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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92 views

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
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27 views

A Question on integration formula on $KAK$ decomposition

The following proposition appears in page 141 in Knapp's book, representation theory of semisimple groups. Let $G$ be linear connected reductive, and fix a positive system $\Sigma^+$ of restricted ...
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30 views

with which binary action is the 2-sphere a Lie group?

show that the 2-sphere is a Lie group. I do not know with which binary action is the 2-sphere a Lie group? and with which binary action is the $\mathbb{R^3}-{0}$ a Lie group? 2-sphere is equal with ...
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116 views

What is the manifold structure of U(n)?

A Lie group is simultaneously a differentiable manifold. As I understand it, the Lie group is generated via exponentiation of the generators of the Lie algebra. It is intuitively clear to me that the ...
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1answer
41 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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23 views

What are the root systems for the n-dimensional torus?

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...
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31 views

Group operation in $SO(3)$ is well-defined

I've just started to read Tapp's Matrix groups for undergraduates and it says: "$SO(3)$ becomes a group under composition of motions (since different motions might place the globe in the same position ...
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56 views

Question on $\mathfrak{sl}(2,\mathbb R)$

I am confused about some facts on $SL(2,\mathbb R)$. The Lie algebra of $SL(2,\mathbb R)$ is $\mathfrak{sl}(2,\mathbb R)$. However, the map $$ \exp:\mathfrak{sl}(2,\mathbb R)\ \rightarrow ...
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29 views

Weyl Group of Parabolic subgroups

Let $G=SL(n,\mathbb R)$ with Lie algebra $\mathfrak{g}=\mathfrak{sl}(n,\mathbb R)$. The classical minimal parabolic subgroup $B$ consists of the upper triangular matrices. The parabolic subgroups $P$ ...
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15 views

Character of a symmetric square

Let $V$ be a representation of $\mathfrak{sl}_2(\mathbb{C})$. As far as I am concerned a character of $V$ is a Laurent polynomial $\sum_{k\in\mathbb{Z}}d_k\cdot t^k$, where $d_k$ is the dimension of ...
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13 views

Representation of $\mathfrak{sl}_2(\mathbb{C})$ corresponding to Lie algebra representation

We have a representation $R$ of a Lie group $\mathrm{SL}_2(\mathbb{C})$ in the space of polynomials $\mathbb{C}[x,y]$ such that $R\begin{pmatrix} a & b \\ c & ...
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19 views

Homogeneous Spaces: The Erlangen Programme

This is a wholly a question of mathematics history. The Klein Erlangen programme is most pithily, if a little tersely, described in modern wording as a homogeneous space: a topological group acting ...
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17 views

Questions about an action of $U(\mathfrak{g})$.

Let $\mathfrak{g}$ be a Lie algebra and $U(\mathfrak{g})$ its universal envoloping algebra. Let $G$ be the Lie group of $\mathfrak{g}$ and $U$, $B^{-}$ the upper unipotent subgroup and lower Borel ...
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21 views

What is $d\mu$ of $\mu:T^*\mathbb{C}^n\rightarrow \mathfrak{gl}_n^*$?

This is an elementary question. Let $\mu:T^*\mathbb{C}^n \longrightarrow \mathfrak{gl}_n^*$ be the moment map given by $(x,y)\mapsto xy$. Then concretely, what is the differential $d\mu$ of $\mu$? ...
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21 views

Real or complex representation

How can one know for a given algebra $\frak{g}$ if a specific representation is real or complex? For example if $\frak{g}=so(10)$ how can one know that the representation $\underline{16}$ is complex? ...
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166 views

Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
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31 views

Compact Lie algebras and Lie groups

A simple or semisimple Lie algebra is said to be compact if the $\mathrm{Tr}\left \{ T^\mathrm{adj}_{a}, T^\mathrm{adj}_{b}\right \}$ is positive definite where $T^\mathrm{Adj}_{a}$ are the generators ...
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118 views

Which subgroup of $\mathrm{SL}(2,\mathbb{C})$ is this?

I am looking into sub-algebras of $\mathfrak{sl}_2(\mathbb{C})$ and the subgroups of $\mathrm{SL}(2,\mathbb{C})$ they generate. The basis of $\mathfrak{sl}_2(\mathbb{C})$ I am using consists of 3 ...
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22 views

Are there some examples of Kac-Moody groups which are not reductive?

Are there some examples of Kac-Moody groups which are not reductive? Thank you very much.
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11 views

How to prove that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$.

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. It is said that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$. Here $\mathcal{O}(\mathfrak{h}^*)$ is the ring of ...
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1answer
22 views

What is the Lie group of $\mathfrak{h}$?

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. What is the Lie group of $\mathfrak{h}$? By definition, the Lie group of $\mathfrak{h}$ consisting of elements of the form $e^{h}$, $h \in ...
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22 views

Subgroups of SO$(2)$

I'm doing an independent study with John Stillwell's Naive Lie Theory and I wanted to know if I'm on the right track. I'm just looking for some confirmation that these are acceptable answers. Find ...
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20 views

Linearly independent skew symmetric complex matrices having the least eigenvalues

Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$ \forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ...
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36 views

Are bounded subsets of Lie groups totally bounded

Let $ G $ be a finite dimensional real Lie group, and take a bounded ball $ B_R(e) \subset G $ in it, coming from the Riemannian metric, which itself is induced from an inner product on $ \mathfrak{g} ...
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31 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
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1answer
27 views

Induced actions.

Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and ...
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20 views

Characters of a unipotent group.

Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some ...
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28 views

Why is U(n) a real form of GL(n)

When $n=1$, we see that $U(1)$ is defined by the equation $z\bar z=1$, hence $a^2+b^2=1$ for $z=a+bi$. Taking complex $a,b$ we see that the solutions are nonzero complex points, hence $U(1)$ is ...
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25 views

Liesubgroups given by representations

Maybe it's a little odd question, but I encountered a classification of homogeneous spaces, and I'm stuck with the following description (c.f. Onishchick, "Topology of transitive transformation ...
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65 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
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1answer
31 views

Lie group and generated ideals

I have this question in my textbook, and I can't seem to solve it on my own: Let $P \subset GL(n,\mathbb{C})$ be a subgroup as following: $P$ consists of all matrices in block ...
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1answer
49 views

Does the set of $n$ by $n$ matrices of rank $q$ form a manifold?

I'm not sure whether the space of all rank-$q$ square matrices of dimension $n$ is a submanifold. I have totally no clue. Can somebody help?
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17 views

Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
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1answer
33 views

Lie groups. How to show that the group operations are smooth.

$N:=\{g\in GL(n,R) : g_{ij}=0 \forall j>i , g_{ii}=1 ∀i\}$. For this matrix group, how can we show that it is a Lie group? I am at the beginning of the subject of Lie groups so I can not ...
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1answer
10 views

Local Lie Groups

Hello i am trying to prove the following proposition : Let $G$ be a connected Lie group, and $U\subset G$ a neighborhood of the identity element. Also, let $U^k = \{g_1 . g_2 . \dots g_k : g_i \in ...
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1answer
84 views

What does it mean to be a real Lie group

What does it mean to be a real Lie-group ? For example it is said that $SU(N)$ is a real Lie-group. While for example for $SU(2)$ the 2 dimensional matrix-representation consists of the Pauli ...
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1answer
37 views

Is $sp(4)$ a subalgebra of $su(5)$?

Is $sp(4)$ a subalgebra of $su(5)$? And how can I prove/disprove this? I know already that it cannot be a regular maximal subgroup of $su(5)$ since the Dynkin diagram (which has two roots of unequal ...
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1answer
61 views

Lie groups, maps and the Weyl group

If I have a map of simple Lie groups $H \to G$, do I get a map of Weyl groups $W_H \to W_G$? If $H$ is the semisimple component of a parabolic subgroup then we can clearly get this (see Ivan's answer ...
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23 views

Decomposition of direct sum representation of a Lie Group

Suppose that $(\varphi,V)$ is an irreducible finite dimensional representation of a Lie group $G$ and let $\psi=\bigoplus_{i=1}^n{\varphi}$ the representation on $W=\bigoplus_{i=1}^n{V}$. I want to ...
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18 views

If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler > structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
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1answer
40 views

Equivalence between vector field and generator of a group of translations

I've been reading Olver's Applications of Lie groups to differential equations and did not understand the excerp where the autor explains that "[...] every nonvanishing vector field is locally ...
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67 views

When do invariant measures arise from smooth differntial forms?

It is well known that the Haar measure of a Lie group $ G $ arises from a invariant differential form density $ |\omega| $ (of top dimension). Also, we know that if we have a closed subgroup $ H \leq ...
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1answer
75 views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
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2answers
51 views

Introduction to discrete subgroups of the euclidean group

I am looking for a general introduction to discrete subgroups of the euclidean group (= group of isometries in euclidean space). Even though I searched quite a bit, I was unable to find a good ...
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1answer
34 views

A maximal subalgebra of $E_6$ !?

I'm puzzeled by the following sentence in one of Baez's posts: The Lie algebra $E_6$ has a subalgebra of maximal rank isomorphic to $\mathfrak{so}(10)\oplus \mathfrak{u}(1)$. However, I thought ...
3
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45 views

Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
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13 views

Assign a root to a irreducible representation

Given a root, e.g. $(-1 0 1 00)$ of $\text{SO}(10)$, how can I see/find to which representation of the Lie group it belongs?
4
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1answer
59 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
2
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32 views

Where does the name “toral” come from?

Where does the name "toral" come from in "toral subalgebra"? I know a little (very little) Lie groups theory, so I guess it could be related to a Lie group whose Lie algebra is the toral one. Is ...
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1answer
26 views

Question on $\mathrm{Ad}(SL(3,\mathbb R))$

The following question appears in an example in page 116, Representation theory of semisimple groups, A. W. Knapp. Consider $G=\mathrm{Ad}(SL(3,\mathbb R))$. It is a subgroup of $GL(8,\mathbb C)$. ...