Skew-symmetry on Lie algebras over a field with characteristic not 2 [closed]
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I´d like to see the proof (I know it could be elemental) of this fact:
Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
Thank you all
lie-algebras
asked Dec 18 '18 at 17:18
LH8LH8
1438
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closed as off-topic by Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste Jan 14 at 19:53
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I´d like to see the proof (I know it could be elemental) of this fact:
Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
Thank you all
lie-algebras
asked Dec 18 '18 at 17:18
LH8LH8
1438
$endgroup$
closed as off-topic by Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste Jan 14 at 19:53
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
I'd like to see your work (I know it could be elemental) on this fact: Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
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– Namaste
Jan 14 at 19:49
2
$begingroup$
... and/or the source of the question (text (name and author and publication date)) and the subject you are studying, and your mathematical background, and/or an explanation as ti what motivated you to ask this question and how it is relevant to you and users on this site. Actually, please read How to ask a good question on math.se, as it well help you edit and improve this question after it is closed.
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– Namaste
Jan 14 at 19:53
add a comment |
$begingroup$
I´d like to see the proof (I know it could be elemental) of this fact:
Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
Thank you all
lie-algebras
asked Dec 18 '18 at 17:18
LH8LH8
1438
$endgroup$
I´d like to see the proof (I know it could be elemental) of this fact:
Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
Thank you all
lie-algebras
lie-algebras
asked Dec 18 '18 at 17:18
LH8LH8
1438
asked Dec 18 '18 at 17:18
LH8LH8
1438
asked Dec 18 '18 at 17:18
LH8LH8
1438
asked Dec 18 '18 at 17:18
LH8LH8
1438
asked Dec 18 '18 at 17:18
LH8LH8
1438
1438
closed as off-topic by Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste Jan 14 at 19:53
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste Jan 14 at 19:53
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Michael Rozenberg, Lee David Chung Lin, A. Pongrácz, José Carlos Santos, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
I'd like to see your work (I know it could be elemental) on this fact: Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
$endgroup$
– Namaste
Jan 14 at 19:49
2
$begingroup$
... and/or the source of the question (text (name and author and publication date)) and the subject you are studying, and your mathematical background, and/or an explanation as ti what motivated you to ask this question and how it is relevant to you and users on this site. Actually, please read How to ask a good question on math.se, as it well help you edit and improve this question after it is closed.
$endgroup$
– Namaste
Jan 14 at 19:53
add a comment |
1
$begingroup$
I'd like to see your work (I know it could be elemental) on this fact: Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
$endgroup$
– Namaste
Jan 14 at 19:49
2
$begingroup$
... and/or the source of the question (text (name and author and publication date)) and the subject you are studying, and your mathematical background, and/or an explanation as ti what motivated you to ask this question and how it is relevant to you and users on this site. Actually, please read How to ask a good question on math.se, as it well help you edit and improve this question after it is closed.
$endgroup$
– Namaste
Jan 14 at 19:53
1
1
$begingroup$
I'd like to see your work (I know it could be elemental) on this fact: Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
$endgroup$
– Namaste
Jan 14 at 19:49
$begingroup$
I'd like to see your work (I know it could be elemental) on this fact: Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
$endgroup$
– Namaste
Jan 14 at 19:49
2
2
$begingroup$
... and/or the source of the question (text (name and author and publication date)) and the subject you are studying, and your mathematical background, and/or an explanation as ti what motivated you to ask this question and how it is relevant to you and users on this site. Actually, please read How to ask a good question on math.se, as it well help you edit and improve this question after it is closed.
$endgroup$
– Namaste
Jan 14 at 19:53
$begingroup$
... and/or the source of the question (text (name and author and publication date)) and the subject you are studying, and your mathematical background, and/or an explanation as ti what motivated you to ask this question and how it is relevant to you and users on this site. Actually, please read How to ask a good question on math.se, as it well help you edit and improve this question after it is closed.
$endgroup$
– Namaste
Jan 14 at 19:53
add a comment |
2 Answers
2
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I suppose that this not what you really want to ask, since the equivalence of those two conditions holds by the simple fact that both conditions hold for any Lie algebra.
However, for any algebra $(A,star)$ over a field $F$ whose characteristic is not $2$, it is true that the conditions
$(forall x,yin A):xstar x=0$;- $(forall x,yin A):xstar y=-ystar x$
are equivalent. In fact, if the second conditions holds and if $xin A$, then $xstar x=-xstar x$, which means that $2xstar x=0$. SInce the characteistic is not $2$, it follows from this that $xstar x=0$. And if the first condition holds, then, if $x,yin A$,begin{align}0&=(x+y)star(x+y)\&=xstar x+xstar y+ystar x+ystar y\&=xstar y+ystar xend{align}and therefore $xstar y=-ystar x$.
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
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add a comment |
$begingroup$
If $[x,x]=0$ for al $x$ then $0=[x+y,x+y]=[x,x]+[x,y]+[y,x]+[y,y]=[x,y]+[y,x],$ so $[x,y]=-[y,x].$
Conversely, if $[x,y]=-[y,x]$ for all $x,y$ then $[x,x]=-[x,x],$ so $2[x,x]=0.$ Since $F$ has characteristic not 2 then $[x,x]=0.$
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I suppose that this not what you really want to ask, since the equivalence of those two conditions holds by the simple fact that both conditions hold for any Lie algebra.
However, for any algebra $(A,star)$ over a field $F$ whose characteristic is not $2$, it is true that the conditions
$(forall x,yin A):xstar x=0$;- $(forall x,yin A):xstar y=-ystar x$
are equivalent. In fact, if the second conditions holds and if $xin A$, then $xstar x=-xstar x$, which means that $2xstar x=0$. SInce the characteistic is not $2$, it follows from this that $xstar x=0$. And if the first condition holds, then, if $x,yin A$,begin{align}0&=(x+y)star(x+y)\&=xstar x+xstar y+ystar x+ystar y\&=xstar y+ystar xend{align}and therefore $xstar y=-ystar x$.
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
$endgroup$
add a comment |
$begingroup$
I suppose that this not what you really want to ask, since the equivalence of those two conditions holds by the simple fact that both conditions hold for any Lie algebra.
However, for any algebra $(A,star)$ over a field $F$ whose characteristic is not $2$, it is true that the conditions
$(forall x,yin A):xstar x=0$;- $(forall x,yin A):xstar y=-ystar x$
are equivalent. In fact, if the second conditions holds and if $xin A$, then $xstar x=-xstar x$, which means that $2xstar x=0$. SInce the characteistic is not $2$, it follows from this that $xstar x=0$. And if the first condition holds, then, if $x,yin A$,begin{align}0&=(x+y)star(x+y)\&=xstar x+xstar y+ystar x+ystar y\&=xstar y+ystar xend{align}and therefore $xstar y=-ystar x$.
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
$endgroup$
add a comment |
$begingroup$
I suppose that this not what you really want to ask, since the equivalence of those two conditions holds by the simple fact that both conditions hold for any Lie algebra.
However, for any algebra $(A,star)$ over a field $F$ whose characteristic is not $2$, it is true that the conditions
$(forall x,yin A):xstar x=0$;- $(forall x,yin A):xstar y=-ystar x$
are equivalent. In fact, if the second conditions holds and if $xin A$, then $xstar x=-xstar x$, which means that $2xstar x=0$. SInce the characteistic is not $2$, it follows from this that $xstar x=0$. And if the first condition holds, then, if $x,yin A$,begin{align}0&=(x+y)star(x+y)\&=xstar x+xstar y+ystar x+ystar y\&=xstar y+ystar xend{align}and therefore $xstar y=-ystar x$.
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
$endgroup$
I suppose that this not what you really want to ask, since the equivalence of those two conditions holds by the simple fact that both conditions hold for any Lie algebra.
However, for any algebra $(A,star)$ over a field $F$ whose characteristic is not $2$, it is true that the conditions
$(forall x,yin A):xstar x=0$;- $(forall x,yin A):xstar y=-ystar x$
are equivalent. In fact, if the second conditions holds and if $xin A$, then $xstar x=-xstar x$, which means that $2xstar x=0$. SInce the characteistic is not $2$, it follows from this that $xstar x=0$. And if the first condition holds, then, if $x,yin A$,begin{align}0&=(x+y)star(x+y)\&=xstar x+xstar y+ystar x+ystar y\&=xstar y+ystar xend{align}and therefore $xstar y=-ystar x$.
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
answered Dec 18 '18 at 17:27
José Carlos SantosJosé Carlos Santos
171k23132240
171k23132240
add a comment |
add a comment |
$begingroup$
If $[x,x]=0$ for al $x$ then $0=[x+y,x+y]=[x,x]+[x,y]+[y,x]+[y,y]=[x,y]+[y,x],$ so $[x,y]=-[y,x].$
Conversely, if $[x,y]=-[y,x]$ for all $x,y$ then $[x,x]=-[x,x],$ so $2[x,x]=0.$ Since $F$ has characteristic not 2 then $[x,x]=0.$
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
$endgroup$
add a comment |
$begingroup$
If $[x,x]=0$ for al $x$ then $0=[x+y,x+y]=[x,x]+[x,y]+[y,x]+[y,y]=[x,y]+[y,x],$ so $[x,y]=-[y,x].$
Conversely, if $[x,y]=-[y,x]$ for all $x,y$ then $[x,x]=-[x,x],$ so $2[x,x]=0.$ Since $F$ has characteristic not 2 then $[x,x]=0.$
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
$endgroup$
add a comment |
$begingroup$
If $[x,x]=0$ for al $x$ then $0=[x+y,x+y]=[x,x]+[x,y]+[y,x]+[y,y]=[x,y]+[y,x],$ so $[x,y]=-[y,x].$
Conversely, if $[x,y]=-[y,x]$ for all $x,y$ then $[x,x]=-[x,x],$ so $2[x,x]=0.$ Since $F$ has characteristic not 2 then $[x,x]=0.$
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
$endgroup$
If $[x,x]=0$ for al $x$ then $0=[x+y,x+y]=[x,x]+[x,y]+[y,x]+[y,y]=[x,y]+[y,x],$ so $[x,y]=-[y,x].$
Conversely, if $[x,y]=-[y,x]$ for all $x,y$ then $[x,x]=-[x,x],$ so $2[x,x]=0.$ Since $F$ has characteristic not 2 then $[x,x]=0.$
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
answered Dec 18 '18 at 17:28
positrón0802positrón0802
4,513520
4,513520
add a comment |
add a comment |
1
$begingroup$
I'd like to see your work (I know it could be elemental) on this fact: Let $L$ be a Lie algebra over a field $mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x in L$ if and only if$ [x,y]=−[y,x]$.
$endgroup$
– Namaste
Jan 14 at 19:49
2
$begingroup$
... and/or the source of the question (text (name and author and publication date)) and the subject you are studying, and your mathematical background, and/or an explanation as ti what motivated you to ask this question and how it is relevant to you and users on this site. Actually, please read How to ask a good question on math.se, as it well help you edit and improve this question after it is closed.
$endgroup$
– Namaste
Jan 14 at 19:53