Ciro Santilli @cirosantilli 34

Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.

It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.

Solution $Z$ for given $X$ and $Y$ of:where $e$ is the exponential map.

If we consider just real number, $Z=X+Y$, but when X and Y are non-commutative, things are not so simple.

Furthermore, TODO confirm it is possible that a solution does not exist at all if $X$ and $Y$ aren't sufficiently small.

This formula is likely the basis for the Lie group-Lie algebra correspondence. With it, we express the actual group operation in terms of the Lie algebra operations.

Notably, remember that a algebra over a field is just a vector space with one extra product operation defined.

Vector spaces are simple because all vector spaces of the same dimension on a given field are isomorphic, so besides the dimension, once we define a Lie bracket, we also define the corresponding Lie group.

Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the Baker-Campbell-Hausdorff formula, we are basically done defining the group in terms of the algebra.

This is a general philosophy that Ciro Santilli, and likely others, observes over and over.

Basically, continuity, or higher order conditions like differentiability seem to impose greater constraints on problems, which make them more solvable.

Some good examples of that:

Basically a synonym for Lie group which is the way of modelling them.

Where derivation == "intuitive routes", since a "law of physics" cannot be derived, only observed right or wrong.

TODO also comment on why are complex numbers used in the Schrodinger equation?.

Some approaches:

Like everything else in Lie group theory, you should first look at the matrix version of this operation: the matrix exponential.

The exponential map links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a Lie algebra, so this map links the two worlds.

The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the product definition of the exponential function.

The definition of the exponential map is simply the same as that of the regular exponential function as given at Taylor expansion definition of the exponential function, except that the argument $x$ can now be an operator instead of just a number.

Examples:

In the classification of finite simple groups, groups of Lie type are a set of infinite families of simple lie groups. These are the other infinite families besides te cyclic groups and alternating groups.

A decent list at: en.wikipedia.org/wiki/List_of_finite_simple_groups, en.wikipedia.org/wiki/Group_of_Lie_type is just too unclear. The groups of Lie type can be subdivided into:

The first in this family discovered were a subset of the Chevalley groups $A_n(q)$ by Galois: $PSL(2,p)$, so it might be a good first one to try and understand what it looks like.

TODO understand intuitively why they are called of Lie type. Their names $A_n$, $B_n$ seem to correspond to the members of the classification of simple Lie groups which are also named like that.

But they are of course related to Lie groups, and as suggested at Video "Yang-Mills 1 by David Metzler (2011)" part 2, the continuity actually simplifies things.

Some sources distinguish "invariant" from "covariant" such that under some transformation (typically Lie group):TODO examples.

Bibliography:

Originally it was likely created to study constrained mechanical systems where you want to use some "custom convenient" variables to parametrize things instead of global x, y, z. Classical examples that you must have in mind include:

When doing lagrangian mechanics, we just lump together all generalized coordinates into a single vector $\overrightarrow{q}(t):\mathbb{R}\to {\mathbb{R}}^n$ that maps time to the full state:where each component can be anything, either the x/y/z coordinates relative to the ground of different particles, or angles, or nay other crazy thing we want.

The Lagrangian is a function ${\mathbb{R}}^n\times {\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$ that maps:to a real number.

Then, the stationary action principle says that the actual path taken obeys the Euler-Lagrange equation:This produces a system of partial differential equations with:

The mixture of so many derivatives is a bit mind mending, so we can clarify them a bit further. At:the $q_i$ is just identifying which argument of the Lagrangian we are differentiating by: the i-th according to the order of our definition of the Lagrangian. It is not the actual function, just a mnemonic.

Then at:

However, people later noticed that the Lagrangian had some nice properties related to Lie group continuous symmetries.

Basically it seems that the easiest way to come up with new quantum field theory models is to first find the Lagrangian, and then derive the equations of motion from them.

For every continuous symmetry in the system (modelled by a Lie group), there is a corresponding conservation law: local symmetries of the Lagrangian imply conserved currents.
Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "The Best Path" mentions that Richard Feynman didn't like the Lagrangian mechanics approach when he started university at MIT, because he felt it was too magical. The reason is that the Lagrangian approach basically starts from the principle that "nature minimizes the action across time globally". This implies that things that will happen in the future are also taken into consideration when deciding what has to happen before them! Much like the lifeguard in the lifegard problem making global decisions about the future. However, chapter "Least Action in Quantum Mechanics" comments that Feynman later notice that this was indeed necessary while developping Wheeler-Feynman absorber theory into quantum electrodynamics, because they felt that it would make more sense to consider things that way while playing with ideas such as positrons are electrons travelling back in time. This is in contrast with Hamiltonian mechanics, where the idea of time moving foward is more directly present, e.g. as in the Schrödinger equation.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

And partly due to the above observations, it was noticed that the easiest way to describe the fundamental laws of particle physics and make calculations with them is to first formulate their Lagrangian somehow: why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics?s.

TODO advantages:

Bibliography:

Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".

Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.

Another important way to think about Lie algebras, is as infinitesimal generators.

Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.

To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.

As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.

Bibliography:

Is the set of all n-by-y square matrices.

Because $GL(n)$ is a Lie group we can use Section "Lie algebra of a matrix Lie group".

For every matrix $x$ in the set of all n-by-y square matrices $M_n$, $e^x$ has inverse $e^{-}x$.

Note that this works even if $x$ is not invertible, and therefore not in $GL(n)$!

Therefore, the Lie algebra of $GL(n)$ is the entire $M_n$.

This makes it clear how the Lie bracket can be seen as a "measure of non-commutativity"

Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.

Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.

Every Lie algebra corresponds to a single simply connected Lie group.

The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.

Bibliography:

The author seems to have uploaded the entire book by chapters at: www.physics.drexel.edu/~bob/LieGroups.html

And the author is the cutest: www.physics.drexel.edu/~bob/Personal.html.

Overview:

Is the solution to a system of linear ordinary differential equations, the exponential function is just a 1-dimensional subcase.

Note that more generally, the matrix exponential can be defined on any ring.

The matrix exponential is of particular interest in the study of Lie groups, because in the case of the Lie algebra of a matrix Lie group, it provides the correct exponential map.

For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.