Timeline for is connected complex Lie group with a trivial center linear?
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14 events
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| Mar 12, 2013 at 6:36 | vote | accept | Dima Sustretov | ||
| Dec 8, 2012 at 14:42 | comment | added | YCor | &Laurent: I haven't checked carefully right now but I think that if $G$ is a connected complex Lie group $f:G\to GL_n(\mathbf{C})$ is an injective continuous homomorphism, then the diagonal homomorphism into $GL_n(\mathbf{C})\times G/\overline{[G,G]}$ is proper (i.e. has a closed image). | |
| Dec 7, 2012 at 22:51 | answer | added | Venkataramana | timeline score: 4 | |
| Sep 23, 2011 at 5:52 | comment | added | Laurent Moret-Bailly | @Alain: this only gives an injective homomrphism into $GL_n$. Is it clearly an embedding, i.e. an isomorphism onto a closed subgroup? | |
| Sep 22, 2011 at 21:00 | comment | added | Alain Valette | @ Dmitry: The answer to your first question ("is a connected complex Lie group with a trivial center, a subgroup of $GL_n(\mathbb{C})$?") is clearly yes: look at the adjoint representation of a Lie group $G$, it is classical that its kernel is $Z(G)$. So if $Z(G)$ is trivial, $G$ is linear. | |
| Sep 22, 2011 at 19:50 | history | edited | Dima Sustretov | CC BY-SA 3.0 |
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| Sep 22, 2011 at 19:44 | comment | added | Dima Sustretov | ...and second, if it is an algebraic subgroup of $GL(n,\mathbb{C})$ (i.e. an algebraic variety over $\mathbb{C}$ with group law a regular map) | |
| Sep 22, 2011 at 19:41 | comment | added | Dima Sustretov | @Jim Humphreys: I must be missing something. In what respect is "complex Lie group" more narrowly defined? Doesn't "complex Lie group" meean a group which is a complex manifold with the group operation a holomorphic map? This definition seems to cover groups that are not subgroups of $GL(n,\mathbb{C})$, like complex tori. I wondered if the Rosenlicht's result generalised from algebraic varieties to complex manifolds. I perhaps mixed two notions together. First, I wonder if a connected complex Lie group with a trivial center is a subgroup of $GL(n,\mathbb{C})$ [continued] | |
| Sep 21, 2011 at 18:03 | comment | added | Dima Sustretov | I have just noticed that I have written "algebraic" where I should have written "complex Lie". I am sorry, the question should read: are connected complex Lie groups with trivial center linear algebraic? | |
| Sep 21, 2011 at 17:55 | history | edited | Dima Sustretov | CC BY-SA 3.0 |
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| Sep 21, 2011 at 13:53 | comment | added | Jim Humphreys | Rosenlicht's term "algebraic groups" includes non-affine examples such as abelian varieties. But a "complex Lie group" is more narrowly defined. So your last formulation of the question is unclear. Early in Chevalley's treatment of affine algebraic groups he shows that such a group is linear, whereas some familiar real Lie groups are not. The detailed structure/classification shows that complex semisimple Lie groups are indeed linear, but for solvable Lie groups you'd have to look further into Hochschild's work including his old book Structure of Lie Groups, etc. | |
| Sep 21, 2011 at 12:03 | history | edited | Dima Sustretov | CC BY-SA 3.0 |
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| Sep 21, 2011 at 11:38 | history | edited | Dima Sustretov | CC BY-SA 3.0 |
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| Sep 21, 2011 at 11:30 | history | asked | Dima Sustretov | CC BY-SA 3.0 |