# What is Lie group: Definition and 82 Discussions

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.

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1. ### I Fundamental representation and adjoint representation

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...
2. ### I Poincare Group as a Lie Group

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...
3. ### Show GL(n,C) is a Lie group

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...
4. ### I How to find the generator of this Lie group?

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...
5. ### A Simple definition of Lie group

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...
6. ### Good introductory book about Lie Group Theory?

Summary:: Good introductory book about Group Theory? Hi, I am looking for a good introductory book about Group Theory for physicists.
7. ### A How to investigate a transformation that might form a Lie group?

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...
8. ### A Invariance of ##SO(3)## Lie group when expressed via Euler angles

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}} =...
9. ### I Why isn't this a Lie group?

"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...
10. ### I Relationship between a Lie group such as So(3) and its Lie algebra

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...
11. ### I Example of a Lie group that cannot be represented in matrix form?

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.
12. ### A Unitary representations of Lie group from Lie algebra

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...
13. ### Lie group global properties

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...
14. ### I Derivative of the Ad map on a Lie 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...
15. ### A Lie Algebra and Lie Group

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...
16. ### A Charge in a Lie Group.... is it always a projection?

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...
17. ### I Rings, Modules and the Lie Bracket

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...
18. ### A Why a Lie Group is closed in GL(n,C)?

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...
19. ### A Example of how a rotation matrix preserves symmetry of PDE

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...
20. A

### All possible inequivalent Lie algebras

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 =...
21. ### A Characterizing the adjoint representation

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...
22. A

### Contractions of the Euclidean Group ISO(3) = E(3)

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...
23. ### A Diagonalization of adjoint representation of a Lie Group

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...
24. A

### Heisenberg algebra Isomorphic to Galilean algebra

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...
25. ### Insights A Journey to The Manifold - Part I - Comments

fresh_42 submitted a new PF Insights post A Journey to The Manifold - Part I Continue reading the Original PF Insights Post.
26. ### A Compact Lie Group: Proof of Discrete Center & Finite Size?

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...
27. ### Find the Lie algebra corresponding to this Lie group

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...
28. ### A Matrix Lie groups and its Lie Algebra

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...
29. ### A Meaning of X in SU(2)XSU(2)

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...
30. ### Parameter of SU(4) and SU(2)

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...
31. ### I Is that the definition of a lie group?

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
32. ### I About Lie group product ($U(1)\times U(1)$ ex.)

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}}...
33. ### A Lie Algebras and Rotations

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...
34. ### Research Orthogonal Lie Group for Physics Applications

[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...
35. ### Definition of regular Lie group action

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...
36. ### Jet Prolongation Formulas for Lie Group Symmetries

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...
37. ### Lie Group v Lie algebra representation

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...
38. ### Lie group multiplication and Lie algebra commutation

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...
39. ### Geodesics in a Lie group

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?
40. ### What does it mean to say a Lie group is real?

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?
41. ### Is sl(3,R) a subalgebra of sp(4,R)?

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...
42. ### Components of adjoint representations

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?
43. ### Definition of Lie Group and its Algebras

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...
44. ### Can Principal Bundles Help with Lie Group Decomposition?

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...
45. ### Linearisation of Lie Group

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}...
46. ### Lie group representations

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...
47. ### Equality involving matrix exponentials / Lie group representations

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...
48. ### Poor Phrasing of a Lie Group Theorem

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"...
49. ### Is $\mathcal{R}$ Lie Group Without 0?

Is it ##(\mathcal{R} without \{0\},\cdot)## Lie group?
50. ### How to find the manifold associated with a Lie Group?

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...