A theorem about semisimple Lie algebra

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I'm reading Lie Groups, Lie Algebras , and Representations (first edition) by Hall and I'm stuck by a theorem (The author did not prove it):

Theorem 6.6 A complex Lie algebra is semisimple if and only if it is isomorphic to the complexification of the Lie algebra of a simply-connected compact matrix Lie group.

Through this theorem (assume it is right) we can prove that $mathfraksl(2,mathbbC)$ is semisimple, but $mathfraksl(2,mathbbC)$ is in fact simple. There is a proof : Example ideal of $mathfraksl(2,mathbbC)$

So whether is the theorem right or wrong? If it is wrong, is there any correct theorem related to it?

lie-algebras

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asked Jul 23 at 15:06

Kokuto

31

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I'm reading Lie Groups, Lie Algebras , and Representations (first edition) by Hall and I'm stuck by a theorem (The author did not prove it):

Theorem 6.6 A complex Lie algebra is semisimple if and only if it is isomorphic to the complexification of the Lie algebra of a simply-connected compact matrix Lie group.

Through this theorem (assume it is right) we can prove that $mathfraksl(2,mathbbC)$ is semisimple, but $mathfraksl(2,mathbbC)$ is in fact simple. There is a proof : Example ideal of $mathfraksl(2,mathbbC)$

So whether is the theorem right or wrong? If it is wrong, is there any correct theorem related to it?

lie-algebras

share|cite|improve this question

asked Jul 23 at 15:06

Kokuto

31

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I'm reading Lie Groups, Lie Algebras , and Representations (first edition) by Hall and I'm stuck by a theorem (The author did not prove it):

Theorem 6.6 A complex Lie algebra is semisimple if and only if it is isomorphic to the complexification of the Lie algebra of a simply-connected compact matrix Lie group.

Through this theorem (assume it is right) we can prove that $mathfraksl(2,mathbbC)$ is semisimple, but $mathfraksl(2,mathbbC)$ is in fact simple. There is a proof : Example ideal of $mathfraksl(2,mathbbC)$

So whether is the theorem right or wrong? If it is wrong, is there any correct theorem related to it?

lie-algebras

share|cite|improve this question

asked Jul 23 at 15:06

Kokuto

31

I'm reading Lie Groups, Lie Algebras , and Representations (first edition) by Hall and I'm stuck by a theorem (The author did not prove it):

Theorem 6.6 A complex Lie algebra is semisimple if and only if it is isomorphic to the complexification of the Lie algebra of a simply-connected compact matrix Lie group.

Through this theorem (assume it is right) we can prove that $mathfraksl(2,mathbbC)$ is semisimple, but $mathfraksl(2,mathbbC)$ is in fact simple. There is a proof : Example ideal of $mathfraksl(2,mathbbC)$

So whether is the theorem right or wrong? If it is wrong, is there any correct theorem related to it?

lie-algebras

share|cite|improve this question

asked Jul 23 at 15:06

Kokuto

31

share|cite|improve this question

share|cite|improve this question

asked Jul 23 at 15:06

Kokuto

31

asked Jul 23 at 15:06

Kokuto

31

asked Jul 23 at 15:06

Kokuto

31

31

add a comment | 

add a comment | 

1 Answer
1

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up vote
1
down vote



accepted

From your question, one could assume that there's some contradiction somewhere. Where is it? Yes, $mathfraksl(2,mathbbC)$ is simple and, yes, $mathfraksl(2,mathbbC)$ is semisimple. That's not a problem, since every simple Lie algebra is also semisimple.

share|cite|improve this answer

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So you mean Theorem 6.6 is correct ? And could you give me a hint on why "every simple Lie algebra is also semisimple"?
  – Kokuto
  Jul 23 at 15:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Kokuto This is more or less trivial, depending on the definition of semisimple Lie algebra. In most cases (but IIRC not in Hall's book) semisimple Lie algebra is defined to be a direct sum of simple Lie algebras, of which simple Lie algebra is obviously an example.
  – Cave Johnson
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Yes, theorem 6.6 is correct.. And you will find a proof of the fact that every simple Lie algebra is also semisimple here.
  – José Carlos Santos
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @José Carlos Santos Well. Thanks a lot :)
  – Kokuto
  Jul 23 at 15:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Cave Johnson Thanks for your comment :)
  – Kokuto
  Jul 23 at 15:45
  

  
  
  
  

add a comment | 

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

From your question, one could assume that there's some contradiction somewhere. Where is it? Yes, $mathfraksl(2,mathbbC)$ is simple and, yes, $mathfraksl(2,mathbbC)$ is semisimple. That's not a problem, since every simple Lie algebra is also semisimple.

share|cite|improve this answer

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So you mean Theorem 6.6 is correct ? And could you give me a hint on why "every simple Lie algebra is also semisimple"?
  – Kokuto
  Jul 23 at 15:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Kokuto This is more or less trivial, depending on the definition of semisimple Lie algebra. In most cases (but IIRC not in Hall's book) semisimple Lie algebra is defined to be a direct sum of simple Lie algebras, of which simple Lie algebra is obviously an example.
  – Cave Johnson
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Yes, theorem 6.6 is correct.. And you will find a proof of the fact that every simple Lie algebra is also semisimple here.
  – José Carlos Santos
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @José Carlos Santos Well. Thanks a lot :)
  – Kokuto
  Jul 23 at 15:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Cave Johnson Thanks for your comment :)
  – Kokuto
  Jul 23 at 15:45
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

From your question, one could assume that there's some contradiction somewhere. Where is it? Yes, $mathfraksl(2,mathbbC)$ is simple and, yes, $mathfraksl(2,mathbbC)$ is semisimple. That's not a problem, since every simple Lie algebra is also semisimple.

share|cite|improve this answer

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So you mean Theorem 6.6 is correct ? And could you give me a hint on why "every simple Lie algebra is also semisimple"?
  – Kokuto
  Jul 23 at 15:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Kokuto This is more or less trivial, depending on the definition of semisimple Lie algebra. In most cases (but IIRC not in Hall's book) semisimple Lie algebra is defined to be a direct sum of simple Lie algebras, of which simple Lie algebra is obviously an example.
  – Cave Johnson
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Yes, theorem 6.6 is correct.. And you will find a proof of the fact that every simple Lie algebra is also semisimple here.
  – José Carlos Santos
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @José Carlos Santos Well. Thanks a lot :)
  – Kokuto
  Jul 23 at 15:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Cave Johnson Thanks for your comment :)
  – Kokuto
  Jul 23 at 15:45
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

From your question, one could assume that there's some contradiction somewhere. Where is it? Yes, $mathfraksl(2,mathbbC)$ is simple and, yes, $mathfraksl(2,mathbbC)$ is semisimple. That's not a problem, since every simple Lie algebra is also semisimple.

share|cite|improve this answer

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

From your question, one could assume that there's some contradiction somewhere. Where is it? Yes, $mathfraksl(2,mathbbC)$ is simple and, yes, $mathfraksl(2,mathbbC)$ is semisimple. That's not a problem, since every simple Lie algebra is also semisimple.

share|cite|improve this answer

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

share|cite|improve this answer

share|cite|improve this answer

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

answered Jul 23 at 15:20

José Carlos Santos

113k1698176

113k1698176

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So you mean Theorem 6.6 is correct ? And could you give me a hint on why "every simple Lie algebra is also semisimple"?
  – Kokuto
  Jul 23 at 15:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Kokuto This is more or less trivial, depending on the definition of semisimple Lie algebra. In most cases (but IIRC not in Hall's book) semisimple Lie algebra is defined to be a direct sum of simple Lie algebras, of which simple Lie algebra is obviously an example.
  – Cave Johnson
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Yes, theorem 6.6 is correct.. And you will find a proof of the fact that every simple Lie algebra is also semisimple here.
  – José Carlos Santos
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @José Carlos Santos Well. Thanks a lot :)
  – Kokuto
  Jul 23 at 15:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Cave Johnson Thanks for your comment :)
  – Kokuto
  Jul 23 at 15:45
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So you mean Theorem 6.6 is correct ? And could you give me a hint on why "every simple Lie algebra is also semisimple"?
  – Kokuto
  Jul 23 at 15:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Kokuto This is more or less trivial, depending on the definition of semisimple Lie algebra. In most cases (but IIRC not in Hall's book) semisimple Lie algebra is defined to be a direct sum of simple Lie algebras, of which simple Lie algebra is obviously an example.
  – Cave Johnson
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Yes, theorem 6.6 is correct.. And you will find a proof of the fact that every simple Lie algebra is also semisimple here.
  – José Carlos Santos
  Jul 23 at 15:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @José Carlos Santos Well. Thanks a lot :)
  – Kokuto
  Jul 23 at 15:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Cave Johnson Thanks for your comment :)
  – Kokuto
  Jul 23 at 15:45
  

  
  
  
  

So you mean Theorem 6.6 is correct ? And could you give me a hint on why "every simple Lie algebra is also semisimple"?
– Kokuto
Jul 23 at 15:32

So you mean Theorem 6.6 is correct ? And could you give me a hint on why "every simple Lie algebra is also semisimple"?
– Kokuto
Jul 23 at 15:32

@Kokuto This is more or less trivial, depending on the definition of semisimple Lie algebra. In most cases (but IIRC not in Hall's book) semisimple Lie algebra is defined to be a direct sum of simple Lie algebras, of which simple Lie algebra is obviously an example.
– Cave Johnson
Jul 23 at 15:35

@Kokuto This is more or less trivial, depending on the definition of semisimple Lie algebra. In most cases (but IIRC not in Hall's book) semisimple Lie algebra is defined to be a direct sum of simple Lie algebras, of which simple Lie algebra is obviously an example.
– Cave Johnson
Jul 23 at 15:35

1

1

Yes, theorem 6.6 is correct.. And you will find a proof of the fact that every simple Lie algebra is also semisimple here.
– José Carlos Santos
Jul 23 at 15:35

Yes, theorem 6.6 is correct.. And you will find a proof of the fact that every simple Lie algebra is also semisimple here.
– José Carlos Santos
Jul 23 at 15:35

@José Carlos Santos Well. Thanks a lot :)
– Kokuto
Jul 23 at 15:44

@José Carlos Santos Well. Thanks a lot :)
– Kokuto
Jul 23 at 15:44

@Cave Johnson Thanks for your comment :)
– Kokuto
Jul 23 at 15:45

@Cave Johnson Thanks for your comment :)
– Kokuto
Jul 23 at 15:45

add a comment | 

 

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