Abstract
The adjoint representations of several small dimensional Lie algebras on their universal enveloping algebras are explicitly decomposed. It is shown that commutants of raising operators are generated as polynomials in several basic elements. The explicit form of these elements is given and the general method for obtaining these elements is described.
Highlights
IntroductionIt is seen that the adjoint action has Un as its invariant subspace (by applying a commutator we cannot obtain an element of higher degree)
Let g be a finite dimensional complex semisimple Lie algebra and U( g) = U its corresponding enveloping algebra
We can see that such a two-sided ideal is invariant with respect to adjoint action of U: if we take adjoint action r: g ® L(U), i.e
Summary
It is seen that the adjoint action has Un as its invariant subspace (by applying a commutator we cannot obtain an element of higher degree) It is completely reducible on each Un, i.e. we can see Un as a direct sum of invariant subspaces generated by certain highest weight vectors. By commuting these vectors we see that The proof of this general formula is based on a dimensional check: First we see that the sum r(U)v1 + r(U)v2, where v1, v2 are highest weight vectors, is direct if and only if v1, v2 are linear independent. C1kE1m2 = E1m2H12k + lower terms where “lower terms“ contain monomials with H12k-1 and lower It is well known from representation theory that the dimension of the irreducible representation with highest weight m is 2m + 1.
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