I need help calculating the exponential map of a general vector.(adsbygoogle = window.adsbygoogle || []).push({});

Definition of the exponential map

For a Lie group [itex]G[/itex] with Lie algebra [itex]\mathfrak{g}[/itex], and a vector [itex]X \in \mathfrak{g} \equiv T_eG[/itex], let [itex]\hat{X}[/itex] be the corresponding left-invariant vector field. Then let [itex]\gamma_X(t)[/itex] be the maximal integral curve of [itex]\hat{X}[/itex] such that [itex]\gamma_X(0)=e[/itex]. Then the exponential map [itex]\mbox{exp}:\mathfrak{g} \to G[/itex] is [itex]\mbox{exp}(A) = \gamma_A(1)[/itex].

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It can be shown that the exponential map when [itex]A[/itex] is a matrix is just the 'exponential taylor series' in matrix form.

However, how do you actually compute the exponential map for a general vector that isn't a matrix?

Say, for example, we have the Lie group [itex]\mathbb{R}^2 - \lbrace (0,0)\rbrace[/itex] with binary operation [itex](a,b) *(c,d) = (ac-bd,ad+bc)[/itex], with identity [itex](1,0)[/itex] and basis [itex]\left(\frac{\partial}{\partial x}\bigg|_{(1,0)},\frac{\partial}{\partial y}\bigg|_{(1,0)}\right)[/itex]. What steps are required to compute [itex]\mbox{exp}\left(\frac{\partial}{\partial x}\bigg|_{(1,0)}\right)[/itex] here?

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# How to compute the exponential map

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