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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Infinitesimal rigidity vs. local rigidity
I am thinking about homomorphisms $\mathrm{Hom}(\Gamma,G)$, where $G$ is a Lie group and $\Gamma$ is a discrete, finitely generated subgroup.
This question talked about the difference of ...
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A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group
Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
\...
- 14.2k
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schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups
I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of ...
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Representation ring and induced representation
Let $i:H \to G$ be a homomorphism of compact Lie groups. The induced representation $\iota_*V := \mathrm{Map}^H(G,V)$ of an $H$-representation $V$ does not give an element of the representation ring $...
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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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Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
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Basics on lattice in classical groups
as a beginner,I am not sure whether this question is too basic to post here./-\。
Many textbook will talk about the prototypical example SL(n,Z)\SL(n,R), which can be identified with the space of ...
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Using topology to characterize embedded Lie subgroups of Lie groups.
Cartan's theorem states that any topologically closed subgroup of a Lie group is an embedded Lie subgroup.
This leads us to ask the following question:
Can we replace "topologically closed" with a ...
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Fixed submanifold of G-manifold
Let $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ the $G$-fixed subset. In general, $F$ has finitely many connected components and ...
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Generator of Lie Group cohomology in degree 3
This is my first question.
Take a simple, connected, compact, simply connected Lie group $ G$ (dim $G\geq 3$).
The cohomology of $G$ with integer coefficients is
$H^{1,2}(G,\mathbb{Z})\cong 0$, $H^{...
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Characterizing left invariant and right-$O_n$ invariant distances on $GL_n$
Consider the group $GL_n(\mathbb{R})$ with its standard topology.
It is not hard to show that there exists Riemannian metrics on it which are left-$GL_n$ and right-$O_n$ invariant. (In fact it's ...
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Any duality between different real forms of a complex Lie group? [closed]
A complex Lie group may have several real forms.
Are there any duality/trinity... between them?
Maybe a trivial question to ask, is $SL(3,\mathbb{C})$ a real form of $SL(3,\mathbb{C})\times SL(3,\...
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Thom-Gysin Sequences and Stratifications
Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
- 4,920
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free loop space and invariant forms
Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the ...
- 2,035
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Different Lie group structures on a vector space with the same Lie algebra structure
This is an eccentric question: recall that a smooth Lie group structure on $\mathbb R^n$ is uniquely identified by a triple $(\mu,\iota,e)$ where $\mu:\mathbb R^n\times\mathbb R^n\to\mathbb R^n$ is ...
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Discrete group action on the sphere
Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$
be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...
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Characterization of $L^1(\text{SL}(3,\mathbb{R}))$ [closed]
Is there a characterisation of the integrable functions on SL($3,\mathbb{R}$) or an explicit expression for the Haar measure?
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Are all orbits semi-Riemannian submanifolds?
Let $M$ be a semi-Riemannian manifold and $G\subset Iso(M)$ a closed connected Lie subgroup which acts properly on $M$. It is known that every orbit of the action is a (closed) submanifold of $M$. My ...
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Manifold approximations to $BO(3)$
We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$
Similarly $BO(2)$ can be approximated by closed, ...
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Is there a Lie II theorem for monoids?
Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of finite-...
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Generator of $\pi_3(SU(4))$ in Mimura-Toda
In this paper of Mimura and Toda, tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi_3(SU(4))$ as $...
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Sobolev spaces defined on non-compact Lie groups
In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ...
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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 ...
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How to deal with the singular reduction of the Hamiltonian n body problem?
I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular.
...
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3
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Intersections of conjugates of Lie subgroups
Let $G$ be a closed, connected Lie group, and let $H$ be a closed (and therefore Lie) subgroup. There is a natural action of $G$ on the space of left cosets $G/H$, for which the stabiliser of $aH$ is ...
- 35.9k
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First Variation of Dyson Series/Magnus Expansion
Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...
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Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
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abelian p- subgroups of E_6(q)
Is there any result about maximal abelian p-subgroups of the exceptional group E_6(q), where q=p^a is prime power?
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Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices
I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to ...
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List of irreducible representations whose weights are in a single Weyl group orbit
Let $\mathfrak g$ be a (finite dimensional) simple (nonabelian) Lie algebra
over $\mathbb C$. I need a complete list of irreducible (nontrivial) representations $V$
of $\mathfrak g$ such that the Weyl ...
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Stratifications and Cohomology Computations
I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...
- 4,920
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Connectedness of Centralisers in Unitary group
I want to understand centralizers of semisimple elements in unitary groups.
Let us begin with example of $GL_n(k)$. Centralizers of semisimple elememts are a product of smaller $GL_m(k)$ thus ...
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is connected complex Lie group with a trivial center linear?
There is a theorem of Rosenlicht ("Some basic theorems on algebraic groups", 1956, Theorem 13) asserting that a quotient of a connected algebraic group by its center is linear. So a connected ...
- 4,070
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Manifold with a quasi-positive curvature
As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature.
But ...
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Homomorphisms of Lie groups preserving regularity
Let $G_1, G_2$ be connected semisimple Lie groups, let us assume for simplicity that both groups are complex (even though, I am interested in the real Lie groups as well). Let $f: G_1\to G_2$ be a ...
- 31.2k
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power log distance between matrices
In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of ...
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Representations of $GL(n)$ containing $S^kV$
Let $V$ be a vector space of dimension $n$.
Let $S^k V$ be a representation of $GL(n)$.
I would like to know if there exists some characterization of finite dimensional $GL(n)$ modules $V_1,V_2$ such ...
- 2,179
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Two matrix Fisher distributions on SO(3)?
After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...
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Approximations of the identity on Lie groups and homogenous spaces
I'm looking for a nice (and preferably classic or book) reference for the following type of result:
Consider a transitive action of a compact Lie group $G$ on a compact manifold $M$ and a continuous ...
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regarding the upper half-plane model for the principal series representations of $\text{GL}_2\big( \Bbb{R}\big)$
Let $B$ be the Borel subgroup of $G = \text{GL}_2\big( \Bbb{R}\big)$, let
${\bf \alpha}:B \longrightarrow \Bbb{C}^*$ be a character, and consider the
induced representation $\text{Ind}_B^G ({\bf \...
- 2,137
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What is the importance of $\pi_{i}G$?
I often see conditons like $\pi_{2}G\not=0$ in reading old papers on Lie groups(no, my memory is wrong, they asked if $\pi_{1}G$ is free). I want to ask why we need this condition and how the higher ...
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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 ...
- 501
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2
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Symmetric and Exterior products of sl(n,C)-module
Let M be the $sl(n,C)$-representation of the inclusion $sl(n,C)\hookrightarrow gl(n,C)$.
Let q be a symbol.
$f(q)=1-M q + \wedge^2Mq^2-...+(-1)^n\wedge^nMq^n$
$g(q)=\sum_{i=0}^\infty Sym^iM \; q^i$
...
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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_\...
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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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sub-tori of a torus, generated by 1-dimensional subgroup
Ok the question is pretty dumb: suppose you have a torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$ and a vector $\bar{v}=(v_1,\ldots,v_n)\in\mathbb{R}^n$.
Consider the torus $T_{\bar{v}}$ given by the closure ...
- 1,452
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Cohomology of Projective Classical Lie Groups
Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...
- 4,920
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Lie group action on a finite dimensional flat manifold
Consider a finite dimensional flat Riemannian manifold $M$ quotiented by an action of a finite dimensional Lie group $G$, giving rise to the quotient $Q$.
First, assume that the action is isometric. ...
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98
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Roots of matrices in $G_2(Z)$
Let $G_2$ denote the exceptional Lie group $G_2$ as a $\mathbb{Q}$-algebraic group. Suppose that is also given a matrix representation $\rho : G_2\rightarrow SO(7)$. Let $M$ be a matrix with integral ...
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A Generalized De Rham cohomology
Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by $L_{\mathbb{C}}^{...
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