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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Compact connected iff semi-simple for Lie Groups?
Are compact & connected Lie Groups in correspondence with semi-simple Lie groups? I think there is a condition on the center (discrete?) but I'm not sure.
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Iwasawa Decomposition & Polar Decomposition related how ?
In an earlier post (Use Lie Sub-Groups of GL(3, R) for elastic deformation ? here), I mentioned polar decompositions as in F = RU where R in SO(3) & U in symmetric positive-semidefinite matrices. ...
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How do Lie groups classify geometry?
I have often heard that Lie groups classify geometry. For example that $O(n)$ is about real manifolds, $U$ is about almost complex manifolds, $SO(n)$ about orientable real manifolds and so on.
I have ...
user2529
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Conjugacy classes with elliptic limit points
Let $G$ be a reductive algebraic group over $\mathbb R$ and $K$ a maximal compact subgroup. Then we refer to the conjugacy class in $G$ of some $k \in K$ as an elliptic conjugacy class.
Question: ...
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Does a finite-dimensional Lie algebra always exponentiate into a universal covering group
Hi,
I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly.
Consider a finite-dimensional Lie algebra, A, spanned by its d generators, X_1,.....
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Explicit isomorphism between distributions and universal enveloping algebra
The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to the algebra of distributions on the Lie group $G$ with support at the identity. The proof I have of this fact uses the ...
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What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?
Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by $$d(...
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Centralizers and Cartan involutions
This should be an easy question about centralizers in reductive lie groups, but I wonder if it is already available from the literature.
Consider $G$ a connected non-compact semi-simple Lie group, ...
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Orthogonal Projections in Lie Theory
I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
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the relation between cohomology and Dynkin graphs of lie groups
I heard it said that the cohomology rings of some Lie groups and Grassmannians can be read from the Dynkin graph. Can someone give me any reference?
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What does the weights of Lie group mean?
Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system
of the exceptional Lie group $G_2$
with $\alpha_1$ is short and $\alpha_2$ is long,
so $\lambda_1=2\alpha_1+\alpha_2,\lambda_2=3\...
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Is there a group-scheme equivalent of the theorem that any Lie group is diff. to a compact one cross R^n?
I'm rather ignorant in both fields, but I would still like to endeavor asking this question. I've just learned that any Lie group is diffeomorphic to a compact Lie group cross $\mathbb{R}^n$. While ...
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how to recognize subgroups through Dynkin diagram?
Fix $\mathbb{C}$ as the base field, and reductive groups are assumed to be connected.
Consider the example $SO_N\subset SL_N$. $SO_N$ is its own normalizer in $SL_N$, and the rank is much smaller ...
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Which covers of Lie groups will I get
Here is a question I get from sitting in my Lie algebra class:
Fix a Lie algebra $\mathfrak{h}$, we know there is a unique simply connected Lie group $H$ which serves as the universal cover of other ...
- 1,160
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Physical Meaning of Constant Velocity Gradient
I'm interested in representing homogeneous elastic deformations using Lie groups/algebras. Homogeneous deformations are those with a deformation gradient F which depends only on time (not position). ...
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What is the outer automorphism group of SU(n)?
All the automorphisms of $SU(2)$ seem to be inner, which would mean that $\mathrm{Out}$ $SU(2)$ is trivial. Is that correct? Is this true in general $SU(n)$? I can't quite see -- any thoughts would be ...
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Groups of Hodge type, hodge structure on Lie algebra
Hi,
Let $W$ be a real algebraic group, and $G$ the associated complex group. Then $W$ is of Hodge type if there is a $\mathbb{C}^*$ action on $G$ such that $U(1)$ preserves $W$ and the action of $-1$ ...
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question about equivariant embeddings of riemannian symmetric domains
Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a ...
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How to calculate the dimensions of representations of SO(6) and SO(10)?
The representation of SO(6) is $[i,j,k]$;
The representation of SO(10) is $[i,j,k,m,n]$.
Is there any analytical formula to calculate the dimensions of those representations?
For example,
for SO(6)...
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Use Lie Sub-Groups of GL(3, R) for elastic deformation ?
I'm interested in representing elastic deformations (e.g. stretching) using Lie groups. There are a few references to using $GL(3,\mathbf{R})$ but I'm wondering if possible to use subgroups of $GL(3,\...
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a question about centralizers in semi-simple groups
I have a question concerning centralizers in real reductive groups. I'd like to know if the following property is available in any references.
Let $L\subset H\subset G$ be an inclusion chain of ...
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Why is the string group not a Lie group?
The string group $String(n)$ is by definition a 3-connected cover of $Spin(n)$. This definition determines the homotopy type of the string group.
[In a previous version of this question I screwed up ...
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evaluating an integral related to the volume of Hessenberg orthogonal matrices
Consider the following integral,
$$
{1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right)
\sin^{2}\left(\theta_{2} \over 2\right)\,}
\,{\rm d}...
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answer
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Is the object we get when we quotient $U(N)$ by $U(N-k)$ familar?
If we quotient $U(N)$ by $U(N-1)$ we get the odd dimensional sphere $S^{2N-1}$. (Here the quotient is in the sense of embedding $U(N-1)$ in the bottom right hand corner (with 1 as the (1,1) entry and ...
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Christoffel symbols on a Lie group in Riemann normal coordinates
Consider a coordinate patch around the identity element in a Lie group given by the exponential mapping (Riemann normal coordinates). We have a Levi-Civita connection corresponding to the bi-...
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Nilpotent Lie algebras and unipotent Lie groups
$\mathbf{n}$ is nilpotent Lie algebra with $N$ being the corresponding algebraic Lie group. Now one neat feature of this setting is that you can take the exponential map to be identity. In other words ...
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"Why" is every polynomial representation of SL(2) selfdual?
Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, ...
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Compatible Iwasawa decomposition for embedding of the orthogonal Lie group
I am looking for an embedding of the orthogonal Lie group
O(n,C) into GL(m,C) such that the standard Iwasawa
decomposition (also known as the QR-decomposition) for the
group GL(m,C) induces an ...
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answer
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An identity for sheaf cohomology of flag varieties
Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ denote the projection.
Given a representation ($\theta,V$) ...
user6311
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Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
- 2,160
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Explicit equations for Schubert varieties
How can one compute the Schubert variety (by compute I mean having actual polynomials that define it) for SL(n)? If this is well known forgive my ignorance and just point me to the right book/paper.
...
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how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6?
how many injective homomorphism between two lie algebra $sl_2 $and $sp_6$ up to conjugate by$Sp_6$ ?
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Reps of groups and reps of algebras
I've got what might be a couple of very basic questions on the fundamental representations of locally isomorphic semi-simple Lie groups and their relationship to representations of the corresponding ...
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Geometric interpretation of exceptional symmetric spaces
Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
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Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
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Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...
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What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?
Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
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module of sections of the horizontal bundle
Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth ...
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Dimensions of Jordan blocks associated to representations
Given a linear representation $\rho$ of $SL_n(\mathbb C)$ of finite dimension $m$,
the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL_n$ decomposes
into generally several Jordan blocks ...
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Maurer-Cartan form
I suppose given a Lie Group ($G$) and its corresponding Lie Algebra ($\mathfrak{g}$) every element in its dual defines a Maurer-Cartan form on the whole Lie Group?
Let $\omega \in \mathfrak{g}^*$ be ...
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Is there an analogue of the hive model for Littlewood-Richardson coefficients of types $B$, $C$ and $D$?
If $V_\lambda$, $V_\mu$ and $V_\nu$ are irreducible representations of $\operatorname{GL}_n$, the Littlewood-Richardson coefficient $c_{\lambda\mu}^\nu$ denotes the multiplicity of $V_\nu$ in the ...
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Can SO_n(R) be approximated arbitrarily well using a discrete subgroup?
Let $G := SO_n(R)$ be equipped with the Euclidean metric on vectors of length $n^2$. Is it true that for any $\epsilon >0$, there is a finite subgroup of $G$ which intersects every metric ball of ...
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Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions
The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...
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votes
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Why the Killing form?
I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
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Has the Lie group E8 really been detected experimentally?
A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum ...
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Left and right eigenvalues
A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...
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Textbook or lecture notes in topological K-Theory
I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require ...
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How big is the center of an orthogonal group?
How big is the center of an arbitrary orthogonal group $O(m,n)$?
In the special case of the "circle group" $O(2)$, it's clear that $|\zeta O(2)|$ = 1. In the case of $O(3)$, it seems clear that the ...
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Todd class and Baker-Campbell-Hausdorff, or the curious number $12$
The number $12$ (or, probably we shall say Bernoulli numbers in general) appears in many places in Mathematics, sometimes leading to unexpected connections between different topics.
For instance, ...
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Some basic questions about Chern-Simons theory
Let the Chern-Simons lagrangian for a group $G$ be,
$$L= k \epsilon^{\mu \nu \rho} Tr[A_\mu \partial _ \nu A_\rho + \frac{2}{3} A_\mu A_\nu A_\rho]$$
Then it is claimed that on "infinitesimal" ...
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