Notation on Lie groups
In this package,the notation introduced in Manifolds.jl Notation is used with the following additional parts.
Symbol | Description | Also used | Comment |
---|---|---|---|
$α$ | a general group action, when it is not specified whether it is a left ($α=σ$) or right ($α=τ$) action. | ||
$∘$ | a group operation | ||
$c_g:\mathcal G → \mathcal G$ | the conjugation map (with g ) | ||
$Df(p)[X]$ | the differential of a map f at point p in direction X | ||
$\mathrm{d}f$ | the differential of a map f as a function on the Lie group and the differential on the Lie algebra. | see also note below | |
$\mathrm{D}_af$ | the differential of a map f . An index is used to indicate a certain parameter. If $f$ is defined on or maps into the Lie group, this differential indicates the one with respect to tangent spaces | see also note below | |
$\mathrm{e}$ | identity element of a group | ||
$\exp_{\mathcal G}(X)$ | The Lie group exponential function | ||
$\exp_g(X)$ | The Lie group exponential map (w.r.t. a Cartan Schouten connection) | ||
$g, h, k$ | elements on a (Lie) group. Sometimes called points. | $g_1, g_2, ...$ | |
$\mathfrak g$ | a Lie algebra | ||
$\mathcal{G}$ | a Lie group | ||
$\operatorname{J}_f(p)$ | the Jacobian of a map f at point p | sometimes left Jacobian, see note below. | |
$λ_g: \mathcal G → \mathcal G$ | the left group operation map $λ_g(h) = g∘h$ | ||
$\log_{\mathcal G}(g)$ | The Lie group logarithmic function | ||
$\log_g(h)$ | The Lie group logarithmic map (w.r.t. a Cartan Schouten connection) | ||
$α: \mathcal M → \mathcal G → \mathcal M$ | a (general) group action | ||
$ρ_g: \mathcal G → \mathcal G$ | the right group operation map $ρ_g(h) = h∘g$ | ||
$σ: \mathcal G × \mathcal M → \mathcal M$ | a left group action | $σ_g(p)$ to emphasize a fixed group element | |
$τ: \mathcal G × \mathcal M → \mathcal M$ | a right group action | $σ_\mathrm{R}$ | $τ_g(p)$ to emphasize a fixed group element |
About differentials and Jacobians
For a function defined on a manifold $f:\mathcal M → \mathcal N$, the differential at a point $p ∈ \mathcal M$ is a map between the tangent spaces
\[Df(p) : T_p\mathcal M → T_{f(p)}\mathcal N.\]
This is the default in Manifolds.jl
.
For the case where $\mathcal M = \mathcal N = \mathcal G$ is an AbstractLieGroup
, the differential can be expressed in terms of the Lie algebra $\mathfrak g$ as
\[\mathrm{d}f(g) : \mathfrak g → \mathfrak g.\]
An alternate way to define this differential on the Lie algebra is to consider the (usual) differential $Dg(p)$ of $g(q) = f(q⋅p)⋅f(p)^{-1}$.
where we use a different notation on purpose. This second notation is the default throughout LieGroups.jl
.
The Jacobian $\operatorname{J}_f(p)$ of $f$ at $p$ is the matrix representation of the differential with respect to a basis of each of the tangent spaces. For the default representation $\mathrm{d}f(g)$ we have to choose a basis of the Lie algebra $\mathfrak g$. Throughout LieGroups.jl
this is the DefaultLieAlgebraOrthogonalBasis
.
About left and right Jacobians
This jacobian using $\mathrm{d}f(g)$ and the DefaultLieAlgebraOrthogonalBasis
is sometimes called the left Jacobian. The other choice using on two tangent space bases of $T_p\mathcal G$ and $T_{f(p)}\mathcal G$, respectively, is sometimes called the right Jacobian. The default throughout LieGroups.jl
is the left Jacobian.