CasimirOperator

Casimir operator is an element of the center of the universal enveloping algebra U(g) of a Lie algebra g. It is constructed from an invariant symmetric bilinear form on g, most commonly the Killing form κ. By choosing a basis {X_i} of g and its dual basis {X^i} with κ(X_i, X^j) = δ_i^j, one forms the quadratic Casimir C = Σ_i X_i X^i. Because it commutes with all elements of g, C lies in Z(U(g)).

For semisimple Lie algebras, the Killing form is nondegenerate, ensuring the existence of Casimir operators. In general, the center of U(g) contains Casimir elements corresponding to invariant symmetric tensors of g. The quadratic Casimir is the simplest and most widely used, while higher-order Casimirs exist for many algebras.

In representation theory, Casimir operators act as scalars on irreducible representations. If V is an irreducible finite-dimensional g-module and the form κ is normalized so that long roots have length squared 2, then the eigenvalue of the quadratic Casimir on V is given by (Λ, Λ + 2ρ), where Λ is the highest weight of V and ρ is the Weyl vector (half-sum of positive roots). This eigenvalue can be expressed in terms of Dynkin labels and the chosen normalization of κ.

Beyond the quadratic Casimir, semisimple algebras of rank l possess l algebraically independent Casimirs of degrees d_i, determined by the exponents of the algebra. These higher Casimirs arise from invariant symmetric tensors and provide additional commuting operators in quantum systems.

Applications include classifying representations, constructing invariants in tensor products, and identifying conserved quantities in quantum mechanics and gauge theories.