariant form whose induced root inner product has long roots of squared length two on each simple factor. The quadratic Casimir then acts on a highest-weight module of weight λ by (λ,λ+2ρ), with the same induced weight pairing. This can be checked on a highest vector and extended through centrality to its cyclic module. For sl(2), taking B(x,y)=tr(xy) gives Ω=h²/2+ef+fe and eigenvalue m(m+2)/2 on L(m). The Killing form is four times this trace form, so its Casimir is one quarter as large.

In higher rank, a quadratic eigenvalue does not generally distinguish all simple modules. For sl(3), the standard module and its dual have highest weights ω₁ and ω₂ and the same normalized quadratic eigenvalue 8/3, yet they are nonisomorphic. Higher central elements provide further information. The Harish-Chandra isomorphism, in a rho-shifted convention, identifies Z(U(g)) with Weyl-invariant polynomial functions on h*. The central character on L(λ) evaluates those polynomials at λ+ρ. This structural theorem is stated here, not derived from the quadratic example.

The shift is essential: ordinary Weyl-invariant polynomials evaluated at λ+ρ correspond to the dot-action organization of highest weights. One must state the shift before comparing central-character formulas. On finite simple modules Schur's lemma gives scalar central action; on reducible or nonsemisimple modules a central operator need not be one scalar on the entire space. Keep the universal element, its action in a particular module, and a chosen numerical normalization separate. That separation lets the same invariant algebra connect representation classification, differential operators, and physical symmetry without conflicting factors.