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Construction of group structure in T^*G where G is a Lie group
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Well, i'm having problems with the follow construction of Arnold's book "Topological Methods of Hydrodynamics", pg. 51:
"The group $G$ acts naturally on itself by left translations, as well as by right ones.
The left and right shifts commute with each other. Hence, right-invariant vector
(or covector) fields are taken to right-invariant ones under left translations, while left-invariant fields are sent to left-invariant ones by right translations.
Extend every covector on $G$, i.e., an element $alpha_g$ of the cotangent bundle $T^*G$ at $g$ ∈ $G$, to the right-invariant section (covector field) $alpha$ on the group. Define the
action of this covector αg on the phase space $T^*G$ as follows. First add to every covector in $T^*G$ at $h$ the value of the right-invariant section $alpha$ at $h$. Then apply
the left shift of the entire phase space $T^*G$ by $g$."
In algebraic language , i understood the following, the covector in the end is
$$ L_g^* (alpha(h) + beta_h)$$
where $alpha(h) = R_g^-1^*(alpha_g)$ a extension of covector $alpha_g$ to entire space and $beta_h$ is any vector of cotangent space at $h$. But the operation only makes sense if we have $h$:
$$ L_h^*( alpha(h) + beta_h)$$
If someone could clarify....
Thanks !
differential-geometry lie-groups lie-algebras co-tangent-space
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Well, i'm having problems with the follow construction of Arnold's book "Topological Methods of Hydrodynamics", pg. 51:
"The group $G$ acts naturally on itself by left translations, as well as by right ones.
The left and right shifts commute with each other. Hence, right-invariant vector
(or covector) fields are taken to right-invariant ones under left translations, while left-invariant fields are sent to left-invariant ones by right translations.
Extend every covector on $G$, i.e., an element $alpha_g$ of the cotangent bundle $T^*G$ at $g$ ∈ $G$, to the right-invariant section (covector field) $alpha$ on the group. Define the
action of this covector αg on the phase space $T^*G$ as follows. First add to every covector in $T^*G$ at $h$ the value of the right-invariant section $alpha$ at $h$. Then apply
the left shift of the entire phase space $T^*G$ by $g$."
In algebraic language , i understood the following, the covector in the end is
$$ L_g^* (alpha(h) + beta_h)$$
where $alpha(h) = R_g^-1^*(alpha_g)$ a extension of covector $alpha_g$ to entire space and $beta_h$ is any vector of cotangent space at $h$. But the operation only makes sense if we have $h$:
$$ L_h^*( alpha(h) + beta_h)$$
If someone could clarify....
Thanks !
differential-geometry lie-groups lie-algebras co-tangent-space
asked yesterday
Inácio
312
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Well, i'm having problems with the follow construction of Arnold's book "Topological Methods of Hydrodynamics", pg. 51:
"The group $G$ acts naturally on itself by left translations, as well as by right ones.
The left and right shifts commute with each other. Hence, right-invariant vector
(or covector) fields are taken to right-invariant ones under left translations, while left-invariant fields are sent to left-invariant ones by right translations.
Extend every covector on $G$, i.e., an element $alpha_g$ of the cotangent bundle $T^*G$ at $g$ ∈ $G$, to the right-invariant section (covector field) $alpha$ on the group. Define the
action of this covector αg on the phase space $T^*G$ as follows. First add to every covector in $T^*G$ at $h$ the value of the right-invariant section $alpha$ at $h$. Then apply
the left shift of the entire phase space $T^*G$ by $g$."
In algebraic language , i understood the following, the covector in the end is
$$ L_g^* (alpha(h) + beta_h)$$
where $alpha(h) = R_g^-1^*(alpha_g)$ a extension of covector $alpha_g$ to entire space and $beta_h$ is any vector of cotangent space at $h$. But the operation only makes sense if we have $h$:
$$ L_h^*( alpha(h) + beta_h)$$
If someone could clarify....
Thanks !
differential-geometry lie-groups lie-algebras co-tangent-space
asked yesterday
Inácio
312
Well, i'm having problems with the follow construction of Arnold's book "Topological Methods of Hydrodynamics", pg. 51:
"The group $G$ acts naturally on itself by left translations, as well as by right ones.
The left and right shifts commute with each other. Hence, right-invariant vector
(or covector) fields are taken to right-invariant ones under left translations, while left-invariant fields are sent to left-invariant ones by right translations.
Extend every covector on $G$, i.e., an element $alpha_g$ of the cotangent bundle $T^*G$ at $g$ ∈ $G$, to the right-invariant section (covector field) $alpha$ on the group. Define the
action of this covector αg on the phase space $T^*G$ as follows. First add to every covector in $T^*G$ at $h$ the value of the right-invariant section $alpha$ at $h$. Then apply
the left shift of the entire phase space $T^*G$ by $g$."
In algebraic language , i understood the following, the covector in the end is
$$ L_g^* (alpha(h) + beta_h)$$
where $alpha(h) = R_g^-1^*(alpha_g)$ a extension of covector $alpha_g$ to entire space and $beta_h$ is any vector of cotangent space at $h$. But the operation only makes sense if we have $h$:
$$ L_h^*( alpha(h) + beta_h)$$
If someone could clarify....
Thanks !
differential-geometry lie-groups lie-algebras co-tangent-space
asked yesterday
Inácio
312
asked yesterday
Inácio
312
asked yesterday
Inácio
312
asked yesterday
Inácio
312
312
add a comment |Â
add a comment |Â
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