Representations of Lie algebras, Casimir operators and their ...

Let us construct an element of the universal enveloping algebra U(g)of the formC =dim∑gk=1ẽ k ⊗ e k =dim∑gk=1e k ⊗ ẽ k (9)(its symmetry comes from the fact that the Killing form is symmetric).Suppressing the tensor product signs and computing mod J , wehave for the commutator between e a ∈ g and C ∈ U(g)[e a , C] ===dim∑gk=1dim∑gk=1dim gk=1)(e a ẽ k e k − ẽ k e k e a =()(e a ẽ k − ẽ k e a ) e k + ẽ k (e a e k − e k e a ) =dim g∑∑([e a , ẽ k ] e k + ẽ k [e a , e k ]) = (c lakẽ l e k + c aklẽ k e l ) = 0.k,l=1We conclude that C is a Casimir operator of g. It is called the quadraticCasimir operator [4]. For its application in the proof of Weyl’s theorem,see [5].We remark that the quadratic Casimir operator does not exhaustall independent Casimir operators of the semisimple Lie algebra gwhen we have rank g > 1. It is known that any semisimple Lie algebraof rank l has l independent Casimir operators which generate the wholecenter of the universal enveloping algebra U(g) through their productsand linear combinations. Their explicit form depends on the detailsof the structure of the considered algebra g.Casimir invariants are of primordial importance in physics. Theyoften represent such important quantities as angular momentum, elementaryparticle mass and spin, Hamiltonians of various physicalsystems etc.Example 1 Let us consider the angular momentum algebrawithso(3) = span{L 1 , L 2 , L 3 }[L j , L k ] =3∑ɛ jkl L l . (10)l=1The quadratic Casimir operator (9) isC = − 1 23∑L 2 l , (11)l=16