AbstractGroupActionType

An abstract supertype for group action types, which are used within a GroupAction.

AbstractLeftGroupActionType <: AbstractGroupActionType

A type representing a (smooth) group action $σ: \mathcal G × \mathcal M → \mathcal M$ of a LieGroup $\mathcal G$ acting (from the left) on an AbstractManifold $\mathcal M$. with the following properties

1. $σ(\mathrm{e}, p) = p$ holds for all $p ∈ \mathcal M$
2. $σ(g, σ(h, p)) = σ(g∘h, p)$ holds for all $g,h ∈ \mathcal G$, $p ∈ \mathcal M$

where $∘$ denotes the group operation of the LieGroup $\mathcal G$. See also [HN12, Definition 9.1.11].

The type of action can be seen a bit better when writing the action as a family $σ_g(p)$: we obtain from the second property as

\[ σ_g(σ_h(p)) = σ_{gh}(p)\]

and see that $g$ appears on the left.

To emphasize the side the group operation is acting from, we sometimes write $σ^{\mathrm{L}}$. If the action is clear from context we write $σ(g, p) = g ⋅ p$.

One notable example of a left action is the inverse of an action of AbstractRightGroupActionType $σ^{\mathrm{R}}$, which is given by $τ_g = (σ^{\mathrm{R}}_g)^{-1} = σ^{\mathrm{R}}_{g^{-1}}$. We obtain

\[τ_g(τ_h(p)) = σ^{\mathrm{R}}_{g^{-1}}(σ^{\mathrm{R}}_{h^{-1}}(p)) = σ^{\mathrm{R}}_{h^{-1}g^{-1}}(p) = σ^{\mathrm{R}}_{(gh)^{-1}}(p) τ_{gh}(p).\]

AbstractRightGroupActionType <: AbstractGroupActionType

A type representing a (smooth) group action $σ: \mathcal M × \mathcal G → \mathcal M$ of a LieGroup $\mathcal G$ acting (from the right) on an AbstractManifold $\mathcal M$. with the following properties

1. $σ(p, \mathrm{e}) = p$ holds for all $p ∈ \mathcal M$
2. $σ(σ(p, g), h) = σ(p, g∘h)$ holds for all $g,h ∈ \mathcal G$, $p ∈ \mathcal M$

where $∘$ denotes the group operation of the LieGroup $\mathcal G$. See also [HN12, Remark 9.1.12].

The type of action can be seen a bit better when writing the action as a family $σ_g(p)$: we obtain from the second property as

\[ σ_g(σ_h(p)) = σ_{hg}(p)\]

and see that $g$ appears on the right.

To emphasize the side the group operation is acting from, we sometimes write $σ^{\mathrm{R}}$. If the action is clear from context we write $σ(p, g) = p ⋅ g$.

One notable example of a right action is the inverse of an action of AbstractLeftGroupActionType $σ^{\mathrm{L}}$, which is given by $τ_g = (σ^{\mathrm{L}}_g)^{-1} = σ^{\mathrm{L}}_{g^{-1}}$. We obtain

\[τ_g(τ_h(p)) = σ^{\mathrm{L}}_{g^{-1}}(σ^{\mathrm{L}}_{h^{-1}}(p)) = σ^{\mathrm{L}}_{g^{-1}h^{-1}}(p) = σ^{\mathrm{L}}_{(hg)^{-1}}(p) τ_{hg}(p).\]

GroupAction{T<:GroupActionType, L<:LieGroup, M<:AbstractManifold}

Specify a group action of AbstractGroupActionType T of a LieGroup G acting on M.

Let $\mathcal M$ be a AbstractManifold and $\mathcal G$ be a LieGroup with group operation $∘$.

A (smooth) action of the group $\mathcal G$ on the manifold $\mathcal M$ is a map

\[σ: \mathcal G × \mathcal M → \mathcal M\]

with the properties

• $σ(\mathrm{e}, p) = p$ holds for all $p ∈ \mathcal M$
• $σ(g, σ(h, p)) = σ(g∘h, p)$ holds for all $g,h ∈ \mathcal G$, $p ∈ \mathcal M$

Fields

• type::T: The type of the group action.
• group::L: The group acting.
• manifold::M: The manifold the group acts upon.

See [HN12, Section 9.1.3] for more details.

inv(::AbstractGroupActionType)

return the inverse group operation action, that is, use the type representing the inverse operation.

 inv(A::GroupAction{T})

Return the GroupAction representing the inverse of an GroupAction of AbstractGroupActionType T. This is usually done by returning the group action with the inverse type of T.

base_lie_group(A::GroupAction)

Return the LieGroup of the GroupAction specifying the action.

default_left_action(G::LieGroup, M::AbstractManifold)

Return the default left action for a Lie group G acting on a manifold M.

default_right_action(G::LieGroup, M::AbstractManifold)

Return the default right action for a Lie group G acting on a manifold M.

diff_apply(A::GroupAction{T, L, M}, g, p, X)
diff_apply!(A::GroupAction{T, L, M}, Y, g, p, X)

Compute the differential $D_p σ_g(p): T_p\mathcal M → T_{σ_g(p)}\mathcal M$, where for a left group action we have $σ_g(p) = σ(g,p)$, for a right action $σ_g(p) = σ(p, g)$.

diff_apply(A::GroupAction{T, L, M}, g, p, X)
diff_apply!(A::GroupAction{T, L, M}, Y, g, p, X)

Compute the differential $D_p σ_g(p): T_p\mathcal M → T_{σ_g(p)}\mathcal M$, where for a left group action we have $σ_g(p) = σ(g,p)$, for a right action $σ_g(p) = σ(p, g)$.

diff_group_apply(A::GroupAction{T, L, M}, g, p, X)
diff_group_apply!(A::GroupAction{T, L, M}, Y, g, p, X)

Compute the differential $D_g σ_g(p): \mathfrak g → \mathfrak g$, where we use the short hand notation $σ_p(g) = σ(g,p)$ for a left action, and for a right action $σ_p(g) = σ(p, g)$.

diff_group_apply(A::GroupAction{T, L, M}, g, p, X)
diff_group_apply!(A::GroupAction{T, L, M}, Y, g, p, X)

Compute the differential $D_g σ_g(p): \mathfrak g → \mathfrak g$, where we use the short hand notation $σ_p(g) = σ(g,p)$ for a left action, and for a right action $σ_p(g) = σ(p, g)$.

switch(T::AbstractGroupActionType)

Return the object representing an AbstractGroupActionType related to a group operation action that switched the side, that is it turns a left action type into its corresponding right action type.

 switch(A::GroupAction{T})

Return the group operation action representing the similar GroupAction of AbstractGroupActionType T but acting from the other side. It switches left to right and vice versa. This is done by returning the group action with the “switched” type of T.

apply(A::GroupAction{T, L, M}, g, p)
apply!(A::GroupAction{T, L, M}, q, g, p)

Apply the group action induced by $g ∈ \mathcal G$ to $p ∈ \mathcal M$, where the kind of group action is indicated by the AbstractGroupActionType T. This can be performed in-place of q.

apply(A::GroupAction{T, L, M}, g, p)
apply!(A::GroupAction{T, L, M}, q, g, p)

Apply the group action induced by $g ∈ \mathcal G$ to $p ∈ \mathcal M$, where the kind of group action is indicated by the AbstractGroupActionType T. This can be performed in-place of q.

base_manifold(A::GroupAction)

Return the AbstractManifold the group action acts upon.

InverseLeftGroupOperationAction <: AbstractRightGroupActionType

A type for the AbstractLeftGroupActionType when acting on the group itself given by the inverse of a LeftGroupOperationAction $σ_h$ as

\[τ_h(g) \coloneqq σ_h^{-1}(g) = σ(h^{-1},g) = h^{-1}∘g\]

Note that while in naming it is the inverse of the left action, it's properties yield that is is an AbstractRightGroupActionType, since

\[τ_g(τ_h(p)) = σ^{\mathrm{L}}_{g^{-1}}(σ^{\mathrm{L}}_{h^{-1}}(p)) = σ^{\mathrm{L}}_{g^{-1}h^{-1}}(p) = σ^{\mathrm{L}}_{(hg)^{-1}}(p) τ_{hg}(p).\]

for its inverse $(σ_h)^{-1}$ see InverseLeftGroupOperationAction.

InverseRightGroupOperationAction <: AbstractLeftGroupActionType

A type for the AbstractLeftGroupActionType when acting on the group itself given by the inverse of a RightGroupOperationAction $σ_h$ as

\[τ_h(g) \coloneqq σ_h^{-1}(g) = σ(h^{-1},g) = g∘h^{-1}\]

Note that while in naming it is the inverse of the right action, it's properties yield that is is an AbstractLeftGroupActionType, since

\[τ_g(τ_h(p)) = σ^{\mathrm{R}}_{g^{-1}}(σ^{\mathrm{R}}_{h^{-1}}(p)) = σ^{\mathrm{R}}_{h^{-1}g^{-1}}(p) = σ^{\mathrm{R}}_{(gh)^{-1}}(p) τ_{gh}(p).\]

for its inverse $(σ_h)^{-1}$ see InverseLeftGroupOperationAction.

LeftGroupOperationAction <: AbstractLeftGroupActionType

A type for the AbstractLeftGroupActionType when acting on the group itself from the left, that is

\[σ_h(g) = σ(h,g) = h∘g\]

for its inverse $(σ_h)^{-1}$ see InverseLeftGroupOperationAction.

RightGroupOperationAction <: AbstractRightGroupActionType

A type for the AbstractLeftGroupActionType when acting on the group itself from the right.

\[σ_h(g) = σ(h,g) = g∘h\]

for its inverse $(σ_h)^{-1}$ see InverseRightGroupOperationAction.

inv(::InverseLeftGroupOperationAction)

Return the inverse of the InverseLeftGroupOperationAction, that is the LeftGroupOperationAction.

inv(::InverseRightGroupOperationAction)

Return the inverse of the InverseRightGroupOperationAction, that is the RightGroupOperationAction.

inv(::LeftGroupOperationAction)

Return the inverse of the LeftGroupOperationAction, that is the InverseLeftGroupOperationAction.

inv(::RightGroupOperationAction)

Return the inverse of the RightGroupOperationAction, that is the InverseRightGroupOperationAction.

GroupOperationAction(action::AbstractGroupActionType, group::LieGroup)

Return a GroupAction for an AbstractGroupActionType action representing the group operation as an action of the group on itself.

switch(::InverseLeftGroupOperationAction)

Return the InverseRightGroupOperationAction, that is, turns $σ_g = g^{-1}∘h$ into $τ_g(h) = h∘g^{-1}$

switch(::InverseRightGroupOperationAction)

Return the InverseLeftGroupOperationAction, that is, turns $σ_g = h∘g^{-1}$ into $τ_g(h) = g^{-1}∘h$

switch(::LeftGroupOperationAction)

Return the RightGroupOperationAction, that is, turns $σ_g = g∘h$ into $τ_g(h) = h∘g$

switch(::RightGroupOperationAction)

Return the LeftGroupOperationAction, that is, turns $σ_g = h∘g$ into $τ_g(h) = g∘h$