Why is this universal cover isomorphic to $textSL_2(mathbbC)$? The 2019 Stack Overflow Developer Survey Results Are InIs the universal cover of an algebraic group an algebraic group?Betti Numbers and the Néron-Severi group.Why aren't those Cartier Divisors equivalent?Gluing sheaves togetherWhy is this a torusStatement of Chow's theoremCovering Maps Abelian Lie groupsDoes the singularities of black holes not being simply connected imply that the universal coverings of quantum field theory fail?Universal homeomorphisms between non-isomorphic schemesAbout the proof of Borel-Weil-Bott theorem: Two derived pushforward sheaves are isomorphic by descent theory
What is the meaning of Triage in Cybersec world?
Should I use my personal e-mail address, or my workplace one, when registering to external websites for work purposes?
What does Linus Torvalds mean when he says that Git "never ever" tracks a file?
Identify boardgame from Big movie
Can one be advised by a professor who is very far away?
Write faster on AT24C32
Earliest use of the term "Galois extension"?
During Temple times, who can butcher a kosher animal?
Can someone be penalized for an "unlawful" act if no penalty is specified?
Why was M87 targetted for the Event Horizon Telescope instead of Sagittarius A*?
What is the motivation for a law requiring 2 parties to consent for recording a conversation
Are there incongruent pythagorean triangles with the same perimeter and same area?
How come people say “Would of”?
Falsification in Math vs Science
Loose spokes after only a few rides
Button changing it's text & action. Good or terrible?
Can you compress metal and what would be the consequences?
Why isn't the circumferential light around the M87 black hole's event horizon symmetric?
How to answer pointed "are you quitting" questioning when I don't want them to suspect
One word riddle: Vowel in the middle
Why did Acorn's A3000 have red function keys?
Is a "Democratic" Oligarchy-Style System Possible?
FPGA - DIY Programming
Why do UK politicians seemingly ignore opinion polls on Brexit?
Why is this universal cover isomorphic to $textSL_2(mathbbC)$?
The 2019 Stack Overflow Developer Survey Results Are InIs the universal cover of an algebraic group an algebraic group?Betti Numbers and the Néron-Severi group.Why aren't those Cartier Divisors equivalent?Gluing sheaves togetherWhy is this a torusStatement of Chow's theoremCovering Maps Abelian Lie groupsDoes the singularities of black holes not being simply connected imply that the universal coverings of quantum field theory fail?Universal homeomorphisms between non-isomorphic schemesAbout the proof of Borel-Weil-Bott theorem: Two derived pushforward sheaves are isomorphic by descent theory
$begingroup$
In Lurie's proof of the Borel-Weil theorem http://www.math.harvard.edu/~lurie/papers/bwb.pdf, he states that the universal cover of a Levi factor of $S$, where $Ccup U'B=SB$, is isomorphic to $textSL_2(mathbbC)$. Why is this the case? Could someone please give a reference for this?
Here $C$ is the Bruhat cell corresponding to a simple root.
algebraic-geometry algebraic-groups
asked Mar 23 at 16:55
theathea
112
$endgroup$
add a comment |
$begingroup$
In Lurie's proof of the Borel-Weil theorem http://www.math.harvard.edu/~lurie/papers/bwb.pdf, he states that the universal cover of a Levi factor of $S$, where $Ccup U'B=SB$, is isomorphic to $textSL_2(mathbbC)$. Why is this the case? Could someone please give a reference for this?
Here $C$ is the Bruhat cell corresponding to a simple root.
algebraic-geometry algebraic-groups
asked Mar 23 at 16:55
theathea
112
$endgroup$
add a comment |
$begingroup$
In Lurie's proof of the Borel-Weil theorem http://www.math.harvard.edu/~lurie/papers/bwb.pdf, he states that the universal cover of a Levi factor of $S$, where $Ccup U'B=SB$, is isomorphic to $textSL_2(mathbbC)$. Why is this the case? Could someone please give a reference for this?
Here $C$ is the Bruhat cell corresponding to a simple root.
algebraic-geometry algebraic-groups
asked Mar 23 at 16:55
theathea
112
$endgroup$
In Lurie's proof of the Borel-Weil theorem http://www.math.harvard.edu/~lurie/papers/bwb.pdf, he states that the universal cover of a Levi factor of $S$, where $Ccup U'B=SB$, is isomorphic to $textSL_2(mathbbC)$. Why is this the case? Could someone please give a reference for this?
Here $C$ is the Bruhat cell corresponding to a simple root.
algebraic-geometry algebraic-groups
algebraic-geometry algebraic-groups
asked Mar 23 at 16:55
theathea
112
asked Mar 23 at 16:55
theathea
112
edited Mar 23 at 17:33
thea
asked Mar 23 at 16:55
theathea
112
asked Mar 23 at 16:55
theathea
112
asked Mar 23 at 16:55
theathea
112
112
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
There is a lot going on in that sentence. A general reference for the structure theory of reductive algebraic groups would be the book by Springer, or Borel, or any of the online notes circulating. The ones by Herzig and Murnaghan are both useful for this stuff. Specifically, see theorem 180 and the rest of Ch. 6 in these notes. The example
to keep in mind that makes reasoning about these things easy is the standard block-upper-triangular parabolic subgroups of $mathrmGL_n$.
Anyway, by construction the Lie algebra of $S$ is isomorphic to $mathfraksl_2,mathbbC)$, so by the theory of Lie groups, $S$ is covered by $mathrmSL(2,mathbbC)$. In more detail, if $alpha=epsilon_1-epsilon_2$ is the positive root in question for $mathrmGL_n$, then I think $S$ is can be taken to be generated by the upper-triangular matrices with 1s on the diagonal and and the matrices with 1s on the diagonal, and a single other entry at $(2,1)$, corresponding to $-alpha$.
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3159567%2fwhy-is-this-universal-cover-isomorphic-to-textsl-2-mathbbc%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
There is a lot going on in that sentence. A general reference for the structure theory of reductive algebraic groups would be the book by Springer, or Borel, or any of the online notes circulating. The ones by Herzig and Murnaghan are both useful for this stuff. Specifically, see theorem 180 and the rest of Ch. 6 in these notes. The example
to keep in mind that makes reasoning about these things easy is the standard block-upper-triangular parabolic subgroups of $mathrmGL_n$.
Anyway, by construction the Lie algebra of $S$ is isomorphic to $mathfraksl_2,mathbbC)$, so by the theory of Lie groups, $S$ is covered by $mathrmSL(2,mathbbC)$. In more detail, if $alpha=epsilon_1-epsilon_2$ is the positive root in question for $mathrmGL_n$, then I think $S$ is can be taken to be generated by the upper-triangular matrices with 1s on the diagonal and and the matrices with 1s on the diagonal, and a single other entry at $(2,1)$, corresponding to $-alpha$.
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
$endgroup$
add a comment |
$begingroup$
There is a lot going on in that sentence. A general reference for the structure theory of reductive algebraic groups would be the book by Springer, or Borel, or any of the online notes circulating. The ones by Herzig and Murnaghan are both useful for this stuff. Specifically, see theorem 180 and the rest of Ch. 6 in these notes. The example
to keep in mind that makes reasoning about these things easy is the standard block-upper-triangular parabolic subgroups of $mathrmGL_n$.
Anyway, by construction the Lie algebra of $S$ is isomorphic to $mathfraksl_2,mathbbC)$, so by the theory of Lie groups, $S$ is covered by $mathrmSL(2,mathbbC)$. In more detail, if $alpha=epsilon_1-epsilon_2$ is the positive root in question for $mathrmGL_n$, then I think $S$ is can be taken to be generated by the upper-triangular matrices with 1s on the diagonal and and the matrices with 1s on the diagonal, and a single other entry at $(2,1)$, corresponding to $-alpha$.
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
$endgroup$
add a comment |
$begingroup$
There is a lot going on in that sentence. A general reference for the structure theory of reductive algebraic groups would be the book by Springer, or Borel, or any of the online notes circulating. The ones by Herzig and Murnaghan are both useful for this stuff. Specifically, see theorem 180 and the rest of Ch. 6 in these notes. The example
to keep in mind that makes reasoning about these things easy is the standard block-upper-triangular parabolic subgroups of $mathrmGL_n$.
Anyway, by construction the Lie algebra of $S$ is isomorphic to $mathfraksl_2,mathbbC)$, so by the theory of Lie groups, $S$ is covered by $mathrmSL(2,mathbbC)$. In more detail, if $alpha=epsilon_1-epsilon_2$ is the positive root in question for $mathrmGL_n$, then I think $S$ is can be taken to be generated by the upper-triangular matrices with 1s on the diagonal and and the matrices with 1s on the diagonal, and a single other entry at $(2,1)$, corresponding to $-alpha$.
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
$endgroup$
There is a lot going on in that sentence. A general reference for the structure theory of reductive algebraic groups would be the book by Springer, or Borel, or any of the online notes circulating. The ones by Herzig and Murnaghan are both useful for this stuff. Specifically, see theorem 180 and the rest of Ch. 6 in these notes. The example
to keep in mind that makes reasoning about these things easy is the standard block-upper-triangular parabolic subgroups of $mathrmGL_n$.
Anyway, by construction the Lie algebra of $S$ is isomorphic to $mathfraksl_2,mathbbC)$, so by the theory of Lie groups, $S$ is covered by $mathrmSL(2,mathbbC)$. In more detail, if $alpha=epsilon_1-epsilon_2$ is the positive root in question for $mathrmGL_n$, then I think $S$ is can be taken to be generated by the upper-triangular matrices with 1s on the diagonal and and the matrices with 1s on the diagonal, and a single other entry at $(2,1)$, corresponding to $-alpha$.
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
answered Mar 26 at 5:18
Stefan DawydiakStefan Dawydiak
37929
37929
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3159567%2fwhy-is-this-universal-cover-isomorphic-to-textsl-2-mathbbc%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
