All Questions
Tagged with lie-groups reference-request
298 questions
3
votes
2
answers
1k
views
Reference for homogeneous spaces
I am a graduate student of differential geometry.
I would like to get an overview over the way, how results are usually obtained for homogeneous spaces by Lie algebraic methods. By definition a ...
- 79
3
votes
1
answer
239
views
What is the curved version of the Tits fibration for $G_2$?
Let
$\require{AMScd}$
\begin{CD}
G_2/(P_1\cap P_2) @= G_2/(P_1\cap P_2)=:\mathbb{I}\\
@V \lambda V V @VV \pi V\\
\mathbb{Q}_5:=G_2/P_1 && G_2/P_2=:\mathbb{N}_5
\end{CD}
be the Tits ...
- 2,527
3
votes
1
answer
234
views
Reference to definition of matrix log with domain SO(3) which is Borel measurable
I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...
- 203
3
votes
1
answer
140
views
Asymptotics of Haar moments on general Lie groups
I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
- 179
3
votes
1
answer
138
views
Frechet Lie groups and their subgroups
1) Let $G$ be a Fréchet Lie group. Let $H$ be a closed subgroup. Is it always true that the centraliser of $H$ is a Fréchet subgroup of the lie group?
2) Is the closed subgroup theorem valid for ...
- 755
3
votes
3
answers
1k
views
Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
- 6,023
3
votes
1
answer
261
views
Subgroups of compact Lie groups generated by a subset of nodes of the Dynkin diagram
Where can I find a reference for the following fact, or as close as possible to it?
Let $G$ be a semisimple compact real Lie group with rank $r$, let $T$ be a maximal torus in $G$, let $\mathfrak{g}...
- 32.5k
3
votes
1
answer
396
views
Choosing canonical representatives of Weyl group elements, some questions
Let $G$ be a connected, reductive group which is quasisplit over a field $k$ of characteristic zero. Let $B$ be a Borel subgroup defined over $k$, containing a maximal torus $T$ defined over $k$. ...
- 6,180
3
votes
0
answers
107
views
Representations of a reductive Lie group vie central character and K-types
Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
- 460
3
votes
0
answers
80
views
Can a semisimple orbit always be identified with a cotangent bundle?
Let $H$ be a semisimple element of the Lie algebra $\mathfrak{g}$ of a semisimple Lie group $G$, and let $M:=\mathrm{Ad}_G(H)\subset\mathfrak{g}$ be the corresponding adjoint orbit. If we choose a ...
- 2,527
3
votes
0
answers
94
views
The tangent bundle of $G \times_H M$
Let $G$ be a Lie group with a closed subgroup $H$, and let $M$ be a smooth $H$-manifold. I am searching for a reference where it is proved that the tangent bundle of $G \times_H M$ is isomorphic to ...
- 198
3
votes
0
answers
142
views
Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces
Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
- 954
3
votes
0
answers
50
views
How to construct lattice points in bounded symmetric domain?
Consider the Hermitian bounded symmetric domain for $k \leq m$:
$$
C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \}
$$
where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken,...
- 8,597
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
3
votes
0
answers
65
views
Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries
Could anyone please suggest related papers or article about the topic related to my one question below?
Reduce PDE to ODE by dilation symmetry
I also cite a paper in the link above.
We know that ...
- 345
3
votes
0
answers
58
views
Isoclinism for Lie groups: existing accounts of basic properties?
Philip Hall introduced the relation of isoclinism between two groups. One statement of the definition (not Hall's original statement) is to introduce a category whose objects are the canonical maps
$$...
- 25.8k
3
votes
0
answers
133
views
Weyl group stabilizer of semisimple element in adjoint group
Let $G$ be semisimple group over $\mathbb{C}$ of adjoint type. Let $T$ be a maximal torus, $s\in T$ semisimple element. Let $W$ be a Weyl group and $W(s)$ be a stabilizer of $s$ in $W$. I am ...
- 31
3
votes
0
answers
136
views
Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
3
votes
0
answers
129
views
Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
3
votes
0
answers
170
views
Finding generators of equivariant cohomology
Let $(M,\omega)$ be a symplectic manifold with symplectic form $\omega$, carrying a Hamiltonian action of a compact connected Lie group $G$ with moment map $\mu:M\to \mathfrak{g}^\ast$, where $\...
- 1,942
3
votes
0
answers
156
views
How large is the intersection of the root system of a subalgebra of a compact Lie algebra with the original root system?
Let $\mathfrak{g}$ be a finite-dimensional real compact Lie algebra and $\mathfrak{t}\subset \mathfrak{g}$ a maximal abelian subalgebra. Let $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})\...
- 1,942
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,087
3
votes
0
answers
116
views
Extension of representations of certain compact Lie groups
Let $G$ be a real, connected, non-compact, semisimple Lie group with finite center and real rank $1$. Let $G=KAN$ be an Iwasawa decomposition, then $K\subset G$ is a maximal compact subgroup, and $\...
- 1,942
3
votes
0
answers
174
views
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 ...
- 646
3
votes
0
answers
61
views
Reference request: table of representation rings and relations
Where can one find a table of generator–relator expressions for representation rings $R(G)$ of simple Lie groups $G$ and explicit maps between them? For example, given a maximal torus $T < G$ or a ...
- 2,995
3
votes
0
answers
108
views
$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 ...
- 3,051
3
votes
0
answers
362
views
Unitary representation of finite-dimensional unitary group
the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...
3
votes
0
answers
214
views
Unitary dual of $Sp_4(\mathbb{R})$
Do we know the unitary dual of $Sp_4(\mathbb{R})$? If so, can someone provide me any references? How about other rank 2 real groups? Thank you!
3
votes
0
answers
151
views
Equivariant Poincare Series of Based Loop Group of SU(2)
Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...
- 4,920
2
votes
2
answers
680
views
Complete representation theory of $\mathrm{SL}(2,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...
- 3,873
2
votes
1
answer
319
views
Reference request: Calculation in exceptional Lie groups
Let $G$ be a compact connected simple exceptional Lie group. Let $G$ be contained in a unitary group ${\rm U}(n)$ by some standard (low dimensional) unitary representation. For example in the case of $...
- 804
2
votes
2
answers
1k
views
Reductive Lie algebra of a Lie group
In the answer of my question:
On the full reducibility of representations of reductive Lie algebras
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in ...
- 671
2
votes
1
answer
119
views
Diagonalization of octonionic Hermitian matrices of size $2\times 2$
The group $Spin(9)$ is a subgroup of $SO(16)$ and acts transitively on the unit sphere $S^{15}$. $Spin(9)$ acts naturally on the space of octonionic Hermitian $2\times 2$-matrices (I do not define ...
- 21.8k
2
votes
1
answer
280
views
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
2
votes
2
answers
311
views
Lie groups and NSS+LC group
Let $G$ be a locally compact group without small subgroups. Is $G$ a "finite" dimensional Lie group? (i.e, $G$ is not infinite dimensional Lie group.)
Are Lie groups precisely the locally Euclidean ...
2
votes
1
answer
315
views
A reductive group is the complexification of a compact subgroup even if not connected?
The definition of a linear algebraic complex reductive group is sometimes using the connectedness hypothesis for the complex algebraic group sometimes not.
Here I use the following definition : a ...
- 1,128
2
votes
1
answer
217
views
A variation of closed-subgroup theorem
$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "...
- 14.3k
2
votes
1
answer
833
views
Which Lie algebras are realised as vector fields on a group?
Fix a Lie group $G$. The vector fields on $G$ form an (infinite-dimensional) Lie algebra with the commutator of vector fields as Lie bracket. What (finite-dimensional) Lie algebras can I realise as ...
- 503
2
votes
1
answer
166
views
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
- 21
2
votes
1
answer
244
views
Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's "Classical Groups"
First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but didn'...
- 339
2
votes
1
answer
165
views
Cocompact lattices in $\mathrm{Sp}(n, 1)$
This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
- 131
2
votes
1
answer
214
views
What is the space parametrising the curved sub-Cartan geometries of a flat Cartan geometry?
I'm basically wondering how to make "curved" the first column of the diagram
$\require{AMScd}$
\begin{CD}
P_1 @>\textrm{inclusion} >> G\\
@V \omega_0 V P_1\cap P_2 V @V\omega V P_2 V\\...
- 2,527
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
2
votes
2
answers
108
views
Hilbert's Finiteness Theorem for connected semisimple Lie groups over $\mathbb{C}$ in Weyl's "Classical Groups" [duplicate]
In Nagata's "Lectures on the 14th problem of Hilbert" I found a reference to Weyl's "Classical Groups". Nagata writes that Weyl gives a positive answer to the original problem
If $G\subseteq\...
- 339
2
votes
2
answers
519
views
How to use that the Hessian is negative definite in this proof
Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
- 139
2
votes
1
answer
335
views
Commutator of radical and Levi factor in Lie algebra
Write a complex finite-dimensional Lie algebra as $L=S\ltimes R$ (Levi decomposition). Then the subalgebra $[S,R]$ (generated by brackets $[s,r]$, $s\in S$, $r\in R$) is an ideal, by a simple ...
- 21
2
votes
1
answer
251
views
Characterization of restricted weights of representations of real semisimple Lie groups
I need to use the following theorem:
Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of ...
- 1,574
2
votes
1
answer
660
views
The quotient of a Lie group by the Levi factor of a parabolic subgroup
I am interested in some references on the quotient spaces obtained by quotienting G, a simple Lie group, by L, the group generated by the Levi factor of a parabolic subalgebra.
Presumably the case ...
- 2,123
2
votes
1
answer
1k
views
Lie derivative on Lie group in the direction of an element of Lie algebra
I want a reference to the definition of the Lie derivative of a smooth function $f:G \to \mathbb R$ on a Lie group $G$ in the direction of an element $\theta$ of the Lie algebra $\mathfrak G$.
I can ...
2
votes
1
answer
85
views
Closure of the set of topological non-generators in compact Lie group
Let $G$ be a (non-abelian) compact connected Lie group. Let $K\subseteq G^2$ be a set of pairs $(g_1, g_2)$ which $\textbf{do not}$ generate $G$ topologically.
For which Lie groups $G$ is it known ...
- 728