Timeline for Lie algebraic Grassmannian
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 17, 2014 at 6:51 | comment | added | abx | Yes, it is an algebraic subvariety (real if you are over $\Bbb{R}$). I would guess that for instance abelian Lie subalgebras would give singularities. I think you have to check examples, e.g. 2-dimensional subalgebras in $\mathfrak{gl}(2,\Bbb{R})$. | |
| Mar 16, 2014 at 22:29 | comment | added | Ali Taghavi | So can we say that it is an algebraic variety? And what would be the singularities? | |
| Mar 16, 2014 at 15:59 | comment | added | abx | I doubt very much that $Gr(k,n)_L$ is a submanifold. All I said is that it is compact, because it is the zero locus of a homomorphism of vector bundles on $Gr(k,n)$. | |
| Mar 16, 2014 at 15:42 | comment | added | Ali Taghavi | concerning the second part of my question, as you said, the characteristic classes of the restricted bundle =the restriction of characteristic classles. Yes it is a general fact. But it would be interesting to compute the ring cohomology of $Gr(k,n)_{L}$, then we restrict the generators of cohomology of ordinary grassmannian to $Gr(k,n)_{L}$. I explain what I am meaning, with the following particular question: assume that $1<k<n$ are given. Is there a Lie structure $L$ such that the canonical bundle is the trivial on $G(k,n)_{L}$ but this spacehas nontrivial cohomolgy? | |
| Mar 16, 2014 at 15:31 | comment | added | Ali Taghavi | thanks for the answer. I do not underestand something in your answer. To prove closedness of $G(k,n)_{L}$ are you defining a continous map from $G(k,n)$ to $\mathcal{Q}$, wh which send $V$ to $[V,V]$? the dimension of $[V,V]$ may vary by choosing different $V$'s. Is your map well defined? what is the exact definition of $\mathcal{Q}$? Morover, as I asked in my question, we want to know wether it is a compact submanifold. Is not possible that the zero locus would be a singular level set? | |
| Mar 16, 2014 at 13:12 | history | edited | abx | CC BY-SA 3.0 |
added 279 characters in body
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| Mar 16, 2014 at 11:58 | history | answered | abx | CC BY-SA 3.0 |