Mar 24, 2017 · Also any rotation has an inverse (rotating it over the opposite angle). This makes the rotations a group. The "Lie" in Lie group means that these rotations can be done arbitrary small. Many small rotations makes for a big rotation. Lie groups capture the concept of "continuous symmetries".

There is a nice book called Matrix Groups — An Introduction to Lie Group theory by Andrew Baker. It starts by talking on Matrix groups, then introduces Lie groups and shows that Matrix groups are in fact Lie groups. The last part is dedicated to the …

Feb 8, 2023 · Cohen already spoke of Lie groups in his book about one-parameter groups 1911 [4], whereas Noether in her famous papers 1918 just called them the group of all analytical transformations of the variables (Lie group), which corresponds to the group of all linear transformations of the differentials (linear representation of the Lie group on its ...

$\begingroup$ What I mean is that if you have a representation of an arbitrary lie algebra $\mathfrak{g}$, then the associated Lie Group is a very specific group constructed by this Lie Algebra, and I've delineated this construction in my answer. $\endgroup$ –

Oct 9, 2021 · If group theory is about symmetry, then Lie group is almost exclusively about the study of symmetry of vector spaces. From another point of view, representation theory can be regarded as a tool to study groups, and it turns out they can tell us almost everything about the groups, just like Cayley's theorem says every abstract group is a ...

$\begingroup$ This is one of my favourite arguments of all time. So simple, yet powerful. I first learnt this one from Sagle & Walde section 3.3 where they scale back their assumptions from analytic Lie groups and deal with general topological groups and was in awe that so much could be gleaned from these more general assumptions.

Jan 3, 2019 · The approximation of curved objects – the Lie group elements – by first-order approximations – the Lie algebra elements – is a standard procedure in physics, which might partially explain the neglect. However, the following lays the emphasis on the algebra part from a terminological point of view.

Feb 24, 2022 · $\begingroup$ I'd also like to mention a case when they agree: compact connected Lie groups over $\mathbb{R}$ are the same thing as connected reductive anisotropic groups over $\mathbb{R}$ -- the former is a Lie group thing (obviously) and the latter is an object of algebraic group theory. This is significant as most people's first introduction ...

Mar 10, 2021 · A Lie group is a group that is also a manifold. Formally the generators are a basis for the tangent space of the identity element in the group. Informally you can think of the generators as corresponding to small group transformations which are "near" the identity.

Note that the Fourier transform of an locally abelian group has a relation to the representation theory, where as your definition above can not be related to representation theory. If you want an example of what comes close to a Fourier transform on a nonabelian Lie group, you can consider the Harish-Chandra transform and the Plancherel measure.