In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.
Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group
SO
(
3
)
{\displaystyle {\text{SO}}(3)}
). Lie groups are widely used in many parts of modern mathematics and physics.
Lie groups were first found by studying matrix subgroups
G
{\displaystyle G}
contained in
GL
n
(
R
)
{\displaystyle {\text{GL}}_{n}(\mathbb {R} )}
or
GL
n
(
C
)
{\displaystyle {\text{GL}}_{n}(\mathbb {C} )}
, the groups of
n
×
n
{\displaystyle n\times n}
invertible matrices over
R
{\displaystyle \mathbb {R} }
or
C
{\displaystyle \mathbb {C} }
. These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.
I have some clarifications on the discussion of adjoint representation in Group Theory by A. Zee, specifically section IV.1 (beware of some minor typos like negative signs).
An antisymmetric tensor ##T^{ij}## with indices ##i,j = 1, \ldots,N## in the fundamental representation is...
The full Lorentz group includes discontinuous transformations, i.e., time inversion and space inversion, which characterize the non-orthochronous and improper Lorentz groups, respectively. However, these groups are excluded from the Poincare group, in which only the proper, orthochronous...
I'd like to clarify a few things; the approach is basically just to show that ##\mathrm{GL}(n,\mathbf{C})## is isomorphic to a subgroup of ##\mathrm{GL}(2n,\mathbf{R})## which is a smooth manifold (since ##\mathbf{R} \setminus \{0\}## is an open subset of ##\mathbf{R}##, so its pre-image...
Hello, there. Consider a Lie group operating in a space with points ##X^\iota##. Its elements ##\gamma [ N^i]## are labeled by continuous parameters ##N^i##. Let the action of the group on the space be ##\gamma [N^i] X^\iota=\bar X^\iota (X^\kappa, N^i)##. Then the infinitesimal transformation...
I'm writing some notes for myself (to read in my rapidly approaching declining years) and I'm wondering if this statement is correct. I"m not sure I am posting this question in the right place.
"Summary: The matrix representations of isometric (distance-preserving) subgroups of the general...
I would like to investigate a function that sends ##f(x)## to ##f(x) - \frac{1}{c}f(x^{c})##, or a function ##g## such that ##g(f(x)) = f(x) - \frac{1}{c}f(x^{c}).## Since symmetries produced by groups are used in physics, I thought there might be someone here who could help explain what the...
I am trying to understand the properties of the ##SO(3)## Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions.
I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of ##SO(3)## dynamics ##\mathbf{\dot{R}} =...
"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0...
I am just starting a QM course. I hope these are reasonable questions. I have been given my first assignment. I can answer the questions so far but I do not really understand what's going on. These questions are all about so(N) groups, Pauli matrices, Lie brackets, generators and their Lie...
I am not sure if this is the right forum to post this question.
The title says it all: are there examples of Lie groups that cannot be represented as matrix groups?
Thanks in advance.
In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...
1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious?
Allow me to give the definitions I am working with.
A Lie group G is a differentiable manifold G which is also a group, such that the group...
Hi,
let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##.
In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a...
Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...
Given a representation of a Lie Group, is there a equivalence between possible electric charges and projections of the roots? For instance, in the standard model Q is a sum of hypercharge Y plus SU(2) charge T, but both Y and T are projectors in root space, and so a linear combination is. But I...
I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups.
Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e,
Lgh = gh where g,h ∈ G
Which corresponds to a translation by g.
Is this an example of a module over a ring...
The Brian Hall's book reads: A Lie group is any subgroup G of GL(n,C) with the following property: If Am is a secuence of matrices in G, and Am converges to some matrix A then either A belongs to G, or A is not invertible. Then He concludes G is closed en GL(n,C), ¿How can this be possible, if...
Good Day
I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc.
But I am making a lot of progress.
There is, however, one issue that continues to elude me.
I often read how Lie developed Lie Groups to study symmetries of PDE's
May I ask if someone...
Homework Statement
How can you find all inequivalent (non-isomorphic) 2D Lie algebras just by an analysis of the commutator?
Homework Equations
$$[X,Y] = \alpha X + \beta Y$$
The Attempt at a Solution
I considered three cases: ##\alpha = \beta \neq 0, \alpha = 0## or ##\beta = 0, \alpha =...
Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as
Ad(U)ta = Λ(U)abtb
I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known...
Homework Statement
Consider the contractions of the 3D Euclidean symmetry while preserving the SO(2) subgroup. In the physics point of view, explain the resulting symmetries G(2) (Galilean symmetry group) and H(3) (Heisenberg-Weyl group for quantum mechanics) and give their Lie algebras...
So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...
Homework Statement
Given for one-dimensional Galilean symmetry the generators ##K, P,## and ##H##, with the following commutation relations: $$[K, H] = iP$$ $$[H,P] = 0$$ $$[P,K] = 0$$
Homework Equations
Show that the Lie algebra for the generators ##K, P,## and ##H## is isomorphic to the...
Hello, let be ##G## a connected Lie group. I suppose##Ad(G) \subset Gl(T_{e}G)## is compact and the center ## Z(G)## of ##G## is discret (just to remember, forall ##g \in G##, ##Ad(g) = T_{e}i_{g}## with ##i_{g} : x \rightarrow gxg^{-1}##.).
I saw without any proof that in those hypothesis...
Homework Statement
The group ##G = \{ a\in M_n (C) | aSa^{\dagger} =S\}## is a Lie group where ##S\in M_n (C)##. Find the corresponding Lie algebra.
Homework EquationsThe Attempt at a Solution
As far as I've been told the way to find these things is to set ##a = exp(tA)##, so...
Hi!
I'm studying Lie Algebras and Lie Groups. I'm using Brian Hall's book, which focuses on matrix lie groups for a start, and I'm loving it. However, I'm really having a hard time connecting what he does with what physicists do (which I never really understood)... Here goes one of my questions...
From my reading, the X between SU(4)XSU(2) mean Cartesian product.
But How the way to mutiply two matrix A in SU(4) and B in SU(2).
Example the matrix
A=\begin{pmatrix} a & b &
c & d \\ e& f &
g & h \\ i & j &
k & l \\ m & n&
o & p \end{pmatrix}
and
B=\begin{pmatrix} 1 &2 \\
3 &4...
Homework Statement
Good day,
From my reading, SU(4) have 15 parameter and SU(2) has 3 paramater that range differently with certain parameter(rotation angle). And all the parameter is linearly independent to each other.
My question are: 1. What the characteristic of each of the parameter? 2...
I learned a lie group is a group which satisfied all the conditions of a diferentiable manifold. that is the real rigour definition or just a simplified one?
thanks
I recently got confused about Lie group products.
Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...
The Lie Algebra is equipped with a bracket notation, and this bracket produces skew symmetric matrices.
I know that there exists Lie Groups, one of which is SO(3).
And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix.
-----------------
First, can someone edit...
[Mentor's Note: Thread moved from homework forums]
Where can I start to research this question? I did not take any course on Group theory before and I know almost nothing about the relationship with this pure maths and physics. I've decided to start with Arfken's book but I'm not sure.
1...
Hello,
in group theory a regular action on a G-set S is such that for every x,y∈S, there exists exactly one g such that g⋅x = y.
I noticed however that in the theory of Lie groups the definition of regular action is quite different (see Definition 1.4.8 at this link).
Is there a connection...
I can never derive the prolongation formulas correctly when I want to prove the Lie group symmetries of PDEs. (If I'm lucky I get the transformed tangent bundle coordinate right and botch the rest.) I've gone through a number of textbooks and such in the past, but I haven't found any clear...
Hi y'all,
This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer.
I have a Lie group homomorphism \rho : G \rightarrow GL(n...
I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements.
I would like to prove this statement for ##SO(3)##.
I know that the commutation relations are ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##.
Can you...
Suppose to have a Lie group that is at the same time also a Riemannian manifold: is there a relation between Christoffel symbols and structure constants? What can i say about the geodesics in a Lie group? Do they have special properties?
What makes a Lie group a real Lie group? I see on the Wikipedia page http://en.wikipedia.org/wiki/Lie_group "A real Lie group is a group that is also a finite-dimensional real smooth manifold" So when is a manifold a real manifold?
As I understand it, the symplectic Lie group Sp(2n,R) of 2n×2n symplectic matrices is generated by the matrices in http://en.wikipedia.org/wiki/Symplectic_group#Infinitesimal_generators .
Does this mean that sl(n,R) is a subalgebra of the corresponding lie algebra, since in that formula we can...
In the way of defining the adjoint representation,
\mathrm{ad}_XY=[X,Y],
where X,Y are elements of a Lie algebra, how to determine the components of its representation, which equals to the structure constant?
Definition/Summary
A Lie group ("Lee") is a continuous group whose group operation on its parameters is differentiable in them.
Lie groups appear in a variety of contexts, like space-time and gauge symmetries, and in solutions of certain differential equations.
The elements of Lie...
Long time reader, first time poster.
Originally, it was my contention that all Lie groups could be written as the semidirect product of a connected Lie group and a discrete Lie group. However, I no longer believe this is true.
The next best thing I could think of was to say that a Lie group is...
Higher dimensional groups are parametrised by several parameters (e.g the three dimensional rotation group SO(3) is described by the three Euler angles). Consider the following ansatz: $$\rho_1 = \mathbf{1} + i \alpha^a T_a + \frac{1}{2} (i\alpha^a T_a)^2 + O(\alpha^3)$$
$$\rho_2 = \mathbf{1}...
The vectors \vec{\alpha}=\{\alpha_1,\ldots\alpha_m \} are defined by
[H_i,E_\alpha]=\alpha_i E_\alpha
they are also known to be the non-zero weights, called the roots, in the adjoint representation. My question is - is this connection (that the vectors \vec{\alpha} defined by the commutation...
We have that A and B belong to different representations of the same Lie Group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations.
A = e^{tX}
B = e^{tY}
We want to show, for a specific matrix M
B^{-1} M B = AM
Does it suffice to...
I found what might be the worst written book on Lie Groups. Ever. Until I find one I like better, I'm going to see if I can persevere through the sludge. I'll write out the theorem word for word and then explain what I can. Hopefully someone can decipher it.
Typically, I use the term "chart"...
Hello,
I have troubles formulating this question properly. So I will explain it through one example.
If we consider the Lie group R=SO(2) of rotations on the plane, we know that we can find a manifold on which the group SO(2) acts regularly: this manifold is the unit circle in ℝ2. In fact...