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2

$begingroup$

$defOmathcalO
defgmathfrakg$
Suppose $G$ is a real or complex Lie group, with Lie algebra $g$. Write $O_G,1$ (resp. $O_g,0$) for the ring of germs of smooth/holomorphic function on $G$ at $1$ (resp. on $g$ at $0$). If we identify $g$ with left-invariant vector fields on $G$, then we get a Lie algebra representation of $g$ on $O_G,1$. The exponential map $exp:g to G$ is a local isomorphism, so induces an isomorphism of $O_g,0$ and $O_G,1$, so we also get a Lie algebra representation of $g$ on $O_g,0$.

Can the representation of $g$ on $O_g,0$ be described in a way that is intrinsic to $g$, without reference $G$?

representation-theory lie-groups lie-algebras

share|cite|improve this question

asked 2 days ago

UserUser

35218

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think it might be more or less the dual of the adjoint action of $mathfrakg$ on its universal enveloping algebra $U(mathfrakg)$.
  $endgroup$
  – Qiaochu Yuan
  2 days ago
  

  
  
  
  

add a comment | 

2

$begingroup$

$defOmathcalO
defgmathfrakg$
Suppose $G$ is a real or complex Lie group, with Lie algebra $g$. Write $O_G,1$ (resp. $O_g,0$) for the ring of germs of smooth/holomorphic function on $G$ at $1$ (resp. on $g$ at $0$). If we identify $g$ with left-invariant vector fields on $G$, then we get a Lie algebra representation of $g$ on $O_G,1$. The exponential map $exp:g to G$ is a local isomorphism, so induces an isomorphism of $O_g,0$ and $O_G,1$, so we also get a Lie algebra representation of $g$ on $O_g,0$.

Can the representation of $g$ on $O_g,0$ be described in a way that is intrinsic to $g$, without reference $G$?

representation-theory lie-groups lie-algebras

share|cite|improve this question

asked 2 days ago

UserUser

35218

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think it might be more or less the dual of the adjoint action of $mathfrakg$ on its universal enveloping algebra $U(mathfrakg)$.
  $endgroup$
  – Qiaochu Yuan
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

$defOmathcalO
defgmathfrakg$
Suppose $G$ is a real or complex Lie group, with Lie algebra $g$. Write $O_G,1$ (resp. $O_g,0$) for the ring of germs of smooth/holomorphic function on $G$ at $1$ (resp. on $g$ at $0$). If we identify $g$ with left-invariant vector fields on $G$, then we get a Lie algebra representation of $g$ on $O_G,1$. The exponential map $exp:g to G$ is a local isomorphism, so induces an isomorphism of $O_g,0$ and $O_G,1$, so we also get a Lie algebra representation of $g$ on $O_g,0$.

Can the representation of $g$ on $O_g,0$ be described in a way that is intrinsic to $g$, without reference $G$?

representation-theory lie-groups lie-algebras

share|cite|improve this question

asked 2 days ago

UserUser

35218

$endgroup$

share|cite|improve this question

asked 2 days ago

UserUser

35218

share|cite|improve this question

asked 2 days ago

UserUser

35218

asked 2 days ago

UserUser

35218

asked 2 days ago

UserUser

35218