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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Doubt in Peter Olver “Applications of Lie groups to differential equations”

Book: Applications of Lie groups to differential equations. Second edition (1993). Page: 117-120. Chapter: 2. Section 2.4: Calculation of symmetry groups. Example: 2.41. The heat equation. Question ...
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Lie subalgebras of $so(3)$ and complex Lie subalgebras of $sl(2,\mathbb{C})$.

I am looking for a reference that: describes all Lie subalgebras of $so(3)$,and describes all complex Lie subalgebras of $sl(2,\mathbb{C})$. Does someone have appropriate references?
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33 views

Zeroth homotopy group of the space $O(3)/H$

Short version of the question: Is $\pi_0(O(3)/C2) = Z_2$ as $\pi_0(O(3)) = Z_2$? Here $C_2$ is a cyclic group of order 2. Long version or the question: The zeroth homotopy group describes the ...
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37 views

positive roots remain positive

Let $W$ be the Weyl group of $SL_{n+1}$ and $w \in W$. Let $R^+$ denote the set of positive roots with respect to the Borel subgroup of upper triangular matrices. Define $R^+(w)=\{\alpha \in R^+: ...
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47 views

Is the exponential map ever not injective?

Let $G\subseteq GL_n(\mathbb R)$ and let $\mathfrak g$ denote its Lie algebra. Let $e: \mathfrak g \to G$ be the map $X \mapsto e^X$. Does there exist an example of $G$ and $\mathfrak g$ such ...
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Smooth manifolds having an analytic structure.

I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are ...
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185 views

Defining an isomorphism that respects the Lie bracket: is my work correct?

I previously determined that if $\mathfrak{sl}$ denotes the Lie algebra of $SL_2(\mathbb C)$ and $\mathfrak o$ denotes the Lie algebra of $O(3,\mathbb C)$ then a basis for $\mathfrak{sl}$ is given by ...
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15 views

Unimodular Lie group property based on the self-adjoint application

Let $\{e_1, e_2, e_3\}$ a pseudo-orthonormal basis of $\mathcal g$, definined the linear transformation $L:\mathcal g \rightarrow \mathcal g$ such that $L(e_1)=[e_2,e_3]$, $L(e_2)=[e_3,e_1]$, ...
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21 views

Determining whether a Lie algebra is also a complex Lie algebra

I am trying to learn Lie theory. In the following I will share my thoughts. Please, can you check my work for correctness and point out any mistakes to me? I am trying to determine whether ...
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21 views

Center of compact lie group closed?

Let me specify that my knowledge about Lie groups/algebras is reduced to bits and pieces I learned from various diff geometry textbooks. I could not find a reference for the following question (I am ...
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39 views

Is a Lie algebra a complex or a real vector space?

I am trying to learn Lie theory and for this purpose I worked out the Lie algebras of some matrix groups. The examples I worked happened to be complex matrix groups and it lead me to wonder whether, ...
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30 views

Determining Lie algebra: are my thoughts correct?

Let $UT_1(n, \mathbb R)$ denote the set of all upper triangular real matrices with diagonal equal to $1$. This is a Lie group. I am trying to determine its Lie algebra. Please can you tell me if ...
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28 views

Lie algebra of the semiorthogonal group $O(p,q)$ [closed]

How do I prove this: If $\mathcal{O}(p,q)$ is a Lie algebra of the semiorthogonal group $O(p,q)$ then $\mathcal O(p,q)$ consist of all matrices of the form: $$X= \left( \begin{matrix} a ...
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23 views

Do I have the right idea for this isomorphism of Lie algebras of matrix groups?

I previously determined that the Lie algebra of $O(3,\mathbb C)$ is the set of skew symmetric matrices and that the Lie algebra of $SL_2(\mathbb C)$ is the set of traceless matrices. I am now trying ...
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39 views

Where is my mistake: determining the Lie algebra of complex orthogonal matrices

I tried to determine the Lie algebra of $O(3, \mathbb C)$ but I think there is a mistake but I can't find it. Here is my work: Let $\mathfrak o$ denote the Lie algebra of $O(3, \mathbb C)$. The ...
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72 views

Example of a linear algebraic group which is not a Lie group

I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to ...
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41 views

Proof that these two definitions are equivalent

Where can I find a proof of or how can I prove that these two definitions are equivalent? Definition 1: The Lie algebra of a Lie group $G \subset GL_n$ is the tangent space at $I$. Definition 2: ...
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33 views

Can someone intuitively describe the fiber bundle and product spaces of SO(3)?

I have zero understanding of differential geometry or topology so the material found online are useless for me. So in light of that can someone use very general terms or analogy to comment about the ...
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50 views

Are complex numbers a trivial lie group of itself? [closed]

Let $z$ be a complex number, then let's define a map $e^{T(*)}$. Let $w = e^{T(z)}$, where $T$ is some real number. Then is $z$ a lie group of $w$?
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51 views

Lie Groups: Differential Operations

Given a Lie group. Multiplication and inversion act infinitesimally at the identity by: $$\mathrm{d}\mu:\mathrm{T}_{(e,e)}(G\times G)\to\mathrm{T}_eG:(u,v)\mapsto u+v$$ ...
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31 views

Injective acton of an Chevalley generator of $\mathfrak{sl}(2)$ on non-intergral weight module

I have a problem as follows. Let $E_{i,j}\in M_2(\mathbb{C})$ be the elementary matrix and $\mathfrak{g}:=\mathfrak{sl}(2) = \{ A \in M_2({\mathbb{C}})| tr(A) =0 \}$ the special linear Lie algebra. ...
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How to “find” this Lie algebra: proof that $\mathfrak{sl}$ is trace zero matrices

I saw this table here on Wikipedia and it states that the Lie algebra of the special linear group $SL_n(\mathbb C)$ is the group of traceless matrices $\mathfrak{sl}_n$. I know the definition of a ...
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20 views

Is there always a g in a compact connected Lie group whose powers equidistributes in G?

I'm starting to understand some basics things about equidistributed sequences and i found my self asking this question on the basis of the example of the torus and Weyl equidistribution theorem: ...
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51 views

Restriction functors between Categories O over semi-simple Lie algebras

I have the following question: Let $\mathfrak{g}:= \mathfrak{gl}(3)$ be the general linear algebra and $\mathfrak{gl}(2) \cong \mathfrak{a} \subset \mathfrak{gl}(3)$ a sub-algebra. Let ...
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20 views

How does the $10$ dimensional irrep (tensor) of $SU(3)$ look like?

We know that for $SU(3)$ the following tensors furnish the $\mathbf{d}$ dimensional irreducible representation: $$\phi^i\hspace{1cm} (\mathbf{3})\\ \phi^{ij}\hspace{1cm} (\text{asymmetric in ...
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1answer
62 views

Typo on Wikipedia? Dimension of $U(n)$

Let $U(n)$ denote the unitary group. That is, $$ U(n) = \{A \in GL_n(\mathbb C)\mid A^\ast A = I\}$$ Wikipedia states: "The unitary group $U(n)$ is a real Lie group of dimension $n^2$. " There ...
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21 views

A question about finite simple groups of Lie type

I have a question about finite simple groups of Lie type. By the classification theorem of finite simple groups, we know that there are four important class of finite simple groups as follows: ...
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30 views

Basic sanity check: dimension of Lie groups / tangent spaces

A potential typo in an exercise prompted me to question my knowledge of manifolds. So what I need is a sanity check. Here is what I used to think before I got unsure: If $M$ is an $n$-manifold then ...
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1answer
59 views

compactness or not of a Lie group

Is the Lie group generated by this Lie algebra compact or not? $$ [X_i,X_j]=0, [H_i,H_j]=f^{ijk} X_k, [X_i,H_j]=0 $$ $f^{123}>0$, and $i,j,k \in \{ 1,2,3\}$. There are 6 generators in ...
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40 views

True or False statements about compactness of Lie group

Several statements I like to know their True or False statements about the compactness of Lie group. Semi-simple Lie algebra: Every semi-simple Lie group generated by the semi-simple Lie algebra is ...
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21 views

Good reference for The Differntiable Slice Theorem

I am looking for a book that will give me a good proof of The Differentiable Slice Theorem - Suppose a compact Lie group $G$ acts smoothly on a manifold $M$. Then every orbit has a $G$-invarient ...
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46 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
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1answer
26 views

Solving a PDE to Yield Determining Equations

I'm going through an example in Peter Hydon's book "Symmetry Methods for Differential Equations" which finds the basis for the Lie Algebra of the point symmetry generators for Burgers' equations. ...
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23 views

Semi-simplicity of tensor product of simple modules over $\mathfrak{sl}(2)$

I have the following questions: $\bf Notations$: Let $\mathfrak{h} \subset \mathfrak{sl}(2)$ be a Cartan subalgebra, and $W$, $V$ be $\mathfrak{h}$-semisimple (i.e. they are weight modules), simple ...
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65 views

Decomposition of a group manifold; is there an associated group decomposition?

The real symplectic group manifold is diffeomorphic to this Cartesian product of manifolds: \begin{equation} \operatorname{Sp}(2n,\mathbb{R}) \simeq \operatorname{U}(n) \times \mathbb{R}^{n(n+1)}. ...
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13 views

Tangent line to the unitary group $U_1$

I have been working through a small project and the last part has me completely stumped. I have just shown that the matrix $\exp\tau X$ is unitary for all $\tau\in\mathbb{R}$ iff $X$ is ...
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how to conclude a subset of $M_n(\mathbb{C})$ is compact from spectral radius?

could any one tell me which of the following is/are compact subset? $S=\{A\in M_n(\mathbb{C}): \rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): A=A^*,\rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): ...
3
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1answer
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set of all $2\times 2$ matrcies having neither eigen value is real

Could any one tell me whether the following subsets of $M_2(\mathbb{R})$ are open, closed or neither open nor closed? set of all $2\times 2$ matrcies having neither eigen value is real. set of all ...
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36 views

An apparent contradiction in SU(3) structure constants?

According to http://www.phys.washington.edu/users/ellis/Phys5578/SU3_5.htm or the related Wikipedia article, the following equation should hold: $[ \frac{\lambda_3}{2}, \frac{\lambda_4}{2}] = i ...
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1answer
26 views

If G is a compact semisimple Lie group and Z is its center, is G/Z always compact?

The title pretty much sums up the question: Suppose $G$ is a compact semisimple Lie group with center $Z$, The question is if $G/Z$ is always compact? or, under which conditions will it be compact?
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17 views

The trivial subgroup is closed (in the sense of topology)

I try to understand the proof of the below statement: Statement: Let $\phi: G \rightarrow H$ be a homomorphism of Lie groups. Then the kernel of $\phi$ is a closed subgroup of $G$. Proof: Put ...
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The space of minimal geodesics on $SU(2m)$

In the proof of Bott periodicity for the unitary group in Milnor's Morse theory (Lemma 23.1, page 128), it is asserted that the space of minimal geodesics from $I$ to $-I$ in the special unitary group ...
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The irreducibility of representations of parabolic induction over $\mathfrak{gl}_n(\mathbb{C})$

I have the following problem: Let $\mathfrak{g} = \mathfrak{gl}(n)$ be the general linear Lie algebra over $\mathbb{C}$. Denote $\Pi = \{\alpha_i:=\epsilon_i-\epsilon_{i+1}\}_{i=1}^{n-1}$ by the ...
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Does the projection from a compact Lie group to its component group split?

This is an elementary question that probably admits an elementary counterexample, but ... Let $G$ be a compact Lie group and $G_0$ its identity component. One then has a short exact sequence $$ 1 \to ...
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Upper triangular matrices $UT_n(\mathbb K)$ is a matrix group: is my proof correct?

I am solving some exercises in Tapp's matrix groups for undergraduates and would be very grateful if someone could check my work (it's exercise 4.12): A matrix $A\in M_n(\mathbb K)$ is called ...
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36 views

Some solvable Lie algebra but not nilpotent

Can someone provide two concrete examples the Lie algebra which is solvable, but not nilpotent? -- And further explain the subtle differences between the solvable Lie algebra and the ...
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36 views

Intuitive understanding of lie group definition

So I have the following definition from the book: Definition: A matrix Lie group is any subgroup $G$ of $GL(n, \mathbb{C})$ with the following property: If $A_m$ is any sequence of matrices in $G$, ...
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25 views

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 $H$ ...
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Prove the manifold of SU(2)/U(1) is the 2-sphere.

I want to demonstrate that the manifold of $SU(2)/U(1)$ is a 2-sphere. In a text-book I've found this way of solution, where there are some unclear points. Let to be $g= a\mathbb{1} + i b_j\sigma_j$ ...
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260 views

Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...