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

learn more… | top users | synonyms

0
votes
1answer
83 views

Hamiltonian potential invariant under lie group action?

Let $(M,g)$ be a riemannian manifold and $G$ a Lie group acting on $M$ by isometries. Taking a $G$-invariant function $f\colon M \to \mathbb{R}$, we have the riemannian gradient $\operatorname{grad} ...
1
vote
1answer
122 views

Lift Lie group action on a small neighborhood

Suppose a manifold $M$ admits a smooth Lie Group action $G$, and $N$ is a closed sub-manifold of $M$ such that $G$ action freely on $N$. Q: Why in a small neighborhood of $N$, $G$ also action ...
3
votes
1answer
136 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
6
votes
1answer
112 views

Chevalley restriction theorem for non-split Cartan

Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...
3
votes
0answers
70 views

Action of longest element of Weyl group on zero weight space

Let: $G$ be a real semisimple Lie group; $\rho$ be an irreducible representation of $G$ on a finite-dimensional real vector space; $A$ be a "Cartan subspace" of $G$ (a Lie subalgebra which is ...
3
votes
3answers
575 views

A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...
2
votes
1answer
76 views

Minimize matrix distance to tensor product

Minimize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
9
votes
2answers
357 views

Integral versus real (universal) characteristic classes

I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to H^n(BG;\mathbb{R}...
10
votes
1answer
542 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
votes
0answers
24 views

Super classical r-matrices and Poisson Lie supergroups

In classical case, given an r-matrix $r$ for $sl_n$, we can compute the corresponding Poisson bracket on $SL_n$ by using the formula $\{L \otimes L\} = [r, L \otimes L]$. For example, let $g=sl_2$ and ...
2
votes
1answer
102 views

Maximize inner product of a tensor of unitary matrices

How can one maximize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$. Both the maximum value of ...
79
votes
7answers
8k views

Homotopy groups of Lie groups

Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
1
vote
4answers
451 views

About structure of parabolic subgroups of finite classical algebraic groups

I am interested about a Fact (if it is right) of the structure of parabolic subgroups of finite classical algebraic groups: Let $G$ be a classical algebraic group over the finite field of order $r^{e}...
23
votes
7answers
6k views

Classification of (compact) Lie groups

I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your ...
1
vote
0answers
103 views

cohomology ring of homogenous manifold

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces : $$ G/H ...
5
votes
1answer
234 views

How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
14
votes
4answers
701 views

Groups of matrices in which all elements have all eigenvalues equal in modulus

I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...
3
votes
1answer
218 views

Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$

Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...
-3
votes
0answers
161 views

The center of a group is equal to the center of its radical?

Given a linear algebraic group $G$, is the connected component of the identity of the center of $G$ equal to the connected component of the identity of the center of its solvable radical? If not, is ...
0
votes
1answer
65 views

Is the toral component of a connected Lie group equal to the toral component of its radical? [closed]

Given a connected Lie group, define its toral component as the maximal connected and compact subgroup of its center. Is the toral component of a connected Lie group equal to the toral component of ...
1
vote
1answer
109 views

Cohen-Macaulayness of the scheme of centralizer

Let $G$ be a simply connected group over an algebraically closed field $k$, and $I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$ the scheme of centralizer. Is $I$ a Cohen-Macaulay scheme ...
2
votes
0answers
102 views

Distance metric on Riemannian quotient manifold

It's well known that if I have a Riemannian manifold $M$ and a Lie group of isometries $G$ that acts freely and properly on $M$, then the quotient $M/G$ is a manifold and inherits the Riemannian ...
17
votes
2answers
423 views

When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?

A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...
1
vote
1answer
149 views

A representation of Spin(9,1)

Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ). Consider the induced ...
2
votes
1answer
67 views

Decomposition into irreducible components of a representation of $Spin(9)$

It is well known that the group $Spin(9)$ acts linearly on the vector space $\mathbb{R}^{16}$ (see for example "Spinors and calibrations" by R. Harvey). Consider the induced representation of $Spin(9)...
3
votes
1answer
66 views

Explicit generators of the Lie algebra $spin(9)$

It is well known that the Lie group $Spin(9)$ acts on the vector space $\mathbb{R}^{16}$ (see e.g. Harvey's book "Spinors and calibrations".) It is convenient to identify this vector space with the ...
11
votes
0answers
294 views

Calculation-free proof of the Weyl Integral formula for U(n)

The Weyl integral formula states that if $f$ is a class function on $U(n),$ $T$ is the torus of diagonal matrices in $U(n)$, and $dU(n)$ and $dT$ are the standard Haar measures on $U(n)$ and $T,$ then:...
2
votes
1answer
211 views

Irreducible representations of Sp(2)

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
1
vote
1answer
143 views

'Accidental' isomorphisms for $Spin^C(n)$

The complex spin groups $Spin^C(n)$ appear in the fibration $Spin(n)\rightarrow Spin^C(n)\rightarrow\ S^1$ which must split since $BSpin(n)$ is 3-connected to give a homotopy equivalence $Spin^C(n)\...
3
votes
1answer
134 views

Isomorphisms between extension group and $\mathfrak{u}$-cohomology

Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a finite-...
14
votes
3answers
544 views

Is SO(2n+1)/U(n) a symmetric space?

I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial. According to https://arxiv.org/abs/1408....
1
vote
1answer
98 views

Convention on Clifford Product

When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign: $vv=Q(v)$ (see, for instance, Wikipedia) $vv=-Q(v)$ (see, for instance, MathWorld ...
1
vote
0answers
104 views

Generating $\mathfrak{so}(7)$

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is actually a subalgebra isomorphic to $\...
3
votes
1answer
57 views

Criterion for convergence of sums for non-continuous functions

The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{...
6
votes
1answer
129 views

Decay of Fourier coefficients for compact Lie groups

Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us ...
6
votes
1answer
97 views

Involutions and Little Adjoint Representations of Simple Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
2
votes
1answer
107 views

If a Weyl element preserves a root, then it has a representative which preserves the root space?

Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\...
2
votes
0answers
33 views

$TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?

Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$. Define $M := K_{n,n} /SU(n)$. Using ...
6
votes
1answer
153 views

Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group?

Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?
1
vote
0answers
111 views

Name for the Quotient $SU(m+1)/(SU(k) \times SU(m-k))$

The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of ...
4
votes
2answers
210 views

Invariant theory for parabolics

Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...
4
votes
1answer
146 views

Differential structures on compact Lie groups

Given a compact Lie group can there be a differential structure on it with respect to which one cannot define a smooth group operation?
1
vote
0answers
58 views

Symplectic gradients whose span doesn't intersect Lie group orbits

I asked a similar question some hours ago. But thinking about my problem, I found a loophole in my arguments, so that my question wasn't the right one. This one is what I wanted to ask: Let $G$ be a ...
0
votes
0answers
61 views

charts for some riemannian embedding

Let $(M,g)$ be some riemannian manifold with some Lie group $G$ acting properly, freely and by isometries. So $M/G$ is a manifold and using the projection $\pi \colon M \to M/G$, we get a smooth ...
5
votes
1answer
122 views

Centreless semisimple Lie group that is not real algebraic

Let $G$ be a connected semisimple Lie group with trivial centre and $\mathfrak{g}$ its Lie algebra. The adjoint representation of $G$ defines an isomorphism of $G$ onto the connected component of the ...
13
votes
3answers
3k views

Curvature of a Lie group

Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
16
votes
0answers
599 views

Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?

The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary, the result of historical accidents. In order to avoid repetitions, the four infinite families $A_\ell, B_\ell, C_\...
1
vote
1answer
81 views

slice theorem for proper actions

I'm trying to understand the slice-theorem for proper Lie-group actions. Having a smooth manifold $M$ and a Liegroup $G$ acting on $M$ in a proper way, we have the slice theorem, saying that at each ...
2
votes
2answers
866 views

The normalizer of a reductive subgroup

Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the ...
3
votes
0answers
137 views

Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$. ...