Are simple Lie algebras complete? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Are there finite-dimensional Lie algebras which are not defined over the integers?Semisimple Lie algebras are perfect.Why, for nilpotent Lie Algebras, is the inclusion to the derivations $x mapsto ad_x$ not surjective?Whitehead's lemma (Lie algebras) for reductive Lie algebras.Solvable Lie algebra with non-characteristic nilradicalMost general definition of Borel and parabolic Lie algebras?Is the theorem of complete reducibility or the abstract Jordan decomposition needed for structure theory of semisimple Lie algebras?Conditions for lie algebras to be semi direct products of lie algebrasIndecomposable Lie algebrasThe opposite of Weyl's theorem on Lie algebras
Is it fair for a professor to grade us on the possession of past papers?
Can a new player join a group only when a new campaign starts?
How do I find out the mythology and history of my Fortress?
Did Deadpool rescue all of the X-Force?
Do any jurisdictions seriously consider reclassifying social media websites as publishers?
Trademark violation for app?
Using audio cues to encourage good posture
Effects on objects due to a brief relocation of massive amounts of mass
Does the Weapon Master feat grant you a fighting style?
How do I use the new nonlinear finite element in Mathematica 12 for this equation?
If Windows 7 doesn't support WSL, then what does Linux subsystem option mean?
Why do we need to use the builder design pattern when we can do the same thing with setters?
What's the meaning of "fortified infraction restraint"?
Chebyshev inequality in terms of RMS
Drawing without replacement: why is the order of draw irrelevant?
Do wooden building fires get hotter than 600°C?
Is there a kind of relay only consumes power when switching?
What is a fractional matching?
Central Vacuuming: Is it worth it, and how does it compare to normal vacuuming?
What would you call this weird metallic apparatus that allows you to lift people?
Why weren't discrete x86 CPUs ever used in game hardware?
Putting class ranking in CV, but against dept guidelines
What was the first language to use conditional keywords?
Can anything be seen from the center of the Boötes void? How dark would it be?
Are simple Lie algebras complete?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Are there finite-dimensional Lie algebras which are not defined over the integers?Semisimple Lie algebras are perfect.Why, for nilpotent Lie Algebras, is the inclusion to the derivations $x mapsto ad_x$ not surjective?Whitehead's lemma (Lie algebras) for reductive Lie algebras.Solvable Lie algebra with non-characteristic nilradicalMost general definition of Borel and parabolic Lie algebras?Is the theorem of complete reducibility or the abstract Jordan decomposition needed for structure theory of semisimple Lie algebras?Conditions for lie algebras to be semi direct products of lie algebrasIndecomposable Lie algebrasThe opposite of Weyl's theorem on Lie algebras
$begingroup$
Let $mathfrakg$ be a finite dimensional simple Lie algebra over a field of characteristic $p >0$.
Is $mathfrakg$ complete?
If not, under what conditions is $mathfrakg$ complete?
A Lie algebra is complete if its center is zero and all its derivations are inner.
lie-algebras positive-characteristic
asked Mar 27 at 15:42
keykey
85
$endgroup$
add a comment |
$begingroup$
Let $mathfrakg$ be a finite dimensional simple Lie algebra over a field of characteristic $p >0$.
Is $mathfrakg$ complete?
If not, under what conditions is $mathfrakg$ complete?
A Lie algebra is complete if its center is zero and all its derivations are inner.
lie-algebras positive-characteristic
asked Mar 27 at 15:42
keykey
85
$endgroup$
$begingroup$
One sufficient condition is that $dim(mathfrakg)<p$, as it ensures that the Killing form is nondegenerate.
$endgroup$
– YCor
Mar 29 at 8:54
add a comment |
$begingroup$
Let $mathfrakg$ be a finite dimensional simple Lie algebra over a field of characteristic $p >0$.
Is $mathfrakg$ complete?
If not, under what conditions is $mathfrakg$ complete?
A Lie algebra is complete if its center is zero and all its derivations are inner.
lie-algebras positive-characteristic
asked Mar 27 at 15:42
keykey
85
$endgroup$
Let $mathfrakg$ be a finite dimensional simple Lie algebra over a field of characteristic $p >0$.
Is $mathfrakg$ complete?
If not, under what conditions is $mathfrakg$ complete?
A Lie algebra is complete if its center is zero and all its derivations are inner.
lie-algebras positive-characteristic
lie-algebras positive-characteristic
asked Mar 27 at 15:42
keykey
85
asked Mar 27 at 15:42
keykey
85
edited Mar 27 at 19:00
key
asked Mar 27 at 15:42
keykey
85
asked Mar 27 at 15:42
keykey
85
asked Mar 27 at 15:42
keykey
85
85
$begingroup$
One sufficient condition is that $dim(mathfrakg)<p$, as it ensures that the Killing form is nondegenerate.
$endgroup$
– YCor
Mar 29 at 8:54
add a comment |
$begingroup$
One sufficient condition is that $dim(mathfrakg)<p$, as it ensures that the Killing form is nondegenerate.
$endgroup$
– YCor
Mar 29 at 8:54
$begingroup$
One sufficient condition is that $dim(mathfrakg)<p$, as it ensures that the Killing form is nondegenerate.
$endgroup$
– YCor
Mar 29 at 8:54
$begingroup$
One sufficient condition is that $dim(mathfrakg)<p$, as it ensures that the Killing form is nondegenerate.
$endgroup$
– YCor
Mar 29 at 8:54
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Examples of simple modular Lie algebras in characteristic $p>2$ having outer derivations are, among others, the Block algebras and Frank algebras, see Seligman. Other examples are certain restricted simple modular Lie algebras of Cartan-type, given in the book by Strade and Farnsteiner.
There is a short argument by Zassenhaus, that a Lie algebra with non-degenerate Killing form has only inner derivations, over a field of arbitrary characteristic.
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
$endgroup$
$begingroup$
A short argument of the latter fact (possibly the same) is given in Proposition 4.4 of these lecture notes: normalesup.org/~cornulier/Lie_cours.pdf
$endgroup$
– YCor
Mar 29 at 8:51
$begingroup$
Thank you Dietrich and YCor for your answer and comments. I am trying to figure out how best to proceed. Given the information that I have received (thank you all) I would change my original question. I think that probably means accepting an answer and then trying to formulate a better question. The examples in Dietrich's answer, answers the original question so I will accept it. Although all comments and answers were useful I can't upvote yet.
$endgroup$
– key
Apr 2 at 8:32
add a comment |
$begingroup$
Not necessarily. However, simple Lie algebras with non-degenerate Killing forms are always complete.
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
$endgroup$
$begingroup$
Thank you for your answer. I will check out what the Killing form is for a small dimensional case of a Lie algebra that I'm interested in.
$endgroup$
– key
Mar 27 at 19:03
$begingroup$
@key Which Lie algebra of low dimension are you interested then?
$endgroup$
– Dietrich Burde
Mar 30 at 9:31
add a comment |
Your Answer
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%2f3164665%2fare-simple-lie-algebras-complete%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Examples of simple modular Lie algebras in characteristic $p>2$ having outer derivations are, among others, the Block algebras and Frank algebras, see Seligman. Other examples are certain restricted simple modular Lie algebras of Cartan-type, given in the book by Strade and Farnsteiner.
There is a short argument by Zassenhaus, that a Lie algebra with non-degenerate Killing form has only inner derivations, over a field of arbitrary characteristic.
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
$endgroup$
$begingroup$
A short argument of the latter fact (possibly the same) is given in Proposition 4.4 of these lecture notes: normalesup.org/~cornulier/Lie_cours.pdf
$endgroup$
– YCor
Mar 29 at 8:51
$begingroup$
Thank you Dietrich and YCor for your answer and comments. I am trying to figure out how best to proceed. Given the information that I have received (thank you all) I would change my original question. I think that probably means accepting an answer and then trying to formulate a better question. The examples in Dietrich's answer, answers the original question so I will accept it. Although all comments and answers were useful I can't upvote yet.
$endgroup$
– key
Apr 2 at 8:32
add a comment |
$begingroup$
Examples of simple modular Lie algebras in characteristic $p>2$ having outer derivations are, among others, the Block algebras and Frank algebras, see Seligman. Other examples are certain restricted simple modular Lie algebras of Cartan-type, given in the book by Strade and Farnsteiner.
There is a short argument by Zassenhaus, that a Lie algebra with non-degenerate Killing form has only inner derivations, over a field of arbitrary characteristic.
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
$endgroup$
$begingroup$
A short argument of the latter fact (possibly the same) is given in Proposition 4.4 of these lecture notes: normalesup.org/~cornulier/Lie_cours.pdf
$endgroup$
– YCor
Mar 29 at 8:51
$begingroup$
Thank you Dietrich and YCor for your answer and comments. I am trying to figure out how best to proceed. Given the information that I have received (thank you all) I would change my original question. I think that probably means accepting an answer and then trying to formulate a better question. The examples in Dietrich's answer, answers the original question so I will accept it. Although all comments and answers were useful I can't upvote yet.
$endgroup$
– key
Apr 2 at 8:32
add a comment |
$begingroup$
Examples of simple modular Lie algebras in characteristic $p>2$ having outer derivations are, among others, the Block algebras and Frank algebras, see Seligman. Other examples are certain restricted simple modular Lie algebras of Cartan-type, given in the book by Strade and Farnsteiner.
There is a short argument by Zassenhaus, that a Lie algebra with non-degenerate Killing form has only inner derivations, over a field of arbitrary characteristic.
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
$endgroup$
Examples of simple modular Lie algebras in characteristic $p>2$ having outer derivations are, among others, the Block algebras and Frank algebras, see Seligman. Other examples are certain restricted simple modular Lie algebras of Cartan-type, given in the book by Strade and Farnsteiner.
There is a short argument by Zassenhaus, that a Lie algebra with non-degenerate Killing form has only inner derivations, over a field of arbitrary characteristic.
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
answered Mar 27 at 19:37
Dietrich BurdeDietrich Burde
82.3k649107
82.3k649107
$begingroup$
A short argument of the latter fact (possibly the same) is given in Proposition 4.4 of these lecture notes: normalesup.org/~cornulier/Lie_cours.pdf
$endgroup$
– YCor
Mar 29 at 8:51
$begingroup$
Thank you Dietrich and YCor for your answer and comments. I am trying to figure out how best to proceed. Given the information that I have received (thank you all) I would change my original question. I think that probably means accepting an answer and then trying to formulate a better question. The examples in Dietrich's answer, answers the original question so I will accept it. Although all comments and answers were useful I can't upvote yet.
$endgroup$
– key
Apr 2 at 8:32
add a comment |
$begingroup$
A short argument of the latter fact (possibly the same) is given in Proposition 4.4 of these lecture notes: normalesup.org/~cornulier/Lie_cours.pdf
$endgroup$
– YCor
Mar 29 at 8:51
$begingroup$
Thank you Dietrich and YCor for your answer and comments. I am trying to figure out how best to proceed. Given the information that I have received (thank you all) I would change my original question. I think that probably means accepting an answer and then trying to formulate a better question. The examples in Dietrich's answer, answers the original question so I will accept it. Although all comments and answers were useful I can't upvote yet.
$endgroup$
– key
Apr 2 at 8:32
$begingroup$
A short argument of the latter fact (possibly the same) is given in Proposition 4.4 of these lecture notes: normalesup.org/~cornulier/Lie_cours.pdf
$endgroup$
– YCor
Mar 29 at 8:51
$begingroup$
A short argument of the latter fact (possibly the same) is given in Proposition 4.4 of these lecture notes: normalesup.org/~cornulier/Lie_cours.pdf
$endgroup$
– YCor
Mar 29 at 8:51
$begingroup$
Thank you Dietrich and YCor for your answer and comments. I am trying to figure out how best to proceed. Given the information that I have received (thank you all) I would change my original question. I think that probably means accepting an answer and then trying to formulate a better question. The examples in Dietrich's answer, answers the original question so I will accept it. Although all comments and answers were useful I can't upvote yet.
$endgroup$
– key
Apr 2 at 8:32
$begingroup$
Thank you Dietrich and YCor for your answer and comments. I am trying to figure out how best to proceed. Given the information that I have received (thank you all) I would change my original question. I think that probably means accepting an answer and then trying to formulate a better question. The examples in Dietrich's answer, answers the original question so I will accept it. Although all comments and answers were useful I can't upvote yet.
$endgroup$
– key
Apr 2 at 8:32
add a comment |
$begingroup$
Not necessarily. However, simple Lie algebras with non-degenerate Killing forms are always complete.
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
$endgroup$
$begingroup$
Thank you for your answer. I will check out what the Killing form is for a small dimensional case of a Lie algebra that I'm interested in.
$endgroup$
– key
Mar 27 at 19:03
$begingroup$
@key Which Lie algebra of low dimension are you interested then?
$endgroup$
– Dietrich Burde
Mar 30 at 9:31
add a comment |
$begingroup$
Not necessarily. However, simple Lie algebras with non-degenerate Killing forms are always complete.
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
$endgroup$
$begingroup$
Thank you for your answer. I will check out what the Killing form is for a small dimensional case of a Lie algebra that I'm interested in.
$endgroup$
– key
Mar 27 at 19:03
$begingroup$
@key Which Lie algebra of low dimension are you interested then?
$endgroup$
– Dietrich Burde
Mar 30 at 9:31
add a comment |
$begingroup$
Not necessarily. However, simple Lie algebras with non-degenerate Killing forms are always complete.
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
$endgroup$
Not necessarily. However, simple Lie algebras with non-degenerate Killing forms are always complete.
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
answered Mar 27 at 15:47
José Carlos SantosJosé Carlos Santos
176k24134243
176k24134243
$begingroup$
Thank you for your answer. I will check out what the Killing form is for a small dimensional case of a Lie algebra that I'm interested in.
$endgroup$
– key
Mar 27 at 19:03
$begingroup$
@key Which Lie algebra of low dimension are you interested then?
$endgroup$
– Dietrich Burde
Mar 30 at 9:31
add a comment |
$begingroup$
Thank you for your answer. I will check out what the Killing form is for a small dimensional case of a Lie algebra that I'm interested in.
$endgroup$
– key
Mar 27 at 19:03
$begingroup$
@key Which Lie algebra of low dimension are you interested then?
$endgroup$
– Dietrich Burde
Mar 30 at 9:31
$begingroup$
Thank you for your answer. I will check out what the Killing form is for a small dimensional case of a Lie algebra that I'm interested in.
$endgroup$
– key
Mar 27 at 19:03
$begingroup$
Thank you for your answer. I will check out what the Killing form is for a small dimensional case of a Lie algebra that I'm interested in.
$endgroup$
– key
Mar 27 at 19:03
$begingroup$
@key Which Lie algebra of low dimension are you interested then?
$endgroup$
– Dietrich Burde
Mar 30 at 9:31
$begingroup$
@key Which Lie algebra of low dimension are you interested then?
$endgroup$
– Dietrich Burde
Mar 30 at 9:31
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%2f3164665%2fare-simple-lie-algebras-complete%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
$begingroup$
One sufficient condition is that $dim(mathfrakg)<p$, as it ensures that the Killing form is nondegenerate.
$endgroup$
– YCor
Mar 29 at 8:54