Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
3,059 questions
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Complex structure on flag manifolds
Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. Then the flag manifold $G/T$ is a complex manifold and a symplectic manifold. One way to see the symplectic structure is to view $G/T$ ...
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Periodic automorphism of nilpotent Lie algebra
Are there a non-abelian nilpotent Lie algebra $\mathfrak{n}$ over $\mathbb{R}$ and an automorphism $\alpha: \mathfrak{n} \to \mathfrak{n}$ such that:
$\alpha$ is periodic,
the fixed subspace of $\...
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A question on Lie algebras, Lie groups and multiplets
I wonder if anyone can help me with this question regarding algebras and multiplets. In a nice review paper (McVoy, Rev Mod Phys 37(1)) the author states the following theorem: “Given any set of ...
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Fréchet manifolds vs ILH manifolds
What is the precise relation between ILH manifolds and Fréchet manifolds? Specifically:
Does any ILH manifold has a canonical structure of a Fréchet manifold?
If so, is it true that any ILH ...
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Linearization of actions of semi-simple groups
What is known about local structure of actions of semi-simple groups? More precisely, suppose I have a semi-simple group $G$ acting on a variety $V$, and $x\in V$. Assume that the stabilizer of $x$ is ...
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How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?
The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...
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Uniform lattices in semisimple Lie groups
Let $\Gamma$ be a uniform lattice in a semisimple Lie group $G$.
Must $\Gamma$ be virtually torsion-free?
If (1) is true, then does this work more generally if $G$ is reductive?
I am motivated by a ...
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Higher order Approximation of Lie groups [closed]
Maybe the following is trivial or folklore, but I can't find any concrete proof of
the theorem, that higher order derivatives of Lie groups don't give any new information
above what is coded in its ...
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Orthogonal group of the lattice $I_{p,q}$?
Here $I_{p,q}$ is the unique-up-to-isometry unimodular lattice of signature $(p,q)$, whose Gram matrix is diagonal with $p$ 1s and $q$ -1s.
In his paper "ON GROUPS OF UNIT ELEMENTS OF CERTAIN ...
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When do reflection groups act discretely on quadrics in indefinite/semi-Riemannian situation?
The hyperbolic case seems to be well understood after work of Vinberg. Given a lattice $L$ with quadratic form $Q$ of signature $(1,n)$, the orthogonal group of $L$ acts discretely on the affine ...
- 1,329
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Special class of bi-hamiltonian systems
A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$.
I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $...
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Reflection groups in O(n+1,n) arising `in nature'?
For a while a friend and I have been thinking about a family of integral symmetric bilinear forms of signature $(n+1,n)$. Such lattices in our case arise 'in nature' (in a certain problem about vector ...
- 1,859
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Lattices in SOL
Consider a semi-direct product $\mathbb{Z}^2\rtimes_A\mathbb{Z}$, where $A\in SL_2(\mathbb{Z})$ and $|Tr(A)|>2$. It is clear that it is isomorphic to a lattice in the 3-dimensional solvable Lie ...
- 11.1k
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pullback diagram of principal bundles
Let $G, G_1, G_2$ be compact Lie groups with homomorphisms $f_1:G_1 \to G$ and $f_2: G_2\to G$. Let $P_1, P_2$ be principal bundles for $G_1,G_2$ and assume that the bundles $P_i\times_{G_i} G$ are ...
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Inverting the integration along a subgroup
Given a locally compact group $G$ and a closed subgroup $H$, one often uses an operator of the form
$$P: C_c(G) \rightarrow C_c(H \backslash G), \qquad Pf(Hg) = \int_H f(hg) d_H h,$$
where $d_H h$ ...
- 11.2k
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Sg: How to Show this Sequence is Exact?
Hi,All:
I am seeing a result in which the following sequence, in the context of the genus-g surface Sg, is described as being exact:
1-->Tg-->$M^{(2)}g$-->$Sp^{(2)}(2g,\mathbb Z)$-->1
Where :
i)Tg ...
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Generating Set for $O(V)$ over $\mathbb Z_2$
I am reading a claim that $O(V)$ — the orthogonal group associated with a finite-dimensional vector space $V$ over $\mathbb Z_2$ and a quadratic form $q$, i.e. the group of linear ...
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Semi-Simple Kahler Groups?
We say that a Kahler manifold is a Kahler group if it is also a Lie group. I would like to know which semi-simple Lie groups are also Kahler groups?
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Is there a general dilogarithm formula for the Cheeger–Chern–Simons class?
I'm looking for a generalization of the calculation of the hyperbolic volume and Chern–Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
Recall ...
- 18.7k
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How does the right regular of GL(n, R) and GL(n,Qp) decompose?
The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right ...
- 11.2k
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generalisation of GL(3,R) polar decomposition
Does polar decomposition work when the 'orthogonal' matrices are not orthogonal wrt to the identity ($O^TO=Id$), but wrt to some other symmetric matrix $K$ (i.e. $O^TKO=K$)?
Specifically, $GL(3,R)$ ...
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Matrix expression for elements of $SO(3)$
Hi all. Is there any explicit matrix expression for a general element of the special orthogonal group $SO(3)$? I have been searching texts and net both, but could not find it. Kindly provide any ...
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Topology on extensions of topological groups
Let $G$ and $H$ be two topological groups and let $\mathcal{E}:0 \to G \to E \to H \to 0$ be an extension of abstract groups.
Is there a way to introduce a topology on $E$ such that $\mathcal{E}$ ...
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Volume of fundamental domain and Haar measure
In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...
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Finite-dimensional subgroups of diffeomorphism groups
This question is a generalization of my previous question about the circle to arbitrary manifolds.
Is there a smooth manifold M with the following property.
There exists a sequence of connected ...
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Finite-dimensional subgroups of circle diffeomorphism group
Is there a sequence of connected finite-dimensional subgroups Gi of the circle diffeomorphism group G with the following properities:
(a) Gi is contained in Gj for i < j
(b) The union of Gi is ...
- 1,368
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Branch cuts of $GL_n^+(\mathbb{R})$
Branch cuts
Let $GL_n^+(\mathbb{R})$ denote the group of $n\times n$ real matrices with positive determinant. Topologically, $GL_n^+(\mathbb{R})$ is connected, and
$$ \pi_1(GL_2^+(\mathbb{R})) = \...
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When did the term "Lie group" first appear?
Does anyone know who was the first to coin the term "Lie group"?
The following thesis from 1928 suggests that the term was already in use by that time: "Systems of Two Differential Equations from the ...
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2
answers
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Lie (and topological) group extensions of $\mathbb{R}^2$ by $\mathbb{R}$
What are all the non-split Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups ...
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Is endoscopy interesting in simply-laced cases?
Let $G$ be a complex algebraic group, and write $Z(g)$ for the centralizer of a semisimple element $g$ in $G$. I will assume $G$ is simply connected, in which case $Z(g)$ is connected.
Let $G^\vee$ ...
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Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
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Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P?
I am an analyst struggling through some geometry used in physics.
Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection ...
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Cohomology of the quotient of a Lie group by a finite subgroup
Let $G$ denote the $\operatorname{Spin}(n)$ group with $n>4$ and let $\Gamma$ be a cyclic subgroup $G$ of a prime order $p >2$. When does the projection $G \to G/\Gamma$ induce a surjection
...
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Is there a Levi decomposition for Lie group and algebraic group?
Let $G$ be a Lie group and $R$ be the largest connected solvable
normal subgroup of $G$.
Question 1
Is there a Lie subgroup $S$ such that: (1) $G=SR$; (2)
every real representation of $S$ is ...
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Computing centralizers in Lie groups
Let $G$ be a real semisimple Lie group. Really, I only care about $\text{SL}(n,\mathbb{R})$ and $\text{Sp}(2n,\mathbb{R})$.
I'd like to perform a computer search for a finite group with a certain ...
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Ricci curvature of the symplectic group
Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...
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Quantization of conjugacy classes in a Lie group
Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\!/G$. Then let $\pi:G\...
- 18.7k
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Dimension of Lie group embedding
Let $G$ be a compact Lie group of dimension $n$. Then we can embed $G$ (topologically) into a connected compact Lie group $H$. (One may choose $H=U(m)$, the unitary group, for example.)
The question ...
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On homeomorphic compact connected topological groups
I wish to thank Professor Claudio Gorodski for his very helpful
answers to my question on the webcite:
If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie ...
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If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie groups?
Let $G_{1}$ and $G_{2}$ be compact connected Lie groups.
If $G_{1}$ and $G_{2}$ are homeomorphic as topological spaces, are
they isomorphic as Lie groups?
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Conjugate points in Lie groups with left-invariant metrics
For any Lie group $G$ there exist many left-invariant Riemannian metrics, namely, one just takes any inner product on the tangent space at the identity $T_eG$ and then left translate it to the other ...
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Torsion for Lie algebras and Lie groups
This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...
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$H_2$ of a simply connected Lie group vanishes
How do I show that the $H_2$ of a simply connected Lie group vanishes? (I don't want to use that $\pi_2(Lie group) = 0$, since this is what I want to prove. And I don't want to use the classification ...
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Can we get fixed points sets of principal isotropy groups of orthogonal representations via iteration of involutions?
Consider an orthogonal representation of a compact Lie group $G$ on an Euclidean space $V$.
Denote by $H$ a fixed principal isotropy group, which we assume to be non-trivial.
Consider the fixed ...
- 4,729
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Which groups admit a unique Lie group structure?
This question is a follow-up on the answer given here Can a Lie group as an abstract group be given more than one topology making it a Lie group?
It is motivated by the following observations:
If $m,...
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A question on random walks on semisimple groups
Let $G$ be a connected semisimple Lie group without compact factor, $\mu$ be a Borel probability measure on $G$ such that the group generated by $\mathrm{supp}(\mu)$ is Zariski dense in $G$. For ...
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Lie subgroups of SU(3)
Apart from images of representations of subgroups of SU(2), what are the Lie subgroups of SU(3)? Where should I look for a reference?
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why are all characters of the maximal torus in a Lie group weights?
Let $G$ be a compact connected Lie group, $T$ maximal torus, identified with $\mathbb{R}^n/\mathbb{Z}^n$, $X^*(T)$
the set of characters of $T$, naturally identified with $\mathbb{Z}^n$. Let next $\...
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Appearances of 'exotic' compact Lie Groups
The structure theorem for compact Lie Groups states that all compact Lie groups are finite central quotients of a product of copies of $U(1)$ and simple compact Lie groups. And yet, as easy as ...
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A question regarding Lie group actions
Can you give me an example of a Lie group acting on a compact metric connected space transitively so that it has a closed finite index subgroup which does not act transitively?
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