About

A vector space $\mathfrak{g}$ over some field $F$ and kitted with a product ("Lie Bracket", "commutator bracket" or simply "commutator") which is bilinear (linear in both arguments of the binary product operator), antisymmetric and fulfills the Jacobi-identity.

In physics, the Lie algebra most often arises as the Lie algebra (tangent space to the identity) of a Lie group $\mathfrak{G}$, which is a group that is also an analytic manifold such that the group product is a continuous function of the manifold's chart co-ordinates, so that the field $F$ is either $\mathbb{R}$ or $\mathbb{C}$. By the solution to Hilbert's fifth problem by Montgomery, Gleason and Zippin, one only needs to assume $C^0$ (continuity alone) of the group product: analyticity of the group product follows from the continuity assumption alone.

In gauge theories, the gauge group (structure group of the fibration arising from assumed gauge symmetry) is often a finite dimensional Lie group $\mathfrak{G}$, so that basis vectors of the gauge group's Lie algebra $\mathfrak{g}$ correspond one-to-one with the Noether currents and conserved quantities in the theory.

In dynamical systems (e.g. those governed by the Schrödinger equation) where the time evolution operator $U(t) \in \mathfrak{G}$ is constrained to be in a Lie group $\mathfrak{G}$ (e.g. the group of unitary transformations in the case of the Schrödinger equation), the Lie algebra $\mathfrak{g}$ is the space of possible time derivatives for the system: the algebra is left or right translated by the time evolution operator in the system's linear dynamical equation $\mathrm{d_t} U(t) = H(t) U(t) = U(t) H^\prime(t)$ where $H(t), H^\prime(t) \in \mathfrak{g}$.

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