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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Question about the definition of the adjoint representation of Lie groups

Let $\mathfrak{g}$ denote the Lie algebra of a Lie group $G\leq GL(n)$. The adjoint representation of $G$ is defined as the function $Ad_g:\mathfrak{g}\rightarrow\mathfrak{g}$ that maps each ...
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What is $\mathbb{S}^{1}/\{\pm {1},\pm {i}\}$ isomorphic to

I am starting to self-study Lie Groups, and came across this question. What is $\mathbb{S}^{1}/\{\pm {1},\pm {i}\}$ isomorphic to To begin with, it seems as if $\{\pm {1},\pm {i}\}$ are ...
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42 views

Manifolds as Homogeneous Spaces

With very little effort one can, for example, show that $S^n$ can be written as a homogeneous space as $S^n\cong G/H$, where $G$ is the group of all rotations in $\mathbb{R}^{n+1}$ about the origin ...
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Left invariant forms on lie groups?

Let $G$ be a Lie group. A vector field $X\in\mathfrak{X}(G)$ is left-invariant if the diagram below is commutative: for every $g\in G$ where $L_g$ stands for the left translation by $g$. Now a ...
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Maurer-Cartan form left invariance

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra, that is, the $\mathbb R$-space of left invariant vector fields on $G$. Recall the isomorphism $\mathfrak{g}\simeq T_eG$. The Maurer-cartan ...
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Is exponential of GUE random matrix Haar random?

Consider the matrix exponential map $H \mapsto e^{i H t}$ acting on the Gaussian unitary ensemble (GUE) of Hermitian matrices. I would expect that for large $t$, the resulting measure on the unitary ...
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25 views

Archimedean Hecke Algebra for number fields

Suppose we have a $GL_1$ over a number field $F$. I am interested in a description for the archimedean Hecke algebra (always taking the maximal compact subgroup). We know will be the tensor product ...
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80 views

Under what mild condition, inclusion of a conjugate subgroup in the initial subgroup gives equality?

Let $G$ be a group, $H \leq G$ a subgroup, and $x \in G$. If $xHx^{-1} \subset H$, then we may not have $xHx^{-1} = H$. Here is a simple counterexample. But this don't prevent the hope to finding ...
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16 views

How to find the absolutely irreducible representations of $D_4$ dihedral group

I want to calculate the irreducible representations of $D_4$ and ultimately the absolutely irreducible representations. Right now what I do is since I know the group order is $2n=2(4)=8$ write down ...
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70 views

Lie groups for beginners: Lie group of hyperbolic geometry

I am trying to understand Lie groups and their relation to (2 dimensional) hyperbolic geometry. as far as I understand it (which is not very far, I am pushing my understanding here) the Lie-group ...
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1answer
14 views

Irreducible Components of Standard representation of SO(2) on $\mathbb{C}^2$

We denote by $SO(2)$ the group of $2 \times 2$ orthogonal matrices of determinant $1$ with real entries. We have a natural representation of $SO(2)$ on $\mathbb{C}^2$ given by matrix multiplication: ...
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Isomorphism between tangentspace of Lie group quotient and quotient of Lie algebras.

Let $G$ be a Lie group and $H$ a closed subgroup with Lie algebras $\mathfrak g, \mathfrak h$. Then the canonical projection $p: G \to G/H$ is a submersion. Fix a $g \in G$. We have a linear ...
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42 views

Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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22 views

Affine group, identification and multiplication law

I have a question about the group of affine transformations of $\mathbb{R^2}$. Where by that I mean the following: $Aff(\mathbb{R^2})=\{AX + b\mid A \in GL_2(\mathbb{R^2}), b \in ...
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31 views

Why a left-invariant Haar measure is $\mu(A)=\int_A \frac{1}{a^2}da\,db$ and a right-invariant Haar measure is $\mu'(A)=\int_A\frac{1}{a}da\,db$?

Let $G$ be the group of affine transformations of $\mathbb R$, $x\mapsto ax+b$, $a>0$. $G$ is the half-plane $(a,b);a>0$. A left-invariant Haar measure is $\mu(A)=\int_A \frac{1}{a^2}da\,db$, ...
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17 views

Lie derivative of Killing form

One may choose the Killing form $Tr(T^aT^b)$ as the metric $g$ on a Lie group $G$. It is known that the Killing form is invariant under the adjoint transformation, i.e., $\delta Tr(XY)= ...
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25 views

Motivation of the definition of principal series.

I am reading the book representation theory of semisimple groups. On page 33, the principal series representation $\mathcal{P}^{k,iv}$ is defined as follows. What are motivations of the above ...
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Why unitary principal series is unitary?

I am reading the book representation theory of semisimple groups. On page 33, I tried to verify that $\mathcal{P}^{k,iv}$ is unitary. We need to verify that $$ \left|\left| ...
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97 views

Rigorous definition of a “generator” for a transformation group

EDIT : I think that the whole question can be summarized as « how do we know that every conformal transformation can be written under the form $e^{tX}$ for some operator $$X ? » I'm reading the ...
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1answer
27 views

Root spaces for symplectic Lie algebra $\mathfrak{s} \mathfrak{p}_n$

Consider the symplectic Lie algebra $\mathfrak{s} \mathfrak{p}_n$ over a field $K$. I know that the root system is given by $C_n=\{\pm 2e_j, \pm e_j \pm e_k:j,k=1 \cdots n, j \neq k\} $ where ...
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Does a coordinate basis exist locally on any manifold?

A holonomic or coordinate basis for a differentiable manifold is a set of basis vector fields $\{e_k\}$ such that some coordinate system $\{x_k\}$ exists such that $e_k=\partial/\partial x_k$. A ...
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Lift of inclusion $U(k)$ into $SO(2k)$ to $Spin^c$ group

In "Clifford Modules" by Atiyah, Bott and Shapiro (p.10) or "Dirac Operators in Riemannian Geometry" by Friedrich (p.28) one finds some sort of a lift of the natural inclusion $\operatorname{U}(k)\to ...
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21 views

Compact factors of Lie groups; possibly varying definitions

Let $G$ be a real connected semisimple Lie group. Are the following equivalent?: (1) $G$ has no proper cocompact Normal subgroups. (2) $G$ has no proper cocompact connected Normal subgroups. In ...
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Action of Lie group on vector fields (of homogeneous space)

1) If I have a Lie group $G$ acting smoothly on a manifold $M$, how does this gives also an action of $G$ on the vector fields on $M$? 2) When I read that the group G acts canonically on the vector ...
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50 views

Is any of these two groups a smooth manifold?

This question came to my mind while I was going through Iian B. Smythe's talk titled A Crash Course in Topological Groups. In the talk it is mentioned that, Lie group G is a group, which is also ...
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1answer
10 views

Induced Lie algebra homomorphisms are equal

Let $\varphi,\psi:G\to H$ be Lie group homomorphisms, with G connected, such that the induced Lie algebra homomorphisms $d\varphi,d\psi:g\to h$ are identical. I want to show that then ...
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32 views

Contraction of representations of universal enveloping algebra

$\quad$ (Following, e.g. SBBM) Given a Lie algebra contraction $\mathfrak{g}\xrightarrow{t(\epsilon)}\mathfrak{g}_0$, one can contract a family $\{\rho_{\epsilon}:\mathfrak{g}\rightarrow ...
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27 views

Adjoint action on quotient space of Lie algebras and vector fields on quotient group

Let $G$ be a Lie group and $H$ a closed subgroup. Then $G/H$ has a unique structure of a smooth manifold with canonical projection $p: G \to G/H$. If $\mathfrak g = T_e(G), \mathfrak h = T_e(H)$ are ...
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“Extension” of orthogonal group

I'm looking for a Lie group $G$, subgroup of $GL(n,\Bbb{R})$, and which contains $O(n,\Bbb{R})$ as a subgroup: $$ O(n,\Bbb{R}) \subseteq G \subseteq GL(n,\Bbb{R}). $$ Obvious examples: the conformal ...
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Deriving SO(3) multiplication rule from so(3) commutation rules

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 ...
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29 views

Understanding Cartan clasification

In class we defined Cartan subalgebra h (of g) as maximally abelian subalgebra containing only ad diagonalizable elements. ad is adjoint map $ad_{H_i}(E) =[H_i, E]$. I have a couple of questions ...
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Two questions on roots of finite, simple, complex lie algebra

Why are there at most two root lengths for a finite, simple, complex lie algebra? I know it is from the constraint that the $2(\alpha,\beta)/(\alpha,\alpha)$ is integer, but what is the argument? ...
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33 views

What's wrong with this derivation about sympletic matrices?

Let $M$ be a $2\times 2$ matrix, the definition of sympletic that I have is that $M$ is sympletic if $$MJM^T = J,$$ being $J$ the matrix $$J = \begin{pmatrix}0 & -1 \\1 & 0\end{pmatrix}.$$ ...
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1answer
36 views

Do we need transpose in the definition of a dual representation?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. There is an action of $G$ on itself given by left multiplication: $G \times G \to G$, $(f,g) \mapsto fg$, $f, g \in G$. There is a ...
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1answer
22 views

low-dim unitary groups and their actions

I need someone to explain for me the unitary groups $U(1)$, $U(2)$ and $U(3)$ and their actions: Specifically: $U(3)/U(2)$ $U(3)/U(2)\times U(1)$ $U(3)/U(1)\times U(1) \times U(1)$ I have seen ...
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1answer
26 views

Trivial representation of a lie Algebra?

Can someone explain why if $\rho:L \rightarrow \text{End}(\mathbb{C})$ is a lie algebra representation then it must be that $\rho(x)=0\ \forall \ x\in L$.
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Affine Kac-Moody Group Isometry of a Manifold

An isometry of a Riemannian manifold is an infinitesimal displacement generated by a Killing vector field $V=\zeta^aV_a=\zeta^aV_a^i\frac{\partial}{\partial x^i}$. If the isometry corresponds to the ...
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Chirality in Lie groups and Lie algebras

Is there an example of a chiral Lie group?In particular, is it true to say that the map $g\mapsto g^{-1}$ is orientation reversing for odd dimensional Lie groups? Moreover is there a concept of ...
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1answer
23 views

Dimension of projective homogenous G-varieties

For a parabolic subgroup of $P$ an algebraic group $G$, symbolized by a set of dots in the Dynkin diagramm of $G$, one obtains a quotient variety $X = G/P$. Given $G,P$ how can one calculate the ...
2
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1answer
19 views

Why $b: V \times \overline{W} \to \mathbb{C}$ is $G$-invariant?

I am reading the book Representations of Compact Lie Groups. On page 79, in the proof of Theorem 4.6, it is said that $b: V \times \overline{W} \to \mathbb{C}$ is $G$-invariant. We have \begin{align} ...
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40 views

How to show that $v \mapsto \pi(f)v$ is differentiable?

Let $G$ be a compact group. Let $(\pi, V)$ be a representation of $G$ and $f$ a smooth function on $G$. Define \begin{align} \pi(f)v = \int_G f(x)\pi(x) v dx. \end{align} We have \begin{align} & ...
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1answer
32 views

Projectivization and stereographic projection, or: Why nonlinear + nonlinear = linear?

This previous question had me thinking about something I've taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line ...
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1answer
26 views

determinant constraint on the dimension of SO(n)

It is very well known that the dimension of $SO(n)$ is $n(n-1)/2$, which is obtained by the number of independent constraint equations we have from the fact that the matrix is orthogonal. However, it ...
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23 views

Prove that the integral curve of a vector field intersects just one time $\partial E$

Suppose $X_1, X_2$ are smooth vector fields in $\mathbb R^n$ with a group law $\cdot$ (the neutral element of $\cdot$ is $O$). We denote with $exp(tX_i)$ the point reached after the time $t\in \mathbb ...
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25 views

Is the log of closed connected subgroups a vector space?

Let $G$ be a simply connected nilpotent Lie group (so that the exponential map $\exp:{\mathfrak g}\to G$ from the Lie agebra ${\mathfrak g}$ to $G$ is a diffeomorphism, and hence so is the inverse ...
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14 views

Deriving the structure constants of the SO(n) group

The commutation relations for the $\mathfrak{so(n)}$ Lie algebra is: $$([A_{ij},A_{mn}])_{st} = -i(A_{j[m}\delta_{n]i}-A_{i[m}\delta_{n]j})_{st}$$ where the generators $(A_{ab})_{st}$ of the ...
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Question 2 in Exercises section 5.8 in the book of Brian Hall's Lie groups,… : an elementary introduction.

Let $\pi$ be an irreducible representation of $\mathfrak{sl}(3,\mathbb{C})$ and let $\pi^*$ be the dual representation of $\pi$ defined by $\pi^*(X) = - \pi(X)^T$, where $T$ stands for transpose. Show ...
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Change of variable for integration with respect to Haar measure

I know how to estimate the integral \begin{gather} \int f(Ub)\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{gather} where $f:S^n(\mathbb{R})\to \mathbb{R}$ ...
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31 views

What exactly is the diagonal subgroup of a group?

In specific consider the example of $SU(2)_a \times SU(2)_b$. What is the definition of the diagonal subgroup and how can one construct it from the generators of the group (or its algebra)? This ...
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41 views

Differential equation and Lie algebra

If I have this differential equation: $$ \frac{d\vec{x}}{dt} = F(\vec{x}) $$ and when $F = A$ is a matrix we can have the solution: $$ \vec{x}(t) = e^{At} \; \vec{x} $$ But what if $F = \mathfrak{g} $ ...