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Is this subalgebra semisimple?

0

$begingroup$

Denote by $mathfrakg$ a complex semisimple Lie algebra and let $mathfrakh$ be a Cartan subalgebra of $mathfrakg$.
Let $Phi$
be the root system of $(mathfrakg,mathfrakh)$ and denote by $mathfrakg_alpha$ the root subspace of $mathfrakg$ corresponding
to a root $alpha$.
We fix a choice of positive roots $Phi^+$, and let $Delta$ be the corresponding subset of simple roots in $Phi^+$. Note that each subset $IsubseteqDelta$ generates a root system $Phi_IsubseteqPhi$, with positive roots $Phi_I^+:=Phi_Icap Phi^+$.

Let
$
mathfrakl_I:=mathfrakhoplussum_alphainPhi_Imathfrakg_alpha
$.

My questions:

1. Is $mathfrakl_I$ complex semsimple? If so, why?




2. Is $mathfrakl_I$ a Levi subalgebra of $mathfrakg$?

lie-algebras

share|cite|improve this question

edited Mar 23 at 1:07

Bernard

124k741117

asked Mar 23 at 1:03

James CheungJames Cheung

1236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Rarely semi-simple, because $mathfrak h$ is too big, but invariably reductive.
  $endgroup$
  – paul garrett
  Mar 23 at 1:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  However, when $I=Delta$, we get $mathfrakl_I=mathfrakg$ is complex semisimple while the $mathfrakh$ is still big. How to explain this?
  $endgroup$
  – James Cheung
  Mar 23 at 1:34
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Well, in the semi-simple case all of $mathfrak h$ is hit by brackets from the root spaces, while this is not so for proper $I$.
  $endgroup$
  – paul garrett
  Mar 23 at 1:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  One thing that helps would be to figure out the centre of $mathfrakl_I$.
  $endgroup$
  – Torsten Schoeneberg
  Mar 23 at 8:57
  

  
  
  
  

add a comment | 

0

$begingroup$

Denote by $mathfrakg$ a complex semisimple Lie algebra and let $mathfrakh$ be a Cartan subalgebra of $mathfrakg$.
Let $Phi$
be the root system of $(mathfrakg,mathfrakh)$ and denote by $mathfrakg_alpha$ the root subspace of $mathfrakg$ corresponding
to a root $alpha$.
We fix a choice of positive roots $Phi^+$, and let $Delta$ be the corresponding subset of simple roots in $Phi^+$. Note that each subset $IsubseteqDelta$ generates a root system $Phi_IsubseteqPhi$, with positive roots $Phi_I^+:=Phi_Icap Phi^+$.

Let
$
mathfrakl_I:=mathfrakhoplussum_alphainPhi_Imathfrakg_alpha
$.

My questions:

1. Is $mathfrakl_I$ complex semsimple? If so, why?




2. Is $mathfrakl_I$ a Levi subalgebra of $mathfrakg$?

lie-algebras

share|cite|improve this question

edited Mar 23 at 1:07

Bernard

124k741117

asked Mar 23 at 1:03

James CheungJames Cheung

1236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Rarely semi-simple, because $mathfrak h$ is too big, but invariably reductive.
  $endgroup$
  – paul garrett
  Mar 23 at 1:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  However, when $I=Delta$, we get $mathfrakl_I=mathfrakg$ is complex semisimple while the $mathfrakh$ is still big. How to explain this?
  $endgroup$
  – James Cheung
  Mar 23 at 1:34
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Well, in the semi-simple case all of $mathfrak h$ is hit by brackets from the root spaces, while this is not so for proper $I$.
  $endgroup$
  – paul garrett
  Mar 23 at 1:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  One thing that helps would be to figure out the centre of $mathfrakl_I$.
  $endgroup$
  – Torsten Schoeneberg
  Mar 23 at 8:57
  

  
  
  
  

add a comment | 

$begingroup$

Denote by $mathfrakg$ a complex semisimple Lie algebra and let $mathfrakh$ be a Cartan subalgebra of $mathfrakg$.
Let $Phi$
be the root system of $(mathfrakg,mathfrakh)$ and denote by $mathfrakg_alpha$ the root subspace of $mathfrakg$ corresponding
to a root $alpha$.
We fix a choice of positive roots $Phi^+$, and let $Delta$ be the corresponding subset of simple roots in $Phi^+$. Note that each subset $IsubseteqDelta$ generates a root system $Phi_IsubseteqPhi$, with positive roots $Phi_I^+:=Phi_Icap Phi^+$.

Let
$
mathfrakl_I:=mathfrakhoplussum_alphainPhi_Imathfrakg_alpha
$.

My questions:

1. Is $mathfrakl_I$ complex semsimple? If so, why?




2. Is $mathfrakl_I$ a Levi subalgebra of $mathfrakg$?

lie-algebras

share|cite|improve this question

edited Mar 23 at 1:07

Bernard

124k741117

asked Mar 23 at 1:03

James CheungJames Cheung

1236

$endgroup$

share|cite|improve this question

edited Mar 23 at 1:07

Bernard

124k741117

asked Mar 23 at 1:03

James CheungJames Cheung

1236

share|cite|improve this question

edited Mar 23 at 1:07

Bernard

124k741117

asked Mar 23 at 1:03

James CheungJames Cheung

1236

edited Mar 23 at 1:07

Bernard

124k741117

edited Mar 23 at 1:07

Bernard

124k741117

asked Mar 23 at 1:03

James CheungJames Cheung

1236

asked Mar 23 at 1:03

James CheungJames Cheung

1236