Mikhail Borovoi
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May 18 |
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Groupoids vs. action groupoids @Mkouboi: I do not assume that there is a $\Gamma$-fixed point in $X$, and therefore $\Gamma$ does not act on the fibres $A(x,-)$ or $A(-,x)$ or $A(x,x)$. |
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May 17 |
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Groupoids vs. action groupoids @Mkouboi: I do not understand your answer. You write: "Moreover, any connected Γ-groupoid is in particular connected and hence in the form G⋉X. By construction of the group G, Γ acts compatibly on it and X..." But the construction of $G$ from the groupoid A⇉X is not functorial, so I do not understand how you get an action of Γ on G. Could you please add details? |
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May 17 |
asked | Groupoids vs. action groupoids |
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May 15 |
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Equivalence and weak equivalence of groupoids @The User: Thank you for the answer and the editing suggestion! |
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May 15 |
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Equivalence and weak equivalence of groupoids added 2 characters in body |
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May 15 |
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Equivalence and weak equivalence of groupoids The braces that I typed in the displayed formula are not visible! Maybe somebody can edit the formula to make the braces visible... Thank you, |
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May 15 |
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Equivalence and weak equivalence of groupoids added 1 characters in body |
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May 15 |
asked | Equivalence and weak equivalence of groupoids |
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May 11 |
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Second nonabelian group cohomology: cocycles vs. gerbes added 191 characters in body |
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May 11 |
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Second nonabelian group cohomology: cocycles vs. gerbes @Urs: Thank you! I will try to read your preprint. Still, I would appreciate if you could write a more explicit answer... |
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May 10 |
awarded | ● Nice Question |
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May 10 |
revised |
Second nonabelian group cohomology: cocycles vs. gerbes edited tags |
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May 10 |
asked | Second nonabelian group cohomology: cocycles vs. gerbes |
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May 10 |
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solve the singularities of parabolic orbits of schubert cells I remark that the OP has never voted up or down any question or answer. |
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Apr 28 |
accepted | gluing gerbes over a spectrum of a field |
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Apr 23 |
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gluing gerbes over a spectrum of a field added 41 characters in body |
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Apr 23 |
answered | gluing gerbes over a spectrum of a field |
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Apr 18 |
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Connected groupoids and action groupoids @Omar: Thank you, it is fine now. |
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Apr 18 |
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Connected groupoids and action groupoids @Sam: Thank you for detailed answer! Unfortunately, I cannot accept two answers, otherwise I would accept your answer as well... |
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Apr 17 |
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Constructing a stack (gerbe) from a connected groupoid @DavidRoberts: Thank you! |
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Apr 17 |
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Connected groupoids and action groupoids @Omar: Thank you, now we have a group $G$ and a action of $G$ on $X$. How can we define an isomorphism between $G\ltimes X$ and $A$? Please kindly add more details when you can. |
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Apr 17 |
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Connected groupoids and action groupoids @Omar: You write: "Any group $G$ with subgroup $H$ of the correct index, the action of $G$ on the set of cosets of $H$ has action groupoid isomorphic to $A$". How can one construct such an isomorphism? |
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Apr 16 |
asked | Connected groupoids and action groupoids |
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Apr 16 |
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Constructing a stack (gerbe) from a connected groupoid @DavidCarchedi: Thank you! However, I am not quite familiar with your notations $Set^{B\Gamma}$, $Gpd(Set^{B\Gamma})$ and $Gpd^{B\Gamma}$. Your links did not help me. Could you please explain me these notations? Also, what is a weak functor? Where can I read about all this stuff? |
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Apr 16 |
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Constructing a stack (gerbe) from a connected groupoid @DavidRoberts: Thank you for your detailed answer. This is exactly the kind of answer that I wanted to get! |
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Apr 15 |
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Constructing a stack (gerbe) from a connected groupoid Probably when you write "such that $\pi(a(p,f))=\pi(a)$", you mean $\pi(a(p,f))=\pi(p)$. When you write "in addition, we demand that the map $\dots$ is an isomorphism", you mean a bijection. Is this correct? |
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Apr 15 |
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Constructing a stack (gerbe) from a connected groupoid I do not understand the line: "isomorphic to $T\times_X S$ for some map $T\to X$". What is the map $S\to X$ in the fibered product? Is it a typo? |
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Apr 15 |
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Constructing a stack (gerbe) from a connected groupoid @David: If $X\rtimes G$ is the action groupoid corresponding to an action of a $\Gamma$-group $G$ on a $\Gamma$-set $X$, then I should take the groupoid of principal $G$-bundles dominating $X$. What are principal $\mathcal{G}$-bundles when $\mathcal{G}$ is a groupoid? |
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Apr 15 |
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Constructing a stack (gerbe) from a connected groupoid @David: Yes, the topology is just surjections. |
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Apr 14 |
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Constructing a stack (gerbe) from a connected groupoid @Simon: I want to construct a gerbe starting from a connected $\Gamma$-groupoid, and to describe the cohomology class of this gerbe in terms of my $\Gamma$-groupoid, thereby explicitly relating the paper of Springer on non-abelian $H^2$ in Galois cohomology with the book by Giraud. |
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Apr 14 |
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Constructing a stack (gerbe) from a connected groupoid @Simon: What you propose looks fine. Why is it not what we want to do?! |
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Apr 14 |
asked | Constructing a stack (gerbe) from a connected groupoid |
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Mar 28 |
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Quotient of a reductive group by a non-smooth subgroup @xunan: Thank you! |
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Mar 28 |
asked | Quotient of a reductive group by a non-smooth subgroup |
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Feb 21 |
awarded | ● Yearling |
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Feb 18 |
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The torsion point count in higher dimension @Will: The Mumford-Tate group should be reductive, and it is never semisimple. For example, the Mumford-Tate group of an abelian variety of CM-type is a torus. |
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Feb 17 |
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On Tamarkin’s proof of Etingof-Kazhdan quantization of Lie bialgebra Spelling corrected |
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Feb 14 |
answered | on z-extensions |
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Feb 11 |
accepted | What are the symmetries of a principal homogeneous bundle? |
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Feb 10 |
answered | What are the symmetries of a principal homogeneous bundle? |
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Feb 9 |
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Algebraic Topology Beyond the Basics:Any Texts Bridging The Gap? Welcome to Math Overflow, Mikhail, and thank you for putting the English version of this wonderful book on the Web! |
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Feb 7 |
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what are the possible CM-fields of PEL type shimura varieties ? @TOM: I agree with your condition (i). Why don't you answer your own question? Try to write explicitly, what semisimple group $G$ you consider, what is a maximal torus $T\subset G$ (compact over $\mathbb{R}$), such a torus gives $L$. You get $E$ from the construction of the family of abelian varieties. Note that the dimension of $L\otimes_F E$ over $\mathbb{Q}$ is $4\cdot[F:\mathbb{Q}]$ - two times the dimension of an abelian variety. |
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Feb 4 |
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what are the possible CM-fields of PEL type shimura varieties ? @TOM: OK, now what do you mean by the possible CM-fields (algebras) of the CM-points? Do you mean the algebras of endomorphisms of the corresponding abelian varieties of CM-type, or maximal commutative subalgebras of these algebras, or maybe the "dual fields"? |
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Feb 2 |
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what are the possible CM-fields of PEL type shimura varieties ? (Continued) Is your question about CM-points of ${\rm Sh}(G,h_0)$ or of this auxiliary Shimura variety of PEL-type, or of ${\rm Sh}(G,h_0)$, but with respect to the weakly canonical model constructed using this embedding? Please elaborate! |
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Feb 2 |
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what are the possible CM-fields of PEL type shimura varieties ? @TOM: OK, Deligne's paper is on my table, opened on Section 6 "Modèles étranges". Still I do not understand your question. Deligne constructs a Shimura variety ${\rm Sh}_{\mathbb{C}}(G,h_0)$, which is not of PEL-type. In order to construct a canonical model ${\rm Sh}(G,h_0)$ of this Shimura variety, he choses a totally imaginary quadratic extension $Z/F$ and embeds ${\rm Sh}_{\mathbb{C}}(G,h_0)$ into a Shimura variety of PEL-type constructed using $Z$. |
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Jan 17 |
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Transitive action on the sphere @Nerd-Math: Not true. Any complex representation $\rho$ of dimension $n=4r$ of the compact group $G$ is a direct sum of irreducible representations $\rho_i$. Clearly ${\rm dim}\ \rho_i\le n=4r$. We have seen that for $r\ge 3$ our group $G$ has no nontrivial irreducible representations of dimension $\le 4r$. Thus each $\rho_i$ is trivial, hence $\rho$ is trivial. |
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Jan 16 |
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Non-trivial representation of second-smallest dimension @Nerd-Math: I don't think you can find details in the literature. You have to perform the calculations yourself! You can find useful formulas for dimensions of certain irreducible representations in Table 5 of the book "Lie Groups and Algebraic Groups" by Onishchik and Vinberg (again, given without proofs). |
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Jan 15 |
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Non-trivial representation of second-smallest dimension @Gabriel-Kj: Since you have accepted my answer and have thanked Jim, you can also vote up both answers... |
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Jan 15 |
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Non-trivial representation of second-smallest dimension added 106 characters in body |
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Jan 15 |
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Non-trivial representation of second-smallest dimension @Jim: You can find the Russian version for free in mathnet.ru/links/419af2a1d9d33839a49ff8898866b056/… . There is just a table, no details of calculations. |
