Timeline for Comparing two similar procedures for quantizing a Casimir Lie algebra
Current License: CC BY-SA 2.5
5 events
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| Feb 23, 2010 at 18:40 | comment | added | Pavel Etingof | There are definitely finite dimensional quasitriangular Hopf algebras (quantum doubles of finite groups with a nontrivial 3-cocycles) which are not twist equivalent to Hopf algebras. You can mimick this construction in the Lie algebra case (take the Drinfeld double of a Lie quasibialgebra with zero cobracket but nontrivial element in $\wedge^3$), which should give you a desired example. So I think the answer is yes, there are Casimir Lie algebras without r-matrices (but I never honestly checked it). | |
| Feb 23, 2010 at 16:23 | comment | added | Theo Johnson-Freyd | So I guess the question I have then is: why are there any classical r-matrices for a given Casimir? (If I understand correctly, your statement is that r-matrices are in bijection with certain equivalences of categories relating the two constructions.) Or are their Casimir Lie algebras that do not support any r-matrices, and hence quasitriangular quasiHopf algebras that are not a twist away from a Hopf algebra? | |
| Feb 23, 2010 at 16:20 | comment | added | Theo Johnson-Freyd | Ah, great. And when I think about it, I think I remember reading both the comments, maybe in your book with Schiffmann. I had only the KRT book handy when writing the question, and they attribute the independence to Le and Murakami. In particular, I copied the braided-category quantization discusion almost directly from KRT. | |
| Feb 23, 2010 at 16:15 | vote | accept | Theo Johnson-Freyd | ||
| Feb 23, 2010 at 13:07 | history | answered | Pavel Etingof | CC BY-SA 2.5 |