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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40 views

$O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$

During a lecture of a Lie Algebras yesterday, the professor of the class stated the following fact without proof $O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$ Note that we are viewing ...
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1answer
27 views

The complex structure of a complex torus

A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex ...
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0answers
20 views

Does the “differential” of a unitary representation give continuous operators on the space of smooth vectors?

Let $\pi : G \rightarrow U(H)$ be a strongly continuous unitary representation of a Lie group, $G$, on a Hilbert space, $H$. Let $H_\infty$ be the space of smooth vectors in $H$, those $v$ for which ...
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0answers
21 views

Representation ring of circle group over complex field

Can someone please describe how to find a representation algebra of circle group over complex field ? I am reading " representation theory of compact Lie group" chapter 3 section 7. It will be great ...
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1answer
26 views

General Element of U(4)

Relating back to previous question about how to write a general element of $U(2)$, I am now wondering about how to write a general element of $U(4)$. Define $\Gamma_{(i,j)}:=\sigma_i\otimes\sigma_j$ ...
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26 views

Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
3
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2answers
29 views

Why do “the Dynkin components of a weight play the role of eigenvalues with respect to the generators $H^i$ of the Cartan subalgebra”?

In the book "Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists" by Jürgen Fuchs,Christoph Schweigert the authors write "In the description of representations, the ...
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0answers
18 views

In what sense are complex representations of a real Lie algebra and complex representations of the complexified Lie algebra equivalent?

In this book I read Proposition A.1. The irreducible complex representations of a real Lie algebra $\mathfrak{g}$ are in one-to-one correspondence with the irreducible complex-linear ...
2
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1answer
49 views

Are the physics and math definitions of a complex representation equivalent?

I was astonished to read at Wikipedia that The term complex representation has slightly different meanings in mathematics and physics. In mathematics, a complex representation is a group ...
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28 views

What does it really mean to complexify the $10$-dimensional representation of $ \mathfrak{so}(10)$?

A commonly used "trick" in $SO(10)$ Grand Unified Theories is to use a "complex" instead of a "real" $10$-dimensional representation for the Higgs fields. My problem is understanding what this ...
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11 views

Complexification of a Lie algebra representation in terms of weights?

EDIT: I found in this book the sentence: The weight system of a real representation of $G$ is defined to be the weight system of its complexification I think if someone can explain what this ...
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36 views

Where to the degrees of freeedom go when a complex representation becomes a real representation of a subalgebra?

As an example consider the complex $16$-dimensional representation of $\mathfrak{so}(10)$. When $\mathfrak{so}(10)$ is reduced to the subalgebra $\mathfrak{so}(9)$, the complex $16$-dimensional ...
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1answer
46 views

Finite normal subgroups of $SO(4)$

What are the finite normal subgroups of $SO(4)$? If these do not exist (or if they are trivial, e.g. from some projection to $SO(2)$), are there different finite normal subgroups of $O(4),$ $U(4)$, ...
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7 views

Regular elements on Lie algebra and on Lie group level

I want to understand (better) the definition and meaning of regular elements of semisimple Lie groups resp. Lie algebras. A regular element $X$ of $\mathfrak g$ is one whose centralizer has smallest ...
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0answers
53 views

About the diameter of a Riemannian manifold

My exact problem is a Riemannian manifold defined on $SU(2^n)$, where the metric is defined as follows: If $U(t)$ is a curve so that $U′(t)=−iH(t)U(t)$ (so just a unitary evolution of a quantum system ...
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2answers
39 views

Decomposition of an unitary operator by simple operators

For quantum computation, it's well known that any unitary operator can be approximated with an arbitrary accuracy by simple operators, for example to approximate an unitary operator on n qubits by no ...
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0answers
22 views

An incorrect proof that the Lie algebra matrix exponential is always injective. What's wrong?

Suppose we have two square complex matrices $X,Y $ in the lie algebra $\mathcal{G}$ of matrix Lie group $\mathbb G$ such that $e^X = e^Y$. Then $e^{tX}$ and $e^{tY}$ define the same one-parameter ...
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25 views

Lie group is parallelizable

While going through the proof given by Alex Youcis at http://math.stackexchange.com/a/308798/86801 , I found the part where the local representation of the map $\ \Phi\ $ is shown to be the identity ...
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0answers
14 views

Reductions of structure groups and sections of coset bundles

I'm looking for a reference for the following proposition: Let $G$ be a Lie group and $H$ a (closed) Lie subgroup of $G$. Let $E \to B$ be a principal $G$-bundle. Then reductions of the ...
2
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1answer
33 views

Orbit closures of real symmetric bilinear form

Let $\alpha$ and $\beta$ be two real symmetric bilinear forms in $\operatorname{sym}(\mathbb{R}^n)$, with signatures $(p_{\alpha},n_{\alpha},z_{\alpha})$ and $(p_{\beta},n_{\beta},z_{\beta})$. I ...
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0answers
19 views

Orbit closures of symmetric bilinear form

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where: ...
1
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1answer
96 views

Parallelizability of Lie groups

I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions: The map $\ G \rightarrow TG \ $ given ...
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0answers
25 views

Lie group action from the Lie algebra

want to find the corresponding lifting f the standard U(n) lifting on $C^n$ to $L=C^n \times C$ with hermitian metric $e^(-|z|^2)$. I try to follow the method in Donaldson, and I find if B in u(n) ...
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0answers
24 views

Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
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0answers
13 views

Do there exist equidistributed countable subgroups in (compact) Lie groups?

By an equidistributed countable subgroup I mean a countable subgroup (with a finite or possibly countable set of generators) that is dense in $G$ such that for any sufficiently nice function (Haar ...
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0answers
20 views

Relation between Iwasawa and Cartan decompositions

Given a real semi-simple Lie group, one have the two decompositions - $$G=KAN \text{ - Iwasawa Decomposition} $$ and $$G=KA^{+}K \text{ - Cartan Decomposition} $$ I'm interested in an explicit ...
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0answers
40 views

Level sets on $SU(n)$

Given $G \in SU(n)$, what are the level sets of the function $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in ...
2
votes
1answer
53 views

Scalar product on Lie algebra of compact Lie group [duplicate]

I am studying Differential Geometry and I am facing with a lemma in which there is a step that I do not understand. In particular, let $G$ be a connected compact Lie group, is used "$\langle\ \cdot , ...
1
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0answers
37 views

Lie group representation and inner product

Let $G$ be a connected semisimple Lie group.Now let $\theta$ be the Cartan involution of $G$ and let $(\pi,V)$ be a finite dimensional representation of $G$. On page 22 of Analysis and geometry on ...
3
votes
1answer
184 views

Lie bracket; confusing proof from lecture

I am having some difficulties understanding this proof. Let $G$ be a closed matrixsubgroup of the general linear group. We have a right translation $Y(g):=dR_g(e) Y(e)$ on the Lie algebra $Y \in ...
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0answers
31 views

The Product map of a Lie Group is a Submersion.

Problem 7.1 of Lee's Introduction to Smooth Manifolds (2nd Edition) reads: Show that for a Lie group $G$, the multiplication map $\mu:G\times G\to G$ is a submersion (Hint: Use Local Sections). ...
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57 views

Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
6
votes
1answer
72 views

Showing $U(n)/Z(U(n))=SU(n)/Z(SU(n))$

I was working on the following problem from Stillwell's Naive Lie Theory. Prove that $U(n)/Z(U(n))=SU(n)/Z(SU(n))$. It was shown earlier in the text that $Z(U(n))=\{ e^{i\theta} \textbf{1}: ...
1
vote
1answer
28 views

A compact, connected, abelian Lie group is a torus?

How to prove that a compact, connected, abelian Lie group is a torus? It seems very intuitive. Any reference?
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0answers
28 views

Easy examples of non-arithmetic lattices

I'm starting to look a bit more at discrete subgroups of Lie groups, particularly lattices. A lot is written about arithmetic lattices of Lie groups, and examples abound. It appears that much less is ...
2
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2answers
47 views

Adjoint representation

I was just wondering why the adjoint representation of the Lie group $Ad$ and Lie algebra $ad$ are called representation. Maybe this word is derived from abstract algebra somehow, but I don't ...
2
votes
2answers
48 views

Homogeneous Isotropic Riemannian Manifolds

In John Lee's book Riemannian Manifolds on page 33, Lee writes "Clearly a homogeneous Riemannian manifold that is isotropic at one point is isotropic at every point". It seems that he means that he ...
2
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0answers
55 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
1
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0answers
25 views

Computing the differential of multiplication by $M$ on $U(n)/O(n)$

Suppose $M\in U(n)$. Then multiplication by $M$ induces a smooth action on $U(n)/O(n)$. How can we compute the differential of this map? If $M$ were acting on a matrix group, then of course the ...
2
votes
1answer
48 views

Two different descriptions of a complex torus

I came across the following description of a (1-dimensional) complex torus while learning about Calabi-Eckmann manifolds: For a fixed $\alpha \in \mathbb{C} \setminus \mathbb{R}$, the subgroup $Z = ...
2
votes
3answers
73 views

How to show that $\int_G f(t) dt = \int_G f(t^{-1}) dt$?

I am reading the lecture notes. On page 34, line 13, it is said that $\int_G f(t) dt = \int_G f(t^{-1}) dt$. How to prove this identity? I think that if we let $s=t^{-1}$, then ...
4
votes
1answer
45 views

Least dimension of Lie group acting transitively of a manifold

I guess that the least possible dimension of a Lie group $G$ acting smoothly and transitively on a compact manifold $M$ is $\operatorname{dim}(M)$. Is this correct and is there a ref?
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0answers
10 views

Compact modulo center

Bit of a naïve question. Suppose that $G$ is a reductive group and that the Lie group $G(\mathbb{R})$ is compact modulo center. Is it true that $G(\mathbb{R})/Z(\mathbb{R})G^0(\mathbb{R})$ is ...
2
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0answers
37 views

Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
3
votes
2answers
72 views

Why is $GL(n, \Bbb R)$ open in the Zariski topology?

The general linear group $GL(n,\mathbb{R})$ is said to have Zariski topology, but I do not understand how. Can anyone explain this to me?
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16 views

Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
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0answers
42 views

How do Lie algebra elements act on symmetric and antisymmetric representations?

The Lie algebra of a group acts on itself through the commutator $ T_a \in ad$: $$ T_a \circ T_b = [T_a,T_b] \in ad $$ I assume the same should be true if we have an antisymmetric adjoint, as for ...
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0answers
23 views

Why aren't all elements of the $45_a$ representation of $SO(10)$ zero?

We can write elements of the $45_a$, where $a$ denotes antisymmetric, as $10 \times 10 $ matrices, because $$ 10 \otimes 10 = 1_s \oplus 54_s \oplus 45_a$$ Here $10$ denotes the fundamental ...
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0answers
37 views

Why is a weight automatically a complex weight?

EDIT: I think the partial answer to my question is that in order to talk about weights we always need a complex Lie algebra. If the Lie algebra is real, we use the complexification, This is necessary, ...
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52 views

A Quotient of the Euclidean Group

$\newcommand{\euc}{\mathscr I}\newcommand{\R}{\mathbf R}$ Let $\euc(n)$ denote the the Euclidean group $\R^n\rtimes O_n(\R)$. Recall that $\euc(n)$ acts on $\R^n$ as $(\mathbf x, T)\cdot \mathbf ...