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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 $\frac{n(n-1)}{2}.$

7686 questions
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What are the simply-connected two-dimensional Lie groups?

I would like to know what the simply-connected Lie groups of dimension $2$ are. It is well-known that for every Lie algebra, there is exactly one simply-connected Lie group having it as its Lie algebra. I know that there are two two-dimensional Lie…
Niklas
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Generators of compact Lie groups

Suppose $G$ is a compact connected Lie group and let $\{X_i\}$ be a basis for its Lie algebra $\mathfrak g$. We know that the exponential $\exp:\mathfrak g \to G$ is surjective but when is it the case that $G$ is generated by $\{\exp(tX_i) : t\in…
Eric O. Korman
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Bi invariant metrics on $SL_n(\mathbb{R})$

Does there exist a bi-Invariant metric on $SL_n(\mathbb{R})$. I tried to google a bit but I didn't find anything helpful.
HenrikRueping
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Need help with proof of $SO(3)$ is path connected

I am working on an exercise in Tapp's matrix groups for undergraduates. It is a proof that $SO(3)$ is path-connected. $SO(3)$ is the group $$ SO(3) = \{A \in O(n)\mid \det A = 1 \}$$ where $O(n)$ is the group of orthogonal matrices. My work so…
learner
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Classification by Dynkin diagram: Why's there no $E_9$?

from Wiki According to Dynkin's classification, we have as possibilities these only, where n is the number of nodes: To me it seems not obvious why there should not be a $E_9$? Further clicking got me to the following: "Roughly speaking, symmetries…
draks ...
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connected $\Rightarrow$ path connected?

Well, so far, I have noticed that whenever a matrix lie group is connected it is path connected, so is it true that in matrix lie group connected $\Rightarrow$ path connected?If yes, could anyone tell me where I can get the proof?or if some one tell…
Myshkin
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Automorphisms of a torus

Why is the set of automorphism of a n-dimensional torus T (also denoted $T^n$) is $GL(n,\mathbb{Z})$ i.e the set of invertible matrices with integral coefficients? In the book by Brocker and Dieck- Representations of Compact Lie Groups, the crucial…
Neeraj Kumar
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Connected, not path-connected subgroup of $\mathbb{T}^2$

I read (in Structure and Geometry of Lie Groups by Hilgert and Neeb, I think) that it is possible for a Lie group to admit a connected yet not-path-connected subgroup. Specifically it said $(\mathbb{R/Z})^2$ admits such a subgroup. I did not manage…
Cronus
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What is the formal mathematical theory behind the concept of the anti commutator used to quantize fermions?

I understand Lie groups are defined by the structure constants associated with the lie brackets, which are treated as commutators in quantum mechanics, but i dont know of a math theory related to group theory to define or use an anti commutator. If…
Fiwel
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Different definitions for semisimple Lie group

I am confused about two definitions for the notion of a semisimple Lie group i found. Lets say for simplicity i am only interested in matrix groups. In this case, do the following two object-classes coincide? 1) A connected Lie group that does not…
Mekanik
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Normalizer of normalizer of maximal torus in a Lie group

I'm stuck at this problem, Let $G$ be a compact connected Lie group and $T$ a maximal torus. Let $H$ be a closed subgroup of $G$. Let $N(T)$ and $N(H)$ denote the normalizers of $T$ and $H$ respectively. Show that if $N(T) \subset H $then $N(H) =…
apurv
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What does $GL_n(R)$ look like?

Exactly as in the title - what does the general linear group "look like" (you are free to interpret this however you like) as submanifold of $R^{n^2}$? What should I imagine when I think of it? (I am aware that the simplest nontrivial case is…
Elle Najt
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Universal enveloping algebra as algebra of differential operators

Let $G$ be Lie group and $g$ be its Lie algebra. Is it true (and if not generally, then under which circumstances) that the the algebra of its differential operators is isomorphic to the universal envelopping algebra $U(g)$ of $g$? If yes, can you…
phil
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First and second homotopy groups of a connected Lie group

I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$? Thanks for anyone who give me references for a 'simple proof' of these…
amine
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Why connected Lie groups are homotopy equivalent to connected compact Lie groups?

I am looking for a simple proof of a Mostow Theroem, which asserts that any connected Lie group $G$ admits a maximal compact subgroup $K$ (which is necessarily connected) such that $$G\simeq K\times\mathbb{R}^d\quad(\text{for certain}\ d).$$ So $G$…
amine
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