Mixing
The action map induces a Lie algebra homomorphism
$begingroup$
Let $G$ be a Lie group and $M$ be a manifold such that $G$ acts on $M$ transitively. Let $$phi: Gtimes Mto M$$
be the action map. Now $phi $ induces a homomorphism
$$alpha:Gto Aut(M)$$
How to show that $alpha$ induces a Lie homomorphism $gamma:mathfrak gto Gamma(M,TM )$? Is $gamma$ injective? what if the action is effective?
differential-geometry lie-groups lie-algebras
asked Jan 21 at 3:14
Amrat AAmrat A
314110
$endgroup$
add a comment |
$begingroup$
Let $G$ be a Lie group and $M$ be a manifold such that $G$ acts on $M$ transitively. Let $$phi: Gtimes Mto M$$
be the action map. Now $phi $ induces a homomorphism
$$alpha:Gto Aut(M)$$
How to show that $alpha$ induces a Lie homomorphism $gamma:mathfrak gto Gamma(M,TM )$? Is $gamma$ injective? what if the action is effective?
differential-geometry lie-groups lie-algebras
asked Jan 21 at 3:14
Amrat AAmrat A
314110
$endgroup$
1
$begingroup$
Take the derivative of $alpha$ at the identity.
$endgroup$
– Amitai Yuval
Jan 21 at 9:03
add a comment |
$begingroup$
Let $G$ be a Lie group and $M$ be a manifold such that $G$ acts on $M$ transitively. Let $$phi: Gtimes Mto M$$
be the action map. Now $phi $ induces a homomorphism
$$alpha:Gto Aut(M)$$
How to show that $alpha$ induces a Lie homomorphism $gamma:mathfrak gto Gamma(M,TM )$? Is $gamma$ injective? what if the action is effective?
differential-geometry lie-groups lie-algebras
asked Jan 21 at 3:14
Amrat AAmrat A
314110
$endgroup$
Let $G$ be a Lie group and $M$ be a manifold such that $G$ acts on $M$ transitively. Let $$phi: Gtimes Mto M$$
be the action map. Now $phi $ induces a homomorphism
$$alpha:Gto Aut(M)$$
How to show that $alpha$ induces a Lie homomorphism $gamma:mathfrak gto Gamma(M,TM )$? Is $gamma$ injective? what if the action is effective?
differential-geometry lie-groups lie-algebras
differential-geometry lie-groups lie-algebras
asked Jan 21 at 3:14
Amrat AAmrat A
314110
asked Jan 21 at 3:14
Amrat AAmrat A
314110
asked Jan 21 at 3:14
Amrat AAmrat A
314110
asked Jan 21 at 3:14
Amrat AAmrat A
314110
asked Jan 21 at 3:14
Amrat AAmrat A
314110
314110
1
$begingroup$
Take the derivative of $alpha$ at the identity.
$endgroup$
– Amitai Yuval
Jan 21 at 9:03
add a comment |
1
$begingroup$
Take the derivative of $alpha$ at the identity.
$endgroup$
– Amitai Yuval
Jan 21 at 9:03
1
1
$begingroup$
Take the derivative of $alpha$ at the identity.
$endgroup$
– Amitai Yuval
Jan 21 at 9:03
$begingroup$
Take the derivative of $alpha$ at the identity.
$endgroup$
– Amitai Yuval
Jan 21 at 9:03
add a comment |
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$begingroup$
Take the derivative of $alpha$ at the identity.
$endgroup$
– Amitai Yuval
Jan 21 at 9:03