Timeline for Explicit formula for the inverse of Harish-Chandra map in case $\mathfrak{g}=\mathfrak{gl}_n$
Current License: CC BY-SA 4.0
13 events
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S Jan 25, 2023 at 23:42 | history | suggested | lukasnor | CC BY-SA 4.0 |
Corrected the triple direct sum to a direct sum of U(h) and (U(g)- + U(g)+), the last one not being direct in general. [Ref: Knapp - Lie Groups Beyond introduction - Prop. 5.34]
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Jan 25, 2023 at 21:28 | review | Suggested edits | |||
S Jan 25, 2023 at 23:42 | |||||
Nov 11, 2022 at 20:11 | comment | added | Alexander Chervov | Concerning the linear basis it itself - we should think about Schur functions - so we can ask what are their appropriate analogues in ZU(gl_n) - that can be found e.g. in arxiv.org/abs/q-alg/9602028 Andrei Okounkov , and his related papers at that time. Though somewhat digging literature similar results can be found in some unknown paper from 1980-ies. | |
Nov 11, 2022 at 20:06 | comment | added | Alexander Chervov | @VítTuček Well... About what part ? The interest to the field re-appeared again in early 90-ies motivated by constructions found by physicists - Yangian in particular. Many works have been done by Molev,Nazarov, Olshanski, Okounkov. See some surveys/books by Molev: arxiv.org/abs/math/0211288 - but it is more general - about Yangians and similar - the paradox insight is that it is more easy to look on classical Lie algebras via the Yangian/related point of view ! Thus exposition kind of not focused on gl_n, but with some experience one can downgrade to gl_n from Yangian without changes | |
Nov 11, 2022 at 18:34 | comment | added | Vít Tuček | @AlexanderChervov Could you please provide a reference? | |
Nov 11, 2022 at 12:16 | comment | added | Alexander Chervov | PS and the MAIN(!) thing about Duflo isomorphism is - it is isomorphism of ALGEBRAS(!) - it respects the multiplication on the center of U(g) and S(g)=(h)^W. (Konstsevich proved it for quantization of any Poisson manifold). Thus working on algebraic generators is enough - e.g. take characteristic polynomial - elementary symmetric functions - Duflo( it) = Cappelli-det - so for generators we are done. (But still there is another option to work on linear bases generators for gl_n - as one prefer). And many other options. | |
Nov 11, 2022 at 11:42 | comment | added | Alexander Chervov | @VítTuček both sides are algebras (and linear spaces), so you can do it only on the generators (linear bases) and there are nice generators (linear bases) for each, so everything is explicit and quite well-known | |
Nov 11, 2022 at 10:51 | comment | added | Vít Tuček | @AlexanderChervov How do you construct $\mathfrak{g}$-invariant polynomial on $\mathfrak{g}$ out of a $W$-invariant polynomial on $\mathfrak{h}$? | |
Nov 11, 2022 at 0:32 | history | edited | richrow | CC BY-SA 4.0 |
added 31 characters in body
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Nov 10, 2022 at 20:50 | comment | added | Alexander Chervov | Duflo isomorphism is general construction but it is not that explicit as these formulas | |
Nov 10, 2022 at 20:45 | comment | added | Alexander Chervov | Particular example for the determinant : mathoverflow.net/questions/92348/… More generally there is bunch of activity to write explicit nice formulas for the center of U(gl_n), such that we can control HC images, related to math. physics. | |
Nov 10, 2022 at 20:31 | comment | added | Alexander Chervov | en.wikipedia.org/wiki/Duflo_isomorphism | |
Nov 10, 2022 at 17:12 | history | asked | richrow | CC BY-SA 4.0 |