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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Existence of diagonalizing coordinates for the metric tensor
Solving for metrics that are Einstein, i.e that satisfy $R_{\mu \nu} = \Lambda g_{\mu \nu}$ is highly non-trivial as soon as $g_{\alpha \beta}$ is allowed to have off-diagonal components. However, ...
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Concept of Facets in the structure of reductive algebraic groups
Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet ...
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Normalizers of maximal compact groups?
Consider a reductive group over a local field. What is the normalizer of a maximal compact subgroup?
If this is to general, what is the normalizer of $GL(n, \mathbb{Z}_p)$ in $GL(n, \mathbb{Q}_p)$, ...
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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-...
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Reference for a statement about upper triangular unipotent matrices
I am revising a paper, and a referee of that paper asked if the following little lemma we proved there is known:
``Let $X$ be an $n\times n$ upper triangular unipotent matrix over $\mathbb R$. There ...
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Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
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Lie group actions with only one orbit type, but not defining a principal bundle
Searched-for situation: A compact connected Lie group acts effectively on a closed Riemannian manifold by isometries, such that there is only one orbit type of dimension strictly less than that of the ...
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About the purpose of introducing '"groups of Heisenberg type"
I would like to know, can we say that the "groups of Heisenberg type" where introduced by A. Kaplan in "Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by ...
- 650
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Bounds on Hilbert-Schmidt norm of difference of products of matrices
I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and $R_{1},\ldots,R_{k}...
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Pedagogical question on Lie groups vs. matrix Lie groups
There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
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Cohomology of boundary of locally symmetric space
Let $S$ be a locally symmetric space, not necessarily compact, and $\overline{S}$ be its Borel-Serre compactification. Let $\partial S$ be the boundary of $S$. Let $\widetilde{\mathbb{C}}$ be the ...
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Laplace-Beltrami operator on a Lie group
For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...
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Are two distinct Weyl chambers always disjoint?
Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian ...
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Mathematica package for Lie algebra computations?
I am interested in performing Lie algebra computations in Mathematica. I did a bit of searching and found several packages (LieART, KILLING, SuperLie, maybe more), and wondered if anyone would ...
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Relation between characters of a representation and characteristic classes
I have a basic question concerning Atiyah and Schmid's paper "A Geometric Construction of the Discrete Series for Semisimple Lie Groups", Inventiones mathematicae 42 (1977): 1-62".
I will use the ...
- 1,903
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Transitive action on the sphere
Hello!
From the book "Einstein manifolds" by Arthur L. Besse (at section 7.B), Lie groups $Sp(n)$, $Sp(n)\cdot U(1)$, $SU(2n)$ and $U(2n)$ constitute the complete list of Lie subgroups of $U(2n)$ ...
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Compactly supported distributions as a projective G-module
For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...
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Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$
Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that
$$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$
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On the proof by Chu-Kobayashi that transformation groups are Lie groups
Chapter I of Kobayashi's Transformation Groups in Differential Geometry contains a very general theorem on transformation groups, due to Palais. I have some questions about its proof (which I attach ...
- 3,146
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Finite groups normalizing a torus
Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...
- 2,384
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Invariants of a set of real unit vectors in 3d space, under SO(3)
I have a set of $n$ real unit vectors, in 3-dimensional space.
(It is a follow up of Sets of vectors related by a rotation.)
Is there a construction providing a complete set of independent*) ...
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A question on an set of 8 matrices related to the SU(3) generators
SU(2) and SU(3) differ quite a bit.
The Lie algebra of SU(2) formed by the three generators $g_n$ is the same as the algebra formed by the SU(2) matrices/elements $F_n=e^{\pi \cdot i \cdot g_n / 2}$. ...
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Iwasawa decomposition of the pseudo-orthogonal group
This is a soft-question, but I haven't found an answer anywhere: do the factors of the Iwasawa decomposition of the pseudo-orthogonal group SO(p, q) have a simple form, in the same way that the ...
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dirichlet problem in the heisenberg group
Good morning everybody.
I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ ...
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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 $\mathfrak{g}^*$.
...
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Are there versions of highly connected covers of Lie groups with highly periodic homotopy groups?
There is much activity around the study of highly connected covers of Lie groups (well, of their "infinite rank" versions like $\displaystyle{\lim_{N\to\infty}} \ O(N)$, say).
Looking at the ...
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G-invariant differential forms on homogeneous space of Lie Groups
Let $G$ be a connected Lie Group and $K<G$ a maximal compact subgroup.
Denote by $\Omega^q(G/K)^G$ the $G$-invariant real-valued $q$-forms on the manifold $G/K$, i.e. those forms $\omega$ s.t. $g^*...
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Right Invariant Randers metrics
I'm hoping to determine the geodesic equation for a right invariant Randers metric $F(x) = \sqrt{a(x,x)} + b(x)$ on $SU(N)$. In my special case the navigation data $(h,W)$ for the Randers metric are ...
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locally compact NSS groups and infinite dimensional Lie groups [closed]
We know that every finite dimensional Lie group has no small subgroups. Is there any infinite dimensional Lie group which has small subgroups?
Please explain your answer by refrence.
Thanks so much.
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Some counter examples in group theory
In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in \{1,2,\ldots,...
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Points with finite stabilizer in Hamiltonian torus actions
Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of $\...
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Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]
As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
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Rational orthogonal matrices
``everybody knows'' that an integral orthogonal matrix is a signed permutation matrix, so there are exactly $2^n n!$ such matrices in $O(n).$ Now, what if we ask for the enumeration of elements of $O(...
- 96.4k
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Number of connected components of the isometry group of a simply-connected lorentzian manifold
Let $(M,g)$ be a finite-dimensional connected lorentzian manifold. Then the group $G$ of isometries of $M$ (i.e., the group of diffeomorphisms $\varphi : M \to M$ with $\varphi^* g = g$) is a Lie ...
- 33.3k
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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 ...
- 3,472
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Is the centralizer $Z_G(A)=\{g\in G| a g= g a\}$ of a finite $A\subset G$ connected for a connected compact Lie group?
Let $G$ be a connected compact Lie group, consider the left/right action on itself.
For any finite $A\subset G$, consider the centralizer
$Z_G(A):=\{g\in G| a g= g a\}$.
Q: is $Z_G(A)$ a connected ...
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Is $SU(\infty)$ amenable?
We can write the finitary special unitary group $SU(\infty)$ as the direct limit
$\varinjlim SU(n)$ of ordinary special unitary groups. These groups $SU(n)$ are compact, thus amenable. In other ...
- 606
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Actions of torsionfree discrete subgroups on hermitian symmetric domains
Let $D$ be a bounded hermitian symmetric domain with automorphism group $G(\mathbb R)$. In the example I have in mind, $D$ is Siegel upper half-space of degree $g$ and $G(\mathbb R) = \mathrm{Sp}(2g,\...
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Fibers of the Bott-Samelson Resolution of Schubert Varieties
Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$.
Also, how would the answer to the ...
- 1,719
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Kodaira dimension of co-adjoint orbit
Let $G$ be a compact Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also ...
user21574
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What is miraculous about the mirabolic subgroup?
I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...
- 2,516
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A (possible) equivalent relation on the space of vector bundles
Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $...
- 356
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Integrating Poisson groups
Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...
- 3,357
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Laplacian on two Lie groups have the same Lie algebra
I know that if $G$ is a Lie group and $\mathfrak g = span\{X_i, 1\leq i \leq n\}$ be its Lie algebra, where $\{X_i\}$ are the vector fields of $G$. Then, the Casimir-Laplacian of $G$ is given by
$$\...
- 650
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Harish-Chandra isomorphism for compact symmetric spaces
I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
Below ...
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Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 1,111
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Representation theory and associated bundles
I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
- 6,025
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Are norm-continuous representations smooth?
Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous
$$
x_i\to x\quad\Longrightarrow\quad ||\varphi(...
- 7,426
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Classifications of projective representations of $SO(N)$ and $SO(N)\times Z_2^T$
This question is motivated by some physics questions: what are the classifications of projective representations of groups $SO(N)$ and $SO(N)\times Z_2^T$? This is equivalent to asking what are $H^2(...
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Infinite dimensional unitary representations of SU(2) for non-half-integer j?
The finite dimensional irreducible unitary representations of $SU(2)$ are labelled by $j$ which needs to be half-integer, the dimension of the representation is $2j+1$. This is well-known, all is good....
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