Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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Brownian bridge on a Lie group as a stochastic differential equation

Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation $$dg_t = dB_t \circ g_t$$ where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...
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263 views

Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...
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21 views

Automorphism groups of symmetric cones

Indecomposable symmetric cones fall into five classes. The automorphism group of any symmetric cone $C$ is a real Lie group $Aut(C)$. What is the associated class of Lie groups $Aut(C)$ for each of ...
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Degree of map into Lie group representation

Suppose $M$ is a smooth manifold with unit volume and that $G$ is a compact Lie group of the same dimension. Given a smooth map $\phi: M\rightarrow G$, we can compute the degree of $\phi$ as: ...
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Compactness and noncompactness of Lie groups [on hold]

I have read that if one complexifies a Lie group, (i.e. a group which has elements written as $G=e^{i\lambda_a T^a}$, where $T^a$ are the Lie algebra elements which generate the group, and $\lambda_a$ ...
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1answer
158 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
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1answer
118 views

Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$

Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that $$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$
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199 views

Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

I'm asking a question about Lie group representation. Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...
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42 views

Representation ring of circle group over complex field [migrated]

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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66 views

Global decomposition of reductive spaces

Consider a reductive homogeneous space $M=G/H$ with corresponding Lie algebra decomposition $\frak{g}=\frak{m}+\frak{h}$. Then there is a local diffeomorphism $$ (exp\, X, h)\mapsto (exp\, X) h\quad ...
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3answers
190 views

Parameterizing rotations of a cube [closed]

For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$ In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$. ...
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100 views

Discrete group action on the sphere

Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$ be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...
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47 views

Orbit closures of symmetric bilinear form [migrated]

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: ...
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1answer
193 views

Generalising the parametric transversality theorem to a foliation

The parametric transversality theorem states that, given a parameterised family of smooth maps of $C^{\infty}$ manifolds $\phi_s:M \rightarrow N$ and a submanifold $R < N$ then for almost all ...
3
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1answer
226 views

Infinite groups of finite exponent inside of SL(2,C)

Fix an integer $n>0$. Are there infinite subgroups of $SL_2(\mathbb{C})$ such that every element is $n$-torsion?
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236 views

Generalization of a theorem of Burnside to non-compact groups

The following two theorems are often attributed to Burnside: Theorem Let $G$ and $H$ be compact groups. Then the irreducible representations of $G\times H$ are precisely the representations ...
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1answer
140 views

Matrix from a homomorphism of simply connected groups

Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$. Let $\rho$ be the standard 7-dimensional complex representation $$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$ We ...
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1answer
72 views

Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...
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156 views

Manifold approximations to $BO(3)$

We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$ Similarly $BO(2)$ can be approximated by closed, ...
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1answer
82 views

How to associate a proper parabolic subgroup of a real s.s Lie group $G$ to a non-trivial unipotent element in a non uniform lattice in $G$?

Let $G$ be a real semi-simple Lie group. Let $\Gamma$ be a non-uniform lattice in $G$. Then it is known that $\Gamma$ contains a non-trivial unipotent element. When $\mathbb{R}$-rank of $G$ is 1, it ...
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82 views

Space of polynomially growing harmonic functions on a Lie group

There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads: If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...
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1answer
172 views

Check symplectomorphism property on infinitesimal generators

I stumbled over the following question: First, let me give the basic definition of a symplectic group action: Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G ...
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74 views

$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here. Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and ...
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79 views

Poincare inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is: symmetric, adapted (in the sense that there is no proper subgroup $H$ such ...
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2answers
167 views

Permutable (Lie) subgroups

Let's recall that, a group $G$ being given, two subgroups $A,B\subset G$ are called permutable iff $AB=BA$ for the Minkowski law. It is straightforward to see that $(A,B)$ are permutable iff $AB$ ...
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109 views

Hasse diagrams of G/P_1 and G/P_2

in the Paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf at the end, we can see Hasse diagrams for several projective, homogeneous $G$-varieties for $G$ ...
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1answer
71 views

Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$ [closed]

Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$? Cross posted from ...
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1answer
99 views

A question about the maximal subgroup of SO(n+1)? [closed]

Is SO(n) a maximal subgroup of SO(n+1)?
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99 views

Gradient on $SU(n)$

I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F_1, F_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F_1(U) = | Tr (G^{\dagger} U) |^2$, $F_2(U) = \Re ...
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Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/). There comes a point in the paper (Lemma 2.8) ...
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155 views

Uniform sub-linearity of sub-additive functions on groups

Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...
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1answer
171 views

Does the Laplacian commutes with elements of the basis of the Lie algebra?

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with ...
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1answer
131 views

Infinite-dimensional admissible representations of SL(2,C)

I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
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3answers
243 views

Jacobson-Morozov theorem

Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL_2$ ...
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1answer
155 views

Non Hamiltonian vector field

Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group. Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...
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45 views

Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
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1answer
90 views

Compact form of symplectic groups defined over the rationals

I am a bit confused regarding the possible constructions/realizations of symplectic groups. Basically I am looking for the following: A linear algebraic group $\mathbb{G}$ defined over $\mathbb{Q}$ ...
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1answer
172 views

Understanding the Weyl Character Formula

Let $G$ be a compact (connected) Lie group with a maximal torus $T$. For each (analytically) integral weight $\lambda$ the Weyl character formula $$\Theta_{\lambda}(H)=\frac{\sum_{w\in ...
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2answers
236 views

Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
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1answer
128 views

How to embed $U(1)$ into a bigger group, using Dynkin diagrams

I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin ...
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1answer
74 views

Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$

I have the following questions: Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra ...
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Curvature tensor for a singular manifold

Given a manifold $M$ with its tangent space $TM$ and frame vector field $e \in TM$. However, the transition functions in this tangent bundle are non-smooth. Therefore, the Lie derivative of $e$ with ...
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134 views

The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$

It is commonly known that we have a chain of embeddings $$SU(4)\subset Spin(7)\subset SO(8)$$ (there is more than one possible $Spin(7)$, just take one). Which is the explicit analog for the Lie ...
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1answer
312 views

Embedding linear algebraic groups of a given dimension into a fixed $\mathrm{GL}_N$

Given $n$, can $n$-dimensional linear algebraic groups over $\mathbb{C}$ be embedded into $\mathrm{GL}(N,\mathbb{C})$ for a uniformly bounded $N$? Thanks so much for your reply!
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291 views

Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin. Now, for self dual ...
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21 views

Restrictions of potential tensor fields to toric subgroups

Let $G$ be a compact connected nonabelian Lie group and let $f$ be a symmetric tensor field of order $m\geq1$ on $G$. Let $T\subset G$ be a translate of a torus subgroup of $G$ with $\dim(T)\geq1$. ...
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190 views

Hyperplane sections of principal homogeneous spaces

Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...
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2answers
163 views

A question on an set of 8 matrices related to the SU(3) generators

SU(2) and SU(3) differ quite a bit. The Lie algebra of SU(2) formed by the three generators $g_n$ is the same as the algebra formed by the SU(2) matrices/elements $F_n=e^{\pi \cdot i \cdot g_n / 2}$. ...
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79 views

a construction on Stiefel manifolds

Are there any references concerning the following space $V(k,N,X)$ and $U(k,N,X)$? And the cohomology of these spaces? Thanks.
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148 views

Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group?

Is the braid group with $n$ strings $\mathcal{B}_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$) Or is it known that it cannot be such a lattice ?