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. | ||
| $\mathrm{D}_af$ | 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 | ||
| $\mathrm{e}$ | 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 | ||
| $λ_g: \mathcal G → \mathcal G$ | 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 |