# What is Lie groups: Definition and 101 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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14. ### A Invariance of ##SO(3)## Lie group when expressed via Euler angles

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15. ### Algebra Book on Lie algebra & Lie groups for advanced math undergrad

Posting for my son (who does not have an account here): He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...
16. ### A Unitary representations of Lie group from Lie algebra

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17. ### (Physicist version of) Taylor expansions

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18. ### Lorentz algebra elements in an operator representation

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image [2] below. I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...
19. ### Algebra Lie Groups and Lie Algebras by Robert Gilmore

Anyone reading Lie Groups and Lie Algebras and Some of Their Applications by Robert Gilmore , might be interested in a series of YouTube videos by "XylyXylyX" that follows the book. The first lecture is:
20. ### I Derivative of the Ad map on a Lie group

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21. ### A Is the Exponential Map Always Surjective from Lie Algebras to Lie Groups?

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...
22. ### I Proving Commutator Identity for Baker-Campbell-Hausdorff Formula

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23. ### A Why does the Lie group ##SO(N)## have ##n=\frac{N(N-1)}{2}## real parameters?

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24. ### I Parametrization manifold of SL(2,R)

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25. ### I Exponential map for Lie groups

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26. ### I Computation of the left invariant vector field for SO(3)

I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward. I have been looking at these notes: https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example...
27. ### I Do the SU(n) generators represent any observables?

Hey there, I've recently been trying to get my head around Yang-Mills gauge theory and was just wandering: do the Pauli matrices for su(2), Gell-Mann matrices for su(3), etc. represent any important observable quantities? After all, they are Hermitian operators and act on the doublets and...
28. ### 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...
29. ### I Question about Haar measures on lie groups

I'm not sure if this question belongs to here, but here it goes Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is...
30. ### I Lie groups left invariant vector fields

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31. ### What is SU*(N)? Definition and Explanation

I've run across a Lie group notation that I am unfamiliar with and having trouble googling (since google won't seem to search on * characters literally). Does anyone know the definition of the "star groups" notated e.g. SU*(N), SO*(N) ?? The paper I am reading states for example that SO(5,1)...
32. ### Generators of Lie Groups and Angular Velocity

I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy) I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix. (I understand how I obtain this equation... that is not the issue.) Now I am making the leap to learning about...
33. ### I Simply-connected, complex, simple Lie groups

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34. ### 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...
35. ### Praise HANK YOU All - 2 Years of Learning Differential Forms & Exterior Algebra

No question this time. Just a simple THANK YOU For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups. My math background was very deficient: I am a 55 year old retired (a good life) professor of...
36. ### A Derivative of smooth paths in Lie groups

Hello, Given a Lie group G and a smooth path γ:[-ε,ε]→G centered at g∈G (i.e., γ(0)=g), and assuming I have a chart Φ:G→U⊂ℝn, how do I define the derivative \frac{d\gamma}{dt}\mid_{t=0} ? I already know that many books define the derivative of matrix Lie groups in terms of an "infinitesimal...
37. ### Definition of chart for Lie groups

Hello, I'm reading a book on Lie group theory, and before giving the definition of a Lie group G, the author defines the concept of chart as a pair (U(g), f) where: i) U(g) is a neighborhood of g∈G ii) f : U(g)→f(U(g))⊂ℝn is an invertible map such that f(U(g)) is an open subset of ℝn. My...
38. ### How Is the Abelianization of a Lie Group Defined?

Hi, the abelianization of a group G is given by the quotient G/[G,G], where [G,G] is the commutator subgroup of G. When dealing with finite groups, the commutator subgroup is given by the (normal) subgroup generated by all the commutators of G. If we consider instead the case of G being a Lie...
39. ### Proving Correspondence between SO(3)/SO(2) and S^2

Homework Statement Take the subgroup isomorphic to SO(2) in the group SO(3) to be the group of matrices of the form \begin{pmatrix} g & & 0 \\ & & 0 \\ 0 & 0 & 1 \end{pmatrix}, g\in{}SO(2). Show that there is a one-to-one correspondence between the coset space of SO(3) by this subgroup and...
40. ### Controllability of non-linear systems via Lie Brackets

In http://www.me.berkeley.edu/ME237/6_cont_obs.pdf , page 65, the controllability matrix is defined as: $$C=[g_1, g_m,\dots,[g_i,g_j],[ad_{g_i}^k,g_j],\dots,[f,g_i],\dots,[ad_f^k,g_i],\dots]$$ where the systems is in general given by $$\dot{x}=f(x)+\sum_i^m{g_i(x)\mu_i}$$ Lets say you have a...
41. ### Lie Groups, Lie Algebra and Vectorfields

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let a and g be elements of a Lie group G, the left translation L_{a}: G \rightarrow G of g by a are defined by : L_{a}g=ag which induces a map L_{a*}...
42. ### What are the algebra prerequisites for Lie groups?

I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract...
43. ### Homomorphism between Lie groups

Hello I've been reading some Group theory texts and would like to clarify something. Let's say we have two Lie groups A and B, with corresponding Lie algebras a and b. Does the fact that a and b share the same Lie Bracket structure, as in if we can find a map M:a->b which obeys...
44. ### Galois Theory, Differential Equations, and Lie Groups?

I apologize for the informal and un-rigourous question. I have heard, in passing, that doing Galois Theory over Lie Groups instead of discrete groups is connected to solutions of differential equations instead of algebraic equations. First of all, is this correct? If so, what is this...
45. ### Why do several Lie groups have the same Lie algebra when acting on a space?

Why is it that several lie groups can have the same Lie algebra? could it have to do with the space where they act transitively? Could two different Lie groups acting transitively on the same space have the same Lie algebra?
46. ### Homogeneous spacetime - Lie groups

All Bianchi type spacetimes have metrics that admits a 3-dimensional killing algebra. They are in general not isotropic. Bianchi type IX have a killing algebra that is isomorphic to SO(3), i.e. the rotation group. But what does it mean? If the fourdimensional spacetime is invariant under the...
47. ### Formulating a Method of Steepest Ascent on Lie Groups

Suppose we have a compact Lie group ##G##, and a differentiable function ##f:G_0\to\mathbb{R}## from the identity component of ##G## to the real numbers. I'm looking to maximize the value of this function. Being something of a neophyte at optimization, especially of this kind, I decided to...
48. ### Why Is U(n) Considered Connected When O(n) Is Not?

I surely am missing something about the notion of connectedness, and I clarify this by means of an example: O(n), the orthogonal group, has two subsets with detO=1 and detO=-1. Now, the maximally connected component of O(n) is SO(n), which is the subgroup with detO=1 including the Identity...
49. ### Differentiation Problem on Lie Groups

Suppose θ is a differential 1 form defined on a manifold and with values in the Lie algebra of a Lie group,G. On MxG define the 1 form, ad(g)θ ,where θ is extended by letting it be zero on the tangent space to G How do you compute the exterior derivative, dad(g)θ ? BTW: For matrix...
50. ### Quantum Gravity: Lie Groups vs. Banach Algebras & Spectral Theory

Quantum Gravity: "Lie Groups" vs. "Banach Algebras & Spectral Theory" I'm interested in researching quantum gravity & non-commutative geometry. I am planning to take one math course outside of my physics classes this Fall to help, but can't decide between two: "Lie groups" or "Banach algebras &...