Notation on Lie groups

In this package,the notation introduced in Manifolds.jl Notation is used with the following additional parts.

SymbolDescriptionAlso usedComment
$α$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 spacessee 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 psometimes 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.