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Questions tagged [lie-groups]

A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group.

Consider using with the (group-theory) tag.

Lie groups are groups that are also differentiable manifolds that represent the best developed theory of continuous symmetry of mathematical objects.

Examples of lie groups are:

1) The Euclidean space $\mathbb{R}^n$ under addition is a lie group.

2) The special orthogonal group of real orthogonal matrices with determinant $1$ (note that $n=3$ is the rotation group in $\mathbb{R}^3$).

3) The spin group, which is the double cover of the special orthogonal group such that $\exists$ a sequence of lie groups:

\begin{equation*} 1\to Z_2\to~\text{Spin}(n)\to SO(n)\to 1. \end{equation*}

Note that it has dimension $\frac{n(n-1)}{2}.$

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Are all Lie groups with a (linear) representation a matrix Lie group?

I am self studying quantum field theories and as many of you know this can not be properly done without at least some knowledge of Lie groups and the corresponding algebra. I am reading the QFT text by Michele Maggiore “A modern introduction to…
William Crawford
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Warner Ex. 3.12; Prove Fundamental Group of a Lie Group is Abelian Using "Discrete normal subgroups are central"

Specifically, Warner says to prove that if a Lie hom $\phi:G\to H$ has a discrete kernel then the kernel is actually contained in the center. (I can do this part.) The exercise then says to use this fact to prove that the fundamental group of a Lie…
user552402
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Is every Lie subgroup of $SU(2)$ closed?

Is there an easy way to prove that every Lie subgroup of $SU(2)$ is closed? That is, is it true that any $K=\exp_{SU(2)}(k)$ is closed for every subalgebra $k \subset \mathfrak{su}(2)$? If there isn't, is there a reference for this fact? I thought…
Max Reinhold Jahnke
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Normal Subgroups of Parabolic Groups in Simple Lie Groups

If G is a noncompact simple (real) Lie group with trivial center and Q is a parabolic subgroup of G, is there a "nice" description of the normal subgroups of Q? In particular, is there any "nice" condition on G that will ensure that Q is simple?
throw12345678
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How to determine if a Lie group is simply connected

I have an explicit description of a Lie group in terms of matrices with entries in a quaternion algebra, and I want to determine if there exist isogenous covers. How can I do this?
Watson Ladd
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Continuous function of GL(n,C)

I was reading a book on Lie groups where they define a function $f: GL(n,C)->GL(n,C)$, by $f(B)=B^\dagger B$ where $B \in GL(n,C)$, and $GL(n,C)$ denotes the general linear group over C (complex field). They mention something like "it's easy to see…
Cami77
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Can a Lie group be a manifold with boundary?

Normally in differential topology, we consider both manifolds with and without a boundary. However, for a Lie group do we only restrict our attention to manifolds without boundary?
z.z
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compactness and fundamental group of $SL(n,\mathbb{C})$

could any one help me to prove the heading? $SL(n,\mathbb{C})$ is closed I can prove only, what are the other tools I need? $SL(n,\mathbb{C})$ connected?simply connected?
Myshkin
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Lie Groups question from Brian Hall's Lie Groups, Lie Algebras and their representations.

In page 60 of Hall's textbook, ex. 8 assignment (c), he asks me to prove that if $A$ is a unipotent matrix then $\exp(\log A))=A$. In the hint he gives to show that for $A(t)=I+t(A-I)$ we get $$\exp(\log A(t)) = A(t) , \ t<<1$$ I don't see how I…
MathematicalPhysicist
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Connected, one-dimensional Lie groups

Show that a connected, one-dimensional Lie group $G$ is isomorphic to $\mathbb{R}$ or $S^1$. So far my approach has been to show a non-trivial, one-parameter subgroup of $G$ is surjective, but I have not really made much progress. I have only just…
user114158
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What do 1, 2, 3 represent in $\operatorname{U}(1)+\operatorname{SU}(2)+\operatorname{SU}(3)$?

What do 1, 2, 3 represent in $\operatorname{U}(1)+\operatorname{SU}(2)+\operatorname{SU}(3)$? If they are dimensions, how they can be added? or plus has another meaning?
Agul
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All left invariant vector fields are right invariant: What's my failure in reasoning

I understand that the conclusion is absurd. But here is my reasoning. Let $G$ be a lie group, let $x \in \mathfrak{g}=T_e(G)$, and let $\gamma(t)$ be a homomorphism $\mathbb{R} \to G$ such that $\gamma'(0)=x$. Then for any $t_0, t$,…
De Yang
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How do evaluate the trace of a member in $SU(n)$?

In Clifford Taubes' differential geometry(if you doubt I might be wrong in my transcription, please check section 5.6) , he seems to be claiming that $$i^{*}w_{q}=e_{I}^{*}w_{q}=\int^{1}_{0}Tr(e^{-as}qe^{as}da)ds$$ is 0 when we are working on…
Bombyx mori
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What is the automorphism group of Lie group $E_6$

I found out that there are outer automorphisms of compact Lie group $E_6$. For example complex conjugation $\tau$ as defined in this Yokota paper is outer automorphism of $E_6$ which fix $F_4$ subgroup. This means that automorphism group of $E_6$ is…
mmm
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Is the set of all Heisenberg Matrices Connected?

The set of all $3\times 3$ matrices of the form $$\begin{pmatrix} 1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1\\ \end{pmatrix}$$ forms a group surely, are they connected?
Myshkin
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