# Tagged Questions

**4**

votes

**3**answers

256 views

### What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields?

I thought they were the same, just different names. Let me make question more precise:
Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the ...

**3**

votes

**0**answers

76 views

### Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...

**4**

votes

**0**answers

172 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**2**

votes

**1**answer

113 views

### A Krull-Schmidt Theorem for Lie groups?

I wondered whether there is an analogue of the Krull-Schmidt theorem for real Lie-groups. More precisely, what conditions do you have to impose on a connected finite dimensional Lie group $G$ so that ...

**0**

votes

**0**answers

59 views

### Bruhat order in homogeneous spaces

Let $R$ be a root system with simple roots $\Delta$. For all $\alpha\in\Delta$ let $\omega_\alpha$ be the fundamental weight associated to $\alpha$. Is then the length of $\alpha$ equal to the length ...

**-4**

votes

**0**answers

52 views

### What are “small” finite groups with “exponentially” large expansion? [closed]

Given an integer $k$ suppose one wants the group to be polynomial in size in $k$ but the expansion to be exponential in size in $k$. I am not sure I am asking a precise question. Kindly add in ...

**1**

vote

**0**answers

177 views

+50

### A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...

**12**

votes

**7**answers

691 views

### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...

**-1**

votes

**1**answer

103 views

### cartan killing metric [closed]

I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...

**4**

votes

**2**answers

195 views

### An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
the first one is the adjoint ...

**1**

vote

**0**answers

117 views

+50

### The geometry of the holomorph of a Lie group

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left ...

**1**

vote

**3**answers

285 views

### What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...

**1**

vote

**0**answers

72 views

### Are lattices in the special real linear group subgroup seperable?

Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : ...

**1**

vote

**0**answers

78 views

### representation of $SO(p)\times SO(q)$ with $p,q$ odd

Assume $p,q$ odd. We denote by $\sigma_p$ the standard representation of $SO(p)$, that is the representation of $SO(p)$ acting on $\mathbf{R}^p$ as matrix. So is $\sigma_q$.
Take $K=SO(p)\times ...

**1**

vote

**1**answer

68 views

### Existence of a fixed-point free map in a manifold [closed]

I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map.
I know ...

**1**

vote

**0**answers

81 views

### homomorphisms from one Lie group to another

I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie"
http://www.ams.org/mathscinet-getitem?mr=0379749
and I was wondering if I could ask for help with understanding one ...

**5**

votes

**1**answer

159 views

### Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and ...

**1**

vote

**0**answers

72 views

### Lattices in $PSL_2(\mathbb{R})$ [migrated]

Can a lattice in $PSL_2(\mathbb{R})$ have a normal abelian subgroup? It looks to me that it doesn't, but where can I read a proof?

**0**

votes

**1**answer

84 views

### query about Jacques Tits' “Homorphismes `abstraits' de groupes de Lie”

I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie" and he seems to be making a claim that if you have a simply connected Lie group then the derived subgroup is always ...

**2**

votes

**0**answers

53 views

### Orthosymplectic group, matrix representations

We have the orthosymplectic $osp(n,m|2k)$. The bosonic part is $so(n,m)\times sp(2k)$. The lie algebra generators are given in eg
http://cds.cern.ch/record/524737/files/0110257.pdf$
where the group ...

**0**

votes

**0**answers

56 views

### adjoint quotient and points in DVRs

Let $G$ be a connected reductive group over an algebraically closed field $k$, $T$ a maximal torus and $W$ its Weyl group.
We have a Steinberg map $\chi:G\rightarrow \mathfrak{C}:=T/W$ if we have a ...

**4**

votes

**0**answers

77 views

### Connected compact Lie groups with Lie algebra so(4n, R)

I am trying to write a complete list of connected compact simple Lie groups (or of connected complex simple Lie groups, both tasks are equivalent). I am missing just one case.
Consider the Lie ...

**3**

votes

**1**answer

160 views

### Are norm-continuous representations smooth?

Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous
$$
x_i\to x\quad\Longrightarrow\quad ...

**3**

votes

**1**answer

156 views

### How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...

**1**

vote

**2**answers

136 views

### Local structure of the quotient of a Lie group action

Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure.
Are there any results for the general case? (a) If the ...

**1**

vote

**0**answers

108 views

### Integrating Poisson groups

Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...

**4**

votes

**2**answers

113 views

### Measurable representations of semi simple Lie groups

Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in
I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple
Lie groups, ...

**4**

votes

**0**answers

75 views

### Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...

**3**

votes

**1**answer

215 views

### Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...

**4**

votes

**1**answer

140 views

### Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky

The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, ...

**2**

votes

**1**answer

164 views

### Lifting one parameter subgroups of algebraic groups

Let $G$ be a linear algebraic group over an algebraically closed field $\mathbb C$ of characteristic zero and $U$ its unipotent radical, then $H:=G/U$ is a reductive group. Assume that I have a one ...

**7**

votes

**1**answer

250 views

### what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...

**6**

votes

**2**answers

394 views

### Do geodesics in SL2R map to geodesics in the hyperbolic plane?

I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...

**2**

votes

**1**answer

112 views

### Fourier transform of a matrix represented compact lie group

In physics, I come across this kind of integration (in the nonlinear sigma model):
\begin{equation}
S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}]
\end{equation}
...

**5**

votes

**0**answers

156 views

### Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...

**1**

vote

**1**answer

129 views

### Equation for non-invertible elements in Clifford algebras

Suppose we have a Clifford algebra $Cl(V,q)$, $V\simeq \mathbb{R}^n$ and $q$ non-degenerate bilinear form. Then every non-zero element of $V\subset Cl(V,q)$ invertible, but they are not the only ones ...

**0**

votes

**0**answers

127 views

### Comparison of two Chevalley basis

Let $G$ be a connected reductive group over an algebraically closed field and $T$ a maximal torus.
Let $H$ be a pseudo-Levi subgroup, say the neutral component of a centralizer of a semisimple element ...

**1**

vote

**0**answers

122 views

### Analogies, Riemann surfaces and Algebraic groups

Let G be a complex simple Lie group of adjoint type. Then, it is well known that every such $G$ contains, unique up to conjugacy, an irreducibly embedded copy of $PSL(2,\mathbb{C}).$ This fact seems ...

**2**

votes

**2**answers

230 views

### Closure relations between Bruhat cells on the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...

**2**

votes

**1**answer

72 views

### Conjugacy class of semisimple elements

Let $G$ be a complex simple Lie group with Lie algebra $\mathfrak g$ and $G_1$ be a complex simple subgroup of $G$ with Lie algebra $\mathfrak g_1$. We may assume $G$ and $G_1$ to be of adjoint type. ...

**2**

votes

**0**answers

39 views

### Conjugacy classes of a triple

In the paper " THE COMPONENT GROUPS OF NILPOTENTS IN EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS" by D. King I am unable to proof the lemma 3.7 which is omitted there.
The lemma is following:
Let $\mathfrak ...

**2**

votes

**1**answer

151 views

### Relationship between Verma modules and delta functions

Reposting my question from math.stackexchange:
What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory ...

**13**

votes

**1**answer

215 views

### Homogeneous spaces that are homotopy tori

Let $G$ be a compact Lie group, and let $H$ be a closed subgroup such that $G/H$ is homotopy equivalent to a torus. Is it true that $H$ is normal and $G/H$ is isomorphic to a torus as a Lie group?
...

**7**

votes

**1**answer

164 views

### $Spin(7)$ as stabilizer of a $4$-form

According to Bryant's work on special holonomy groups, $G_2\subset SO(7)$ may be defined as the group preserving the following 3-form:
...

**-2**

votes

**1**answer

105 views

### finitely generated subgroups of SO(3) [closed]

Is it known whether there is any example of a pair of rotations in $SO(3)$ about orthogonal axes such that the group that they generate is not a free product of the two cyclic groups generated by each ...

**1**

vote

**1**answer

139 views

### abelian p- subgroups of E_6(q)

Is there any result about maximal abelian p-subgroups of the exceptional group E_6(q), where q=p^a is prime power?

**1**

vote

**1**answer

89 views

### Reducible reductive Lie subalgebras of so(p,q)

Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...

**0**

votes

**0**answers

39 views

### Discrete subgroup of complex orthogonal group

Is there any reference for the discrete subgroup of complex orthogonal group SO(n,C)? Any classification or examples?

**3**

votes

**1**answer

161 views

### Relationship between Laplacian and Hessian on compact Lie groups

If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has
$$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|_F^2\,d\mathbf{x}\,,$$
where $\Delta$ ...

**12**

votes

**2**answers

343 views

### Langlands duality and multiplying cocharacters

Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation)
of the Langlands dual group ...