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 ...

learn more… | top users | synonyms

0
votes
0answers
10 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, ...
1
vote
0answers
18 views

Finding basis for skew-hermitian matrices: is my work correct?

Consider the Lie algebra $$ \mathfrak o = \{A \in GL_3(\mathbb C) \mid A^\ast = -A\}$$ This is the Lie algebra of complex orthogonal matrices $O(3,\mathbb C)$. I am trying to find a basis for ...
2
votes
0answers
16 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 ...
0
votes
1answer
11 views

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

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 ...
2
votes
0answers
16 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 ...
0
votes
0answers
16 views

self-adjoint operator over a three dimensional vector space [on hold]

How do I prove that a self-adjoint operator over a three dimensional vector space, is a matrix $$X= \left( \begin{matrix} a & x\\ x^t & B \\ \end{matrix} \right),$$ ...
2
votes
1answer
23 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 ...
1
vote
1answer
40 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 ...
0
votes
1answer
24 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: ...
1
vote
0answers
25 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 ...
1
vote
1answer
42 views

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

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 $w$?
0
votes
1answer
42 views

Lie Groups: Differential Operations

Given a Lie group. Multiplication and inversion act infinitesimally at the identity by: $$\mathrm{d}\mu:\mathrm{T}_{(e,e)}(G\times G)\to\mathrm{T}_eG:(u,v)\mapsto u+v$$ ...
0
votes
1answer
24 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. ...
1
vote
2answers
36 views

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 ...
2
votes
0answers
40 views

Lie Group: Lie Algebra Structure

Application For groups of endomorphism one has: $$\mathfrak{gl}(V)\cong \langle GL(V)\rangle,\,\mathfrak{sl}(V)\cong\langle SL(V)\rangle,\,\mathfrak{so}(V)\cong\langle SO(V)\rangle,\,\ldots$$ ...
0
votes
0answers
15 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: ...
1
vote
1answer
28 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 ...
0
votes
0answers
16 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 ...
1
vote
1answer
54 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 ...
0
votes
0answers
18 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: ...
-1
votes
2answers
25 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 ...
1
vote
1answer
22 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 ...
1
vote
1answer
32 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 ...
0
votes
0answers
13 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 ...
1
vote
0answers
37 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 ...
0
votes
0answers
21 views

Is $BU^-$ open in GL(n,C)? [on hold]

Given $G= GL(n, \mathbb{C})$ seen as a Lie Group, let B be the Borel subgroup of upper triangular matrices and $U^-$ be the subgroup of unipotent lower triangular matrices (i.e. lower triangular ...
1
vote
0answers
21 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. ...
0
votes
0answers
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 ...
1
vote
0answers
48 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)}. ...
0
votes
0answers
12 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 ...
1
vote
0answers
28 views

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}): ...
2
votes
1answer
61 views

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 ...
2
votes
0answers
30 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 ...
2
votes
1answer
25 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?
0
votes
2answers
15 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 ...
3
votes
0answers
20 views

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 ...
3
votes
1answer
32 views

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 ...
1
vote
0answers
10 views

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 ...
1
vote
0answers
36 views

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 ...
2
votes
2answers
28 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 ...
2
votes
2answers
33 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 $G$, ...
2
votes
0answers
23 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 $H$ ...
0
votes
1answer
66 views

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 $g= a\mathbb{1} + i b_j\sigma_j$ ...
5
votes
1answer
149 views
+100

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) ...
3
votes
0answers
34 views

The bijection between central characters and linkage classes over a semisimple Lie algebra

I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in ...
1
vote
1answer
33 views

Non-integral blocks of category $\mathcal{O}$ over $\mathfrak{sl}_2$ are semisimple.

Hi: I have a problem as follows. Consider the category $\mathcal{O}$ of $\mathfrak{g}: = \mathfrak{sl}_2(\mathbb{C})$. Let $r\in \mathbb{C}$ but $r\notin\mathbb{Z}$. Let $s_\alpha$ be the simple ...
1
vote
1answer
27 views

Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)\times SU(2)$, or their Lie algebras

I have seen sources claim that $SO^+(1,3) \cong SU(2) \times SU(2)$, but have seen others claim that only their Lie algebras are isomorphic. Is it true that $SO^+(1,3) \cong SU(2) \times SU(2)$? If ...
2
votes
1answer
58 views

Can someone direct me to prove the following three claims about a rotation group $SO(3)$ and its Lie Algebra?

In class my prof made three claims about a group and its Lie algebra. I cannot find direct reference to these claims because they are delivered in verbatim (im not even sure if I have them jogged down ...
0
votes
0answers
24 views

Proving subgroups of the unitary group U(n) are Lie groups

The problem: Let $K$ be a complex $n \times n$ matrix such that $K$ and $K^\dagger$ is non-singular and $K + K^\dagger$ is positive definite. Show that the set of complex $n \times n$ matrices $G$ ...
1
vote
2answers
156 views

Are there algebraic subgroups of Lie groups which happen to be Lie groups on their own but not Lie subgroups?

Let $G$ and $H$ be Lie groups and $i:H\to G$ an injective group homomorphism, not necessarily continuous. Once $i$ is continuous, it follows easily that $i$ is differentiable and even an immersion. ...