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 ...

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226 views

Minimal parabolic subgroups of a reductive group - Bruhat type decomposition

Let $G$ be a reductive group, $B$ a Borel subgroup, $P$ a minimal parabolic subgroup having a Levi decomposition $P = UL$, let $\alpha$ be one of the two roots of $L$ relative to $T$, and $U_\alpha, ...
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1answer
115 views

Is multiplicative group of real numbers a Lie group?

Is group $\mathbb{R} \setminus \lbrace 0\rbrace$ together with multiplication operation a Lie group? I'm just learning group theory and I would appreciate it if you could explain in greater detail or ...
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1answer
47 views

Quaternion techniques for a geometric description of the composition of two rotations

Let $q \in S^3$. Therefore $q$ can be represented as $q=\cos(\alpha/2) + \sin(\alpha/2)u$ for some $\alpha \in \mathbb{R}$ and some $u \in S^3$ with it's real part zero. Recall that the quaternions ...
7
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164 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
3
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1answer
77 views

When is the Lie algebra of the center of Lie group the center of its Lie algebra

Suppose that $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and the center of $G$ is denoted by $Z(G)$ with its Lie algebra denoted $Z(\mathfrak{g})$. It's easy to show that ...
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26 views

Maximal noncompact forms in classical Lie algebra?

In this short note on Lie algebra, discussing about classical Lie algebra A,B,C,D class, in page 4 after Eq.(7), on the part of B,D class of O(2n,F) and O(2n+1,F) group (or algebra?), there is a ...
7
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1answer
180 views

Is every Lie group the automorphism group of a riemannian manifold?

Given a finite-dimensional Lie Group $G$, is there always a Riemannian manifold $M$, such that $G$ is the group of isometries of $M$?
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60 views

Quaternion, Dihedral groups and A-D-E classification

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification? $\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or ...
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0answers
65 views

SU(2) representations and differential equations in physics.

I studied that $SU(2)$ has a spin $j$ representation $U_j$on a homogeneous space of 2 variables with dimension $2j+1$. Now I am trying to understand the following sentences. Suppose $\phi: ...
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1answer
102 views

Map $n(g,h) = gh^{-1}$ is smooth implies $G$ is a Lie Group.

$G$ is a smooth manifold with group structure. The map $n(g,h) = gh^{-1}$ is smooth implies $G$ is a Lie Group (exercise 2.8, John Lee). We are using the definitions of smooth and etc from John Lee's ...
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1answer
70 views

Euler characteristic of 2-dimensional compact Lie Groups

I'd like to know why the Euler characteristic of $G$, a compact Lie Group of dimension 2, is zero. I'm aware of the fact that this is true not only for dimension 2. The point is that I'm not familiar ...
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2answers
292 views

Is $SO_n({\mathbb R})$ a divisible group?

The title says it all ... Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is ...
3
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0answers
91 views

Lie groups with structure constant $f_{abc} \neq f_{bca}$.

The structure constant $f_{abc}$ of Lie group is defined by the commutators of generators, $$[T^a,T^b]=i f_{abc}T_c$$ automatically $f_{abc}=-f_{bac}$. Can someone give a list of explicit examples ...
4
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2answers
75 views

$\pi_n(SU(2)/Z_N)\simeq?$, $\pi_n(SO(3)/Z_N)\simeq?$, $\pi_n(U(1)/Z_N)\simeq?$

So based on the tool, I have attempted to compute the following threes homotopy groups. $\pi_n(SU(2)/Z_N)\simeq?$ $\pi_n(SO(3)/Z_N)\simeq?$ $\pi_n(U(1)/Z_N)\simeq?$ With a fixed positive integer ...
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0answers
34 views

$A_{2}^{T} + A_{2} < 0$ for $A_{2} =(A_{1}A_{0}^{-1})^{\alpha}A_{0}$?

Given are two matrices $A_0, A_1$, whose symmetric part is negative definite: $A_{0}^{T} + A_{0} < 0$, $A_{1}^{T} + A_{1} < 0$ How could one proof that: $A_{2}^{T} + A_{2} < 0$ for ...
5
votes
2answers
231 views

Homotopy groups of some magnetic monopoles

This is a list of homotopy groups which I (as a physics researcher) encounter when studying magnetic monopole under certain configuration of gauge field profiles. \begin{gather} \pi_2(SU(2)/U(1)) ...
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2answers
419 views

A left invariant vector field on a Lie group

Let $G$ be a matrix Lie group. Let $v$ be a left invariant vector field on $G$ and $v_1 \in \frak g$, where $\frak g$ is a Lie group of $G$. Let $v_1$ be its value at the identity. We define $\phi_t ...
7
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0answers
187 views

ODE system and Lie symmetries

The ODE system (see below), where $F$ is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some $k$ constant. The result that $y$ is a translation ...
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1answer
171 views

The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of ...
2
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1answer
91 views

$X\in \mathfrak{g}$ means flow commutes with left-translation

Suppose $X\in \mathfrak{g}$ is a left invariant vector field on a Lie group G. In this article it mentions that The fact that our vector fields satisfy $L^*_gX = X$ implies that the flow ...
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1answer
67 views

how can I show $H^1(g , Hom_C(g,M))=0$?

For a simple Lie algebra $g$ and a finite dimensional vector space $M$ with a trivial $g-$action, how can I show $H^1(g , Hom_C(g,M))=0$?
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2answers
85 views

Characterization of differentiability via Lie derivatives

Yesterday I asked this question in MathOverflow but did not receive an answer yet. I want to try my chance here too, since I am in kind of a hurry. Answers will be much appreciated. I intend to ...
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1answer
59 views

An existence of exponential function for a Lie algebra.

Let $G$ be a Lie group (given by a matrix). Let $\frak g$ be its Lie algebra. I would like to know if the following is true. "Let $X$ be a matrix in $\frak g$. Then $\gamma(t)=\exp(tX)$ is a curve ...
2
votes
1answer
134 views

How to show that $ G $ is a Lie group?

Problem: Let $ G $ be a group that is also a smooth manifold, and suppose that the mapping $ (x,y) \mapsto x y $ is smooth. How can we show that $ G $ is a Lie group? This is a problem from ...
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0answers
127 views

A representation of $SU(2)$ is self dual

Let $SU(2)$ be a set of $2 \times 2$ unitary matrices over $\mathbb{C}$ with determinant $1$. Let $H_j$ be a $2j+1$ dimensional vector space with basis $x^ay^b$ with $a+b=2j$. A representation $U_j$ ...
0
votes
1answer
35 views

Matrix multiplication in $SO(3)$ that fixes row

I want to find all matrices $G \in SO(3)$ that do not change the first row of elements in $SO(3)$ when right multiplying by $G$, i.e. $$ \{ G \in SO(3): \forall A \in SO(3) \quad A = \begin{pmatrix} ...
2
votes
1answer
37 views

How to show that $\mathbb{Z}[1/p]$ is a discrete subgroup of $\mathbb{R} \times \mathbb{Q}_p$?

By definition, a discrete subgroup $\Gamma$ of a Lie group $G$ is a subgroup such that there is some open neighborhood $U$ of the identity $e$ such that $\Gamma \cap U = \{e\}$. How to show that ...
2
votes
1answer
36 views

How to show that $H(\mathbb{R})/ZH(\mathbb{R}) \cong SO(3, \mathbb{R})$?

Let $H(\mathbb{R})=\{a+bi+cj+dk \mid i^2=j^2=k^2=-1, ij=-ji=k, a, b, c,d \in \mathbb{R}\}$. Let $ZH(\mathbb{R})$ be the center of $H(\mathbb{R})$. How to show that $H(\mathbb{R})/ZH(\mathbb{R}) ...
3
votes
1answer
132 views

Is there a Peter-Weyl theorem for the quasi-invariant measure on a homogeneous space of a compact semisimple group?

Let $H \hookrightarrow G$ be an inclusion of semisimple, compact Lie groups. There is a measure on the homogeneous coset space $G/H$ by pulling back the Haar measure on $G$ via the projection $G ...
0
votes
1answer
132 views

Does local flow of left-invariant vector field commute with the left-translation operator?

Let $G$ be a Lie group and $X$ a left-invariant vector field over $G$ (i.e. $\forall g,p\in G: (D_p l_g)(X_p) = X_{gp}$ whereby $l_g$ is the map $G\rightarrow G:p\mapsto gp$). Let $\phi_t$ be the ...
1
vote
2answers
123 views

Vector field invariant under transitive action: restricts to free transitive action?

Thinking about how you can put vector fields on homogeneous spaces that respect the homogeneity, I'm interested in the following situation: Let $V$ be a nonzero vector field on a manifold $M$, let ...
4
votes
3answers
1k views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
0
votes
1answer
260 views

Classify the compact abelian Lie groups

It's a classical theorem of Lie group theory that any compact connected abelian Lie group must be a torus. So it's natural to ask what if we delete the connectedness, i.e. the problem of ...
2
votes
1answer
76 views

On the Lie bracket of the Lie algebra of the group of invertible elements of an algebra

Assume $A$ is a finite dimensional associative $\mathbb{R}-$algebra, with identity $1( \neq 0)$, let $A^{\times}$ be the set of all invertible elements of $A$, then it's easy to see that ...
0
votes
2answers
161 views

Lie algebra adjoint representation

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Then there are representations $$Ad : G \rightarrow GL(\mathfrak{g}), \; \; ad : \mathfrak{g} \rightarrow GL(\mathfrak{g}).$$ Subrepresentations ...
1
vote
1answer
70 views

A slice orthogonal to each orbit

Assume that a compact (connected) Lie group $G$ acts on a manifold $M$. We choose a $G$-invariant Riemannian metric on $M$ and a point $p \in M$. Then using the exponential map at $p$, we can obtain a ...
0
votes
1answer
56 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
2
votes
1answer
255 views

What are the length of the longest element in a Coexter group for every type?

What are the length of the longest element in a Coxeter group for every type? Thank you very much.
5
votes
0answers
131 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
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votes
1answer
64 views

Is O(n) a compact manifold?

I guess O(n) is a compact manifold How can show that it is?
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0answers
152 views

How do I find the Cartan subalgebra?

I know the definition of a Cartan subalgebra, but how do I actually find it explicitly for a particular Lie algebra over the complex numbers?
2
votes
1answer
149 views

Jacobi identity and Leibniz rule - the same thing?

Is there any formal connection between the Jacobi identity $$[[a,b],c] = [a,[b,c]] + [b,[c,a]]$$ and the Leibniz rule $$d(a \cdot b) \cdot c = a \cdot d(b) \cdot c + b \cdot c \cdot d(a) ~\text{?}$$
2
votes
1answer
184 views

Flag varieties and representation theory

I've recently been reading about flag varieties and their cohomology. I'm mainly interested in representation theory, and I've heard that flag varieties are important objects, especially in Lie ...
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0answers
57 views

The Lie subgroup of the compact Lie group

$G$ is a compact connected Lie group with Lie algebra $g$ whose center is $h$. Let $h^{\bot}$ be the orthogonal complement of $h$ where the inner product is chosen to be invariant under the adjoint ...
3
votes
1answer
167 views

With the branching rules of subalgebra, how can I write down explicit matrix elements for a representation?

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
3
votes
2answers
118 views

Riemannian Manifolds with $n(n+1)/2$ dimensional symmetry group

Given a $n$-dimensional connected Riemannian manifold $(M,g)$, its symmetry group $G$ can be considered as a subbundle of orthonormal frame bundle of $M$ (which I call $F_OM$), yielding: $$\dim G\le ...
3
votes
1answer
164 views

Unitary representations of noncompact groups

Does there exist a noncompact connected Lie group with a finite-dimensional, unitary, faithful, irreducible representation over $\mathbb{C}$? If you remove any of these hypotheses except that of ...
3
votes
1answer
162 views

Subgroups of $SO(4)$ with free transitive action on $S^3$

By considering $S^3$ as the group manifold of $SU(2)$, the ordinary action of $SO(4)$ on the three sphere can be written as the $SU(2)\times SU(2)/\mathbb{Z}_2$ given by the group action of ...
3
votes
2answers
248 views

How to show that exp is a diffeomorphism between symmetric reals and positiv definite matrices?

I am looking for an easy proof of the fact that the exponential function is a diffeomorphism between the finite dimensional vector space of symmetric real nxn-matrices and the open subset of positive ...
1
vote
1answer
175 views

Lie algebra homomorphism and action on a manifold

In Introduction to smooth manifolds Lee says on page 527: If $\mathfrak{g}$ is an arbitrary finite-dimensional Lie algebra, any Lie algebra homomorphism ...