= Lie theory
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<Lie theory> studies <continuous symmetries> through <Lie groups>, <Lie algebras>, and their <representations>. A <rotation> can change by an arbitrarily small angle: its parameter varies continuously, yet composing two <rotations> obeys an exact <group operation>. The central question is how such finite transformations are controlled by their velocities at the <identity element>.
An <infinite group> need not require infinitely many independent parameters. The planar <rotation> <group> has infinitely many elements but only one local angle coordinate. More generally, an $n$-dimensional <Lie group> has an $n$-dimensional <tangent space> $\mathfrak g=T_eG$ at its <identity element>. A <basis> of this <Lie algebra>, together with the <Lie brackets> between its <basis vectors>, supplies finite data for infinitesimal composition. These brackets record the first correction to adding small transformation parameters. The <Exponential map of a Lie group> turns a chosen infinitesimal direction into an actual motion.
Finite tangent data consists of a finite <basis> and a finite table of <Lie brackets>. For planar <rotation>, every generator is $aJ$ with $a\in\mathbb R$, using the quarter-turn <matrix> $J$ computed below: one real coefficient specifies any <angular velocity>. This <Lie algebra> has infinitely many <vectors>, but its <dimension> is one. Linear combinations of the chosen <basis vectors> supply all the possible infinitesimal motions.
This passage between geometry and algebra is useful because <linear maps> and <matrix commutators> are easier to compute than an entire family of transformations. The local information does not determine every global identification: the additive <real line> and the <circle group> have isomorphic one-dimensional <Lie algebras>, but only the latter wraps around. For the local-versus-global viewpoint and the connection with <differential equations>, see https://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf[Meinrenken, Lie groups and Lie algebras, introduction and Section 4].
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