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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on the open bruhat cell
Let $G$ a connected reductive group and $S=U^{-}TU$ the open cell.
Do we have $G=\bigcup\limits_{g\in G}gSg^{-1}$?
And also if I assume that $G$ is adjoint and $\overline{G}$ is the de Concini-...
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Curves in $\mathfrak{su}(n)$ with specific property
Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ ...
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Connected compact Lie groups with Lie algebra so(4n, R)
I am trying to write a complete list of connected compact simple Lie groups (or of connected complex simple Lie groups, both tasks are equivalent). I am missing just one case.
Consider the Lie ...
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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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$Pin^{+}(4k)$ 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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Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
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Classification of Compact Symplectic Homogeneous Spaces
Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...
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When representation of two different coadjoint orbits are equivalent?
Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where $\mathfrak{t}$...
user21574
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From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?
Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of $...
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Can this Lie group written as a direct product?
Let $G=G_1.G_2$ be a Lie Subgroup of $SO(k) \times SO(2) \subset SO(k,2)$, where
$G_1=SU(k/2) \subset SO(k)$ and $G_2$ is a Lie subgroup of $SO(k) \times SO(2)$ isomorphic to $SO(2)$.
Let $G_1 \cap ...
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How to compute SE(2) group exponential and logarithm?
I want the rodrigues like formula using sin and cos , not a matrix series expansion.
I've found some references for se(n) , n > 3 in :
ftp://ftp.cis.upenn.edu/pub/papers/gallier/rodrig.pdf
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Singularities of the Quantum propagator (baby version)
Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...
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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 $K$...
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Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$
I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon_{ijk} \theta^j \...
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exponential and anisotropic torus
Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...
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A statement on a connected semisimple non-compact Lie group with finite center
In the book `Ergodic Theory and Semisimple groups' (1984 Edition, page 64, Proposition 4.1.8), Zimmer has made this statement "if G is a connected semisimple non-compact Lie group with finite center, ...
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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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Springer isomorphisms and parabolics
Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $...
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Are there explicit formulas for spherical functions on oriented real grassmannians?
Let $p$ and $q$ be integers. The group $K=SO(p) \times SO(q)$ can be naturally seen as a subgroup of $G=SO(p+q)$. The quotient space $G/K$ is identified with the space of oriented $p$-dimensional ...
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Lie Algebras and Simple Connectivity for general algebraic groups
In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
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Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
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good choice of extension of equivariant map
Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that $...
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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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Is there a generalization of Schur - Weyl duality and plethysm for direct product of special unitary groups?
Consider the semisimple compact group $K=SU(N_1)\times SU(N_2) \times \ldots \times SU(N_S)$ acting naturally on $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...
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Stationary curves on homogeneous spaces
Consider $M \cong G/K$ ($G$ a lie group with a transitive action on $M$ and $K$ a subgroup) and consider a Lagrangian $\mathcal{L}: TM \rightarrow \ \mathbb{R}$ (no time dependence). Consider also ...
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Diophantine elements in SU(2)
Following notions from [1], call a set of elements $g_1, \dots, g_k \in G = SU(2)$ Diophantine if it satisfies the following property: there exists a constant $D$ such that for every word $W_m$ of ...
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Polycyclic group not of type $FP_\infty$
In finitely presented groups, the question of the existence of a projective resolution $P_i$ (with each $P_i$ finitely generated) of $\mathbb{Z}G$ is equivalent to the existence of a $K(G,1)$ which ...
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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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Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
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Equivariant exponential map on Hilbert manifolds
Let $M$ be a Hilbert Riemannian manifold and $G$ a finite-dimensional Lie group acting on $M$. It is well-known that when $M$, too, is finite-dimensional
$$\exp_p: U \subset T_pM \rightarrow \exp_p(U) ...
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Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups?
For brevity, let $LG=\mathbb{T}\ltimes \tilde{L}G$, the affine loop group and let $G$ be a simple simply conneceted Lie group. I have a map $\phi:L\mathfrak{g} \to L\mathfrak{g}$ that is equivariant. ...
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The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular
This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We ...
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Action of the endomorphism monoid on an irreducible GL-module
Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
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Reconstructing a Lie group Banach representation from the Lie algebra rep. on analytic vectors
Dear all,
I have some difficulties with the following assertion in the book of Kirillov.
Let $G$ be a connected Lie group, and T a given (!) representation of G on a Banach space V.
Let $V^\omega$ ...
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Lie Automorphisms and Isotopy
Let $X$ be a Lie group, $Aut(X)$ be the Lie automorphism group of $X$ (group automorphisms which are also diffeomorphisms), and $Homeo(X)$ be the homeomorphism group of the underlying manifold. For ...
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A question about G-Manifolds
I am looking for a clear reason for following fact:Is there any reference ?
Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...
user21574
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Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups
Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak R=\mathrm{Hom}(\Gamma,G)...
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The normalizer a maximal compact subgroup of a semi-simple Lie group
Let $G$ be a semi-simple real Lie group such that $|\pi_0(G)|<\infty$ and let $K$ be a maximal compact subgroup of $G$.
Q1: How does one prove that $N_G(K)=K$?
So I know a nice (and low-tech) ...
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when $g^*$ is invariant under $Ad(G)$?
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
user21574
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Lie's third theorem via differential graded algebras?
Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following.
Let $\mathfrak{g}$ be a (finite-dimensional, real)...
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Subgroups of $E(n) = \mathbb{R}^n \rtimes O(n)$ with trivial orbit space
Let G be a subgroup of $E(n) = \mathbb{R}^n \rtimes O(n)$(the rigid motions of $\mathbb{R}^n$ ) with orbit space as a point.
Example: the group of all translations of $\mathbb{R}^n$ and of course any ...
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Quotient of 3-sphere by binary octahedral group?
Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...
- 3,059
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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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Maximal nilpotent subgroups of SO(n,1)
For the Lie group $SO(n,1)$ I believe the maximal nilpotent subgroups are conjugate to either a diagonal group times a compact group or a unipotent group times a compact group. In either case the ...
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Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)
I have difficulties finding an appropriate reference for the following question (which I hope that it to be true).
Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexification and $\tau: U^...
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Citation for positive Ricci curvature
Does anyone know a citeable source where it is shown that the Ricci curvature of SU(n) is strictly positive? I can sketch the proof but I need to shorten my notes.
Thanks,
Stefan
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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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symmetric points on symmetric spaces
Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on $1,2,\...
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equations over (some) lie groups
To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the ...
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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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