Der Zentralisator einer Liealgebra in einer einhüllenden Algebra

Let g be a complex semisimple Lie algebra and k a reductive subalgebra of g. The paper is concerned with the centralizer U(g)^k of k in the enveloping algebra U(g). Denote by z(g), z(k) the center of U(g), U(k) respectively. Then there is a canonical homomorphism p:z(g)\otimes z(k)-->U(g)^k.

Theorem: Assume k does not contain a non-zero ideal of g. Then the following statements are equivalent:

  1. p is an isomorphism;
  2. U(g)^k is commutative;
  3. the pair (g,k) is either (sl(n), gl(n-1)) n>=2 or (so(n), so(n-1)) n>=4.
The method is using the canonical filtration of U(g) and then proving an equivalent theorem for the associated graded algebra.

Appeared in: Journal für die reine und angewandte Mathematik 406 (1990) 5-9

Available files:

Remark: At the time of writing the note I was blatantly unaware of the existing (mostly unpublished) literature and that the theorem was sort of "folklore" among expert. Therefore, my apologies to all people not mentioned in the references of the paper and who proved all or part of the theorem before me. I compiled the literature I am now aware of although I am afraid that it is still incomplete.

Finally, I should note that in a later paper I did contribute some new result on U(g)^k, namely a generalization of (1. equivalent to 2.).

Theorem. Assume k does not contain a non-zero ideal of g. Then p is injective and its image is the center of U(g)^k.

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Last updated: October 26, 2000