Stability under conjugation of a subvector space of Lie algebra of an affine linear algebraic group (in...
$begingroup$
Let $G$ a matrix Lie group and $mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true and can be seen via the adjoint representation and the exponentiation map. Does the same holds when $G$ is an affine linear algebraic group over a field $K$, $text{char}(K)>0$? If yes, how can I see this?
I do not have much experience with both Lie and algebraic groups. Please be gentle :)
EDIT: By conjugation I mean the action of the adjoint representation. In the matrix case this is equal to conjugation by an elemeng $gin G$ and stable subspaces under this action seem to be ideals, at least for classic Lie groups (manifolds). But what if $G=G(mathbb{F}_q)$ is a linear algebraic group?
lie-groups algebraic-groups matrix-exponential positive-characteristic
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
$endgroup$
add a comment |
$begingroup$
Let $G$ a matrix Lie group and $mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true and can be seen via the adjoint representation and the exponentiation map. Does the same holds when $G$ is an affine linear algebraic group over a field $K$, $text{char}(K)>0$? If yes, how can I see this?
I do not have much experience with both Lie and algebraic groups. Please be gentle :)
EDIT: By conjugation I mean the action of the adjoint representation. In the matrix case this is equal to conjugation by an elemeng $gin G$ and stable subspaces under this action seem to be ideals, at least for classic Lie groups (manifolds). But what if $G=G(mathbb{F}_q)$ is a linear algebraic group?
lie-groups algebraic-groups matrix-exponential positive-characteristic
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
$endgroup$
$begingroup$
A Lie algebra ideal $I$ is a subspace which is closed under the Lie bracket, i.e. $[I,I]subseteq I$ (so not stable under conjugation), see here.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 12:03
$begingroup$
@DietrichBurde: First of all thank you for your time!. You are telling being an ideal does not imply closure under conjugation, although I'm asking the opposite direction. This seems to be true, see here (proof of cor 4.2). Then one finds out that the adjoint rep is essentially conjugation by elements of G (see here). My question is how to obtain such analogy in positive characteristic. Do you think a should edit my question to be more clear?
$endgroup$
– Filippo Sneakerhead
Dec 22 '18 at 13:09
add a comment |
$begingroup$
Let $G$ a matrix Lie group and $mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true and can be seen via the adjoint representation and the exponentiation map. Does the same holds when $G$ is an affine linear algebraic group over a field $K$, $text{char}(K)>0$? If yes, how can I see this?
I do not have much experience with both Lie and algebraic groups. Please be gentle :)
EDIT: By conjugation I mean the action of the adjoint representation. In the matrix case this is equal to conjugation by an elemeng $gin G$ and stable subspaces under this action seem to be ideals, at least for classic Lie groups (manifolds). But what if $G=G(mathbb{F}_q)$ is a linear algebraic group?
lie-groups algebraic-groups matrix-exponential positive-characteristic
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
$endgroup$
Let $G$ a matrix Lie group and $mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true and can be seen via the adjoint representation and the exponentiation map. Does the same holds when $G$ is an affine linear algebraic group over a field $K$, $text{char}(K)>0$? If yes, how can I see this?
I do not have much experience with both Lie and algebraic groups. Please be gentle :)
EDIT: By conjugation I mean the action of the adjoint representation. In the matrix case this is equal to conjugation by an elemeng $gin G$ and stable subspaces under this action seem to be ideals, at least for classic Lie groups (manifolds). But what if $G=G(mathbb{F}_q)$ is a linear algebraic group?
lie-groups algebraic-groups matrix-exponential positive-characteristic
lie-groups algebraic-groups matrix-exponential positive-characteristic
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
edited Dec 22 '18 at 13:19
Filippo Sneakerhead
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
asked Dec 22 '18 at 10:46
Filippo SneakerheadFilippo Sneakerhead
285
285
$begingroup$
A Lie algebra ideal $I$ is a subspace which is closed under the Lie bracket, i.e. $[I,I]subseteq I$ (so not stable under conjugation), see here.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 12:03
$begingroup$
@DietrichBurde: First of all thank you for your time!. You are telling being an ideal does not imply closure under conjugation, although I'm asking the opposite direction. This seems to be true, see here (proof of cor 4.2). Then one finds out that the adjoint rep is essentially conjugation by elements of G (see here). My question is how to obtain such analogy in positive characteristic. Do you think a should edit my question to be more clear?
$endgroup$
– Filippo Sneakerhead
Dec 22 '18 at 13:09
add a comment |
$begingroup$
A Lie algebra ideal $I$ is a subspace which is closed under the Lie bracket, i.e. $[I,I]subseteq I$ (so not stable under conjugation), see here.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 12:03
$begingroup$
@DietrichBurde: First of all thank you for your time!. You are telling being an ideal does not imply closure under conjugation, although I'm asking the opposite direction. This seems to be true, see here (proof of cor 4.2). Then one finds out that the adjoint rep is essentially conjugation by elements of G (see here). My question is how to obtain such analogy in positive characteristic. Do you think a should edit my question to be more clear?
$endgroup$
– Filippo Sneakerhead
Dec 22 '18 at 13:09
$begingroup$
A Lie algebra ideal $I$ is a subspace which is closed under the Lie bracket, i.e. $[I,I]subseteq I$ (so not stable under conjugation), see here.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 12:03
$begingroup$
A Lie algebra ideal $I$ is a subspace which is closed under the Lie bracket, i.e. $[I,I]subseteq I$ (so not stable under conjugation), see here.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 12:03
$begingroup$
@DietrichBurde: First of all thank you for your time!. You are telling being an ideal does not imply closure under conjugation, although I'm asking the opposite direction. This seems to be true, see here (proof of cor 4.2). Then one finds out that the adjoint rep is essentially conjugation by elements of G (see here). My question is how to obtain such analogy in positive characteristic. Do you think a should edit my question to be more clear?
$endgroup$
– Filippo Sneakerhead
Dec 22 '18 at 13:09
$begingroup$
@DietrichBurde: First of all thank you for your time!. You are telling being an ideal does not imply closure under conjugation, although I'm asking the opposite direction. This seems to be true, see here (proof of cor 4.2). Then one finds out that the adjoint rep is essentially conjugation by elements of G (see here). My question is how to obtain such analogy in positive characteristic. Do you think a should edit my question to be more clear?
$endgroup$
– Filippo Sneakerhead
Dec 22 '18 at 13:09
add a comment |
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$begingroup$
A Lie algebra ideal $I$ is a subspace which is closed under the Lie bracket, i.e. $[I,I]subseteq I$ (so not stable under conjugation), see here.
$endgroup$
– Dietrich Burde
Dec 22 '18 at 12:03
$begingroup$
@DietrichBurde: First of all thank you for your time!. You are telling being an ideal does not imply closure under conjugation, although I'm asking the opposite direction. This seems to be true, see here (proof of cor 4.2). Then one finds out that the adjoint rep is essentially conjugation by elements of G (see here). My question is how to obtain such analogy in positive characteristic. Do you think a should edit my question to be more clear?
$endgroup$
– Filippo Sneakerhead
Dec 22 '18 at 13:09