Timeline for The Killing form on quantized enveloping algebras and reduction to the classical case
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| Aug 3, 2012 at 15:40 | comment | added | Chuck Hague | Thanks for the comment -- I'll look at Dixmier's book and Joseph's papers. My hope with such an invariant form is to construct an appropriate adjoint action of the positive part on the negative part. (The zero Verma structure won't do because I want a graded module structure). Perhaps there is an easier way of doing such a thing. (Also, you're right to be unsure about roots of unity -- upon closer inspection of Jantzen's book, the quantum pairing is NOT nondegenerate at roots of 1). | |
| Aug 3, 2012 at 14:30 | comment | added | Jim Humphreys |
For the complex enveloping algebra, a direct route should be possible: Dixmier's 1974 book on enveloping algebras has some steps in that direction, involving study of the center. Further papers by A. Joseph may help too, but are not easy to dig into. For the hyperalgebra, I'm less certain what is possible. In any case I wonder what new information you expect to get from existence of a suitable ad-invariant bilinear form? (I'm also unsure about the role of quantum groups at a root of unity in this framework. The symbol $q$ has more than one meaning.)
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| Aug 2, 2012 at 19:22 | history | asked | Chuck Hague | CC BY-SA 3.0 |