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Feb
19
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 thank you ! At the moment I am afraid that even in the case of S^2 desired quantization does not exists :( because it seems generators x_i which form su(2), but su(2) seems does not have appropriate infinite-dime irep - I am not sure - probably I will write separate MO question on it...
Feb
18
comment A representation similar to coadjoint representation?
@whitejet it would be enough prove existence or NON-existence of Hilbert space representation for non-integer "R" for the algebra in question. Unfortunately it seems it does NOT exist :( Berezin-Toeplitz does not give any new information - because for integer "R" it is "obvious" that there is finite-dim irrep, just basic rep-theory for su(2).
Feb
16
comment Distinguishing the Duflo star product
Why " Duflo theorem can be rephrased by saying that there exists a star-product mdmd whose restriction to the Poisson center is undeformed, " ?
Feb
16
comment A representation similar to coadjoint representation?
What is " This is the well-known fuzzy sphere" ? See question: mathoverflow.net/questions/231072/…
Feb
16
comment Are all the Lie bialgebra structure on $sl_n$ coboundary?
As Makoto Yamashita writes it is because H^2(semisimpleLieAlg) = 0 - "Whitehead's lemma" en.wikipedia.org/wiki/Whitehead%27s_lemma_(Lie_algebras)
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 there are other properties like that existence of Trace (mentioned in main body of question), quantization of Lagrangian submanifolds, probably symplectomorphisms should be quantized somehow, some functoriality ...
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 there are other properties, but may be they follow from the previous one. Properties: 1) in well-known examples like R^2n, g^*, NC-torus we should get the same as we know 2) Hilbert space representation if \omega is INTEGRAL cohomology class and dimention of that Hilbert space shoudl be equal to Fedosov's index formula
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 you are welcome for questions. Yes there are some properties which are expected to hold true. Probably the main property: "agreemet with deformation quantization" - I mean any quantization should give rise to family of algebras depending on "h", as a linear space these algebras should be of the same size as original algebra, so using some identification of vector spaces we "can" get star product depending on "h" - this star product should be equivalent to Kontsevich's ones. Power of Kontsevich theorem is that he establish bijection: Poisson-StarProduct.
Feb
15
comment Mirror Symmetry for Flag Supermanifolds
Good question. Is there any examples of mirror for any supermanifold ? nor particulary flag ?
Feb
15
comment Low-dimensional classical r-matrices
For quantum YBE there is paper: "All solutions to the constant quantum Yang-Baxter equation in two dimensions" Jarmo Hietarinta sciencedirect.com/science/article/pii/037596019290044M , about classical I do not know off hand
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 I just beleive "correct" quantization is unique ! that is why the answer would be unique, even if I do know the correct procedure... So it is reasonable to ask about the properties of that "unknown" procedure if we believe that it exists and unique...
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 deformation quantization is very good , but it does not give analysis , "correct" analytical quantizatin should agree with deformation quantization but add some more - C^* or whatever analytical structure ...
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 I guess that not a quantization we would expect as "correct quantization" as that paper by Cattaneo analyzes ...
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 See mathoverflow.net/questions/231072/… - I do not know what is correct "non-pertrubative" quantization in general, but in that particular example it must give that answer - that is I am sure.
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 "In what kind of quantization are you interested? " you will be surpised, but I think that nobody knows how to marriage analysis and quantization in the right way - or may be someone knows like Nick Landsman but it is not widely accepted. There are several properties which I would expect from the "correct quantization" and several examples where we think we know the answer - like NC tori - without knowledge what is correct non-pertrubative quantization in general ...
Feb
15
comment Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?
@user40276 "Every symplectic manifold can be quantized using symplectic groupoids " in what sense in can be quantize ? As far as I heard this quantization is not even deformation quantization , but some approximation to it: arxiv.org/abs/math/0312380 . Although I am not expert - can you clarify in what sense it "quantizes" ?