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

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7
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311 views

Do compact groups acting irreducibly have finite subgroups which do the same?

Let $G$ be a closed subgroup of $U(n,{\bf C})$, not necessarily connected. Regard ${\bf C}^n$ as a complex $G$-module $M$. Q. Suppose $M$ is irreducible as a $G$-module (equivalent, I think, to ...
3
votes
1answer
141 views

Rational homogenous spaces and symmetric spaces

What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...
3
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2answers
163 views

reference help indecomposable representations of SL(2,R)

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible ...
0
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1answer
75 views

equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
11
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2answers
400 views

Who originated the standard symbols for Lie groups GL, SL, SU, etc.?

Who was first to use symbols GL, SL, O, SO, U, SU, Sp and their projective versions, and how did this notation become standard? The notation appears in fairly modern form in Weyl's "The Classical ...
1
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1answer
48 views

Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...
5
votes
3answers
402 views

Characterising the adjoint representation of SU(N)

One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an ...
1
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1answer
61 views

Bi-invariant one forms on compact Lie groups

I'm hoping to learn of a criterion for the existence of a bi-invariant one form on a Lie group $G$. I'm looking for a reference that there are no such one forms son $SU(n)$ (as long as this is in fact ...
1
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0answers
91 views

adjoint representation of 2-Lie groups

Baez and Crans in their paper on Lie 2-algebras refer to adjoint representations of Lie 2-groups but don't say much, as far as I can tell, except to say that such a representation acts on a 2-Lie ...
5
votes
1answer
178 views

Special linear groups contained in symplectic groups

Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and ...
8
votes
1answer
197 views

Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...
0
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0answers
67 views

Reference request: How you can reach any point in the vector space of vector fields generated by Lie brackets

By a Theorem of Chow, you can reach any point in the vector space of vector fields generated by Lie brackets. Do you know any reference for this theorem?
6
votes
1answer
212 views

Laplace-Beltrami operator on a Lie group

For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by $$\Delta f = \delta^{i j} X_i X_j f$$ for ...
1
vote
1answer
133 views

Compact subgroups of linear groups over nonarchimedean fields

Let $n \in \mathbb{N}$, $K$ a (nonarchimedean) local field, $\overline{K}$ its algebraic closure. Take a compact subgroup $G \leq \text{GL}_n(\overline{K})$. Must there be a finite extension $F$ of ...
1
vote
1answer
140 views

Euler-Poincare equations with constraints

It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} ...
6
votes
2answers
232 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ Here, $H$ is an NxN skew-Hermitian matrix (for very ...
8
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0answers
368 views

Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...
10
votes
2answers
452 views

Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find? In the adjoint representation, one can express the Killing form as a matrix and consider it as ...
11
votes
2answers
544 views

Multiplicative Identity for all elements in SU(n)

Let $\{P_i\}$ be a subset of $SU(n)$ such that for any $U$ in another subset (or perhaps subgroup) $H$ of $SU(n)$: $$P_1UP_2U\cdots P_mU=I$$ where $I$ is the identity element. Is there a sequence ...
2
votes
1answer
189 views

Maximal compact subgroups of a semisimple Lie group are conjugate

I'm trying to go through the proof that all maximal compact subgroups of a semisimple Lie group $G$ are conjugate. I know that a possible proof follows the following steps: Take one maximal compact ...
4
votes
1answer
111 views

$C^\infty$-vectors in general representations of Lie groups on locally convex spaces

This question is related to this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In ...
2
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0answers
79 views

$G$-invariant part of products of determinants of minors

Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for ...
7
votes
3answers
720 views

nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
2
votes
2answers
188 views

Bound on the index of an abelian subgroup in discrete subgroup of the euclidean group?

I asked the following question on math.SE a couple of days ago. Dietrich Burde gave an answer for the case that the subgroup is not only discrete but also acts cocompactly. What about the general ...
8
votes
1answer
194 views

Integrals of representations over geodesics

Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the ...
7
votes
1answer
951 views

Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition, with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant ...
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0answers
39 views

Symmetry analysis of differential equations

What is the connected component of the identical transformation in the pseudogroup of local diffeormorphism on the real line? similar question Let $\tilde t=T(t)\quad T_t>0,$ be a local ...
2
votes
1answer
153 views

Unipotent conjugacy classes

Consider a connected reductive group G over the complex numbers. Is there a `simple' formula for the number of conjugacy classes of unipotent elements in G?
5
votes
1answer
331 views

The surjectivity of the exponential map for the isometry group

Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective. Let $M$ be a noncompact connected Riemann manifold, and ...
1
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1answer
162 views

Orbits of an action of maximal compact subgroups of p-adic orthogonal groups

Let $Q$ be a non-degenerate indefinite quadratic form on ${\mathbb R}^n$ and write $G=SO(Q)$ for the associated special orthogonal group. Let $K$ be a maximal compact subgroup of $G$ and consider the ...
4
votes
1answer
227 views

Calculation with weights of $E_6$

Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...
6
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1answer
379 views

Differences in philosophy between Lie Groups and Differential Galois Theory

As far as I have heard,Sophus Lie's aim was to construct an analogue of galois theory for differential galois theory. I am familiar with lie group but not with differential galois theory. What is the ...
2
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1answer
158 views

Fibers of the Bott-Samelson Resolution of Schubert Varieties

Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$. Also, how would the answer to the ...
6
votes
2answers
226 views

Union of conjugates of a closed subgroup of a compact group

Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$. Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of ...
4
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0answers
155 views

Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here. Is there ...
4
votes
2answers
117 views

Hermitian Symmetric Subspaces of Siegel Space

Let $\mathbb{H}_g$ denote Siegel space, and $M$ denote an order 4 element of the unitary subgroup $U(n)(\mathbb{R})$with $p$ eigenvalues equal to $i$, and $q$ eigenvalues equal to $-i$, $p+q=g$. ...
3
votes
1answer
172 views

Lie group GL(4) representation decomposition

Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$. Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation. My question: How does $$V\otimes ...
4
votes
3answers
258 views

Is the group of isometries of a homogeneous Riemannian manifold maximal?

I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that: Iso is a proper subgroup of ...
1
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1answer
85 views

A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]

Consider the following system of ODEs $$ Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t) , $$ where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...
0
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1answer
168 views

Compact Lie groups with only 3 dimensional cohomology generators

Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd. For which $M$, ...
0
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0answers
51 views

Godement-Jacquet and L-functions

Let $M_{r}(F)$ be the matrices with coefficients in a local nonarchimedean field $F$ and $q$ the cardinal of the residue field. We have a Fourier tansform on $M_{r}(F)$ with kernel ...
5
votes
2answers
193 views

Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices

I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...
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0answers
80 views

Representations of $\mathfrak{so}(3)$ ($\mathfrak{so}(2,1)$) and $SO(3)$ ($SO(2,1)$)

(Apologies if this question is too basic!) I have explicit 5-dimensional real representations of $\mathfrak{so}(3)$ and $\mathfrak{so}(2,1)$, and I want to know whether it's necessarily true that the ...
2
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0answers
46 views

Locus maximizing the holomorphic sectional curvature in a non-compact Hermitian symmetric space

Is there a quick way to prove the following statement, if possible without resorting to the classification of simple Lie groups? Let $G$ be a simple Lie group of non-compact Hermitian type of rank ...
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63 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 $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
2
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1answer
196 views

R-linear representations of sl(2,C)

Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$? Equivalently, what ...
1
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1answer
78 views

A canonical G_m (or G) action on the Slodowy slice

Question By Slodowy slice I mean a transverse slice at a subregular nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ (in particular I am not intersecting with the nilpotent cone). Consider the ...
5
votes
2answers
390 views

How to estimate the Haar measure on $G_2$

I have a sequence of real numbers. I want to know whether this sequence looks like the traces in the standard representation of a random sequence of elements of $G_2$. (Here random is according to the ...
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0answers
221 views

Relationship between algebraic groups and Lie groups? [closed]

In the literature, e.g. in representation theory, there seems to be a passage from Lie groups to (linear) algebraic groups. It is clear, particularly over $\mathbb R$ and $\mathbb C$ that they are ...
0
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84 views

Is every bi-invariant Finsler metric on $SU(N)$ necessarily Riemannian?

Is every bi-invariant Finsler metric on $SU(N)$ necessarily Riemannian? If possible I'd also like to know if the right translation of the Shatten $p-norm$ on the Lie algebra gives rise to a ...