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
3
votes
0
answers
142
views
Solvability of a matrix exponential equation - generalized matrix logarithm
For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation
$$
G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .
$$
Basic ...
- 1,081
1
vote
0
answers
67
views
Poisson bracket on $T^*T\mathrm{SU}(1,1)$
Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \...
- 829
2
votes
0
answers
81
views
Complex semisimple Lie algebra modules with non-semisimple Cartan action
Let $\frak{g}$ be a complex semisimple Lie algebra. I would like to know about infinite-dimensional representations $M$ of $\frak{g}$ for which the Cartan $\frak{h} \subseteq \frak{g}$ does not act ...
- 257
3
votes
1
answer
111
views
Fusing conjugacy classes II
(Followup to this question)
Consider a finite-dimensional Lie group $G$ and two conjugacy classes $H$ and $I$ of isomorphic subgroups of $G$.
Question. Is there some finite-dimensional Lie overgroup ...
- 2,773
0
votes
0
answers
124
views
Calculation of first correction to Selberg type integral
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\arcsinh{arcsinh}$Let $U \in G$, where $G$ is $\SU(N)$ matrix.
$\Tr U$ will denote the character ...
2
votes
1
answer
136
views
Generalization of $G/T \simeq G_\mathbb{C}/B$
Let $G$ be a compact Lie group and Let $G_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$.
Consider $H$ to be a Lie subgroup of $G$ and denote ...
- 139
53
votes
5
answers
8k
views
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 1,422
18
votes
2
answers
1k
views
Emergence of the orthogonal group
Do we know what mathematician first considered, and perhaps named, what we call the group $\mathrm O(n)$, or $\mathrm{SO}(n)$, for some $n>3$?
I mean it specifically as group (not Lie algebra) ...
- 31.5k
2
votes
1
answer
94
views
Unitary dual of universal cover
The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...
user473423
9
votes
0
answers
164
views
Parallelizability of Lie monoids
A Lie monoid is a monoid, together with a structure of a smooth manifold (possibly with a boundary), such that the monoid multiplication is smooth.
If all left (or right) translations in a Lie monoid $...
- 61
6
votes
1
answer
644
views
Torus bundles and compact solvmanifolds
I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...
- 4,081
6
votes
0
answers
147
views
Maximum symmetry metric on irreducible compact symmetric space
Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
- 4,081
3
votes
3
answers
582
views
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
- 143
3
votes
2
answers
298
views
Transitive action on non-orientable $ M $ lifts to orientable double cover
Suppose that $ M $ is non-orientable with transitive action by a Lie group $ G $. Does that imply that some Lie group $ G' $ acts transitively on the orientable double cover $M'$?
This is true for ...
- 4,081
4
votes
1
answer
348
views
Verma modules and Borel–Weil
Let $\mathfrak{g}$ be a semisimple Lie algebra and fix a root system. Let $\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$. The complex irreducible representation of $\...
- 181
0
votes
0
answers
97
views
Methods for calculating (one-parameter subgroup) actions
For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form
\begin{equation}
\mathrm{e}^{t L(z)} f(z)
\end{equation}
...
- 679
1
vote
1
answer
106
views
A lattice in $ \operatorname{SL}_n $ is Ad-irreducible
$\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ \SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by ...
- 4,081
3
votes
1
answer
203
views
Free $S^1$-action on compact homogeneous spaces
Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
If $r(G) > r(K)$ (...
- 143
13
votes
3
answers
1k
views
Extending group actions to vector bundles
Let $G$ be a group acting on a manifold $M$. Suppose $V$ is a rank $n$ vector bundle on $M$.
Is there any obstruction to extending the action of $G$ to $V$? In how many ways can the action be extended ...
- 139
8
votes
1
answer
429
views
Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?
I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that $\infty$-groupoids, with $\...
- 1,198
77
votes
7
answers
21k
views
What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
- 54.6k
10
votes
1
answer
434
views
Viewing exceptional Lie algebras via the classical ones
I've been trying to understand the exceptional Lie algebras through the classical ones that I am more familiar with. In particular I wanted to get a handle on the root spaces and most discussions that ...
- 954
6
votes
1
answer
445
views
Is every finite subgroup the integer points of a linear algebraic group?
Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link Is every finite subgroup the integer points of a linear algebraic group?
Let $ ...
- 4,081
9
votes
2
answers
523
views
Is the group of integer points of a simple real linear algebraic group a maximal closed subgroup?
Let $ G $ be a simple linear algebraic group. Let $ G_\mathbb{R} $ be the real points of $ G $. Let $ G_\mathbb{Z} $ be the integer points of $ G $. Is $ G_\mathbb{Z} $ a maximal closed subgroup? In ...
- 4,081
1
vote
0
answers
46
views
Weight of adjoint action on a lower central series extension
Let $\mathcal{U}$ be a unipotent Lie $\mathbb{Q}_p$-group scheme, whose associated gradeds from the lower central series filtration are $\mathcal{U}_0 = \mathcal{U}^{\text{ab}}$, $\mathcal{U}_1 = [\...
- 2,907
3
votes
1
answer
195
views
A representation of $\frak{sl}_n$ as partial derivatives on polynomials
As is known to all, the Lie algebra $\frak{sl}_2$ admits a very nice representation on
$$
\mathbb{K}[X,Y]
$$
the polynomials in two variables, given by
$$
E \mapsto X\frac{\partial }{\partial Y}, ~~ F ...
- 1,144
0
votes
1
answer
169
views
Question about coadjoint orbits of compact connected Lie groups
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Denote by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $r$ be an element of $\mathfrak{g}^*$ such that $G_r$ the stabilizer of ...
- 139
5
votes
0
answers
218
views
$C^1$ isometries of pseudo-Riemannian metrics are smooth?
It is well known that $C^1$ (actually even just differentiable) isometries of Riemannian manifolds are actually $C^\infty$. The proof is based on the metric structure generated by the Riemannian ...
- 51
12
votes
2
answers
849
views
Groups associated with infinite dimensional Lie algebras
There is a classical correspondence between Lie algebras (over $\mathbb{R}$ or $\mathbb{C}$) and Lie groups in the finite dimensional case: to every Lie group $G$ there is an associated Lie algebra $\...
- 1,089
0
votes
0
answers
84
views
Determinant of SU(N) elements, and radius of associated manifold
I'm wondering if the fact $SU(2)$ group elements have $det = 1$ is connected with the radius of the unitary $S^{3}$ manifold associated.
The context is demonstration of dU being an Haar invariant ...
- 1
4
votes
0
answers
178
views
The homotopy type of the space of symplectic structures
While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
- 516
0
votes
0
answers
172
views
The largest abelian subgroups of a Lie group
Let $G$ be a semisimple Lie group. Denote $d(G)$ as the maximal integer $p$ such
that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\...
12
votes
0
answers
247
views
Symmetric spaces are quandles. Is this important?
For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
- 2,516
10
votes
0
answers
292
views
Each simple real Lie algebra as a representation of its maximal compact subalgebra
I am interested in a detailed description of the Cartan decomposition of each type of simple, real, finite-dimensional Lie algebra. (This is essentially a question about the classification of simple, ...
- 56.6k
12
votes
1
answer
655
views
Which Lie groups have finitely many conjugacy classes of subgroups of fixed isomorphism type?
Let $G$ be a real Lie group.
What conditions must $G$ satisfy so that the following is true:
For any finite group $\Gamma$ there exist finitely many conjugacy classes of subgroups of $G$ that are ...
- 577
27
votes
10
answers
2k
views
Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?
In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic.
What are some good examples of Kan extensions, adjunctions, and (co)...
3
votes
0
answers
143
views
Summing over roots of a simple Lie algebra and Deligne series
For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...
- 857
5
votes
1
answer
234
views
A group in a neighbourhood of a Zariski dense subgroup
By Borel's theorem, lattices in simple Lie groups are Zariski dense. I expect that a small (in metric sense) deformation of a lattice in a Lie group is also Zariski dense.
Suppose we have a Zariski ...
- 9,237
0
votes
1
answer
175
views
Centralizer of a reductive subgroup
Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
- 833
2
votes
1
answer
160
views
Is the restriction of the Cartan 3-form on conjugacy classes exact?
Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
...
- 443
14
votes
5
answers
1k
views
History of the notion of $(G,X)$-structure
I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...
- 403
7
votes
1
answer
358
views
Automorphism group of compact almost complex manifold
Does the automorphism group of a compact almost complex manifold carry a (canonical) Lie group structure? Part 3 of Theorem 4.1 in
*"The automorphism group of a homogeneous almost complex ...
- 271
7
votes
3
answers
599
views
Root system of fixed point Lie sub-algebra
It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
- 103
7
votes
1
answer
221
views
Is a quotient of real linear algebraic groups always a Cartesian product of compact and contractible factors?
Let $ G $ be the real points of a linear algebraic group and $ G' $ a Zariski closed subgroup. Then is $ G/G' $ a Cartesian product
$$
(K/K') \times F
$$
where $ F $ is contractible? Here $ K,K' $ ...
- 4,081
0
votes
1
answer
130
views
Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules
Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial?
...
- 327
2
votes
0
answers
137
views
Visualisation of general 3x3 matrices, with applications to the pedagogy of linear algebra?
I've got a method for visualising non-zero $2 \times 2$ real matrices (modulo non-zero scalar factor) using the fact that:
Nonnegative determinant matrices (modulo non-zero scalar factor) are in 1-to-...
- 4,943
8
votes
1
answer
566
views
Importance of third homology of $\operatorname{SL}_{2}$ over a field
$\DeclareMathOperator\SL{SL}$I am reading some papers about the third homology of linear groups. In particular for the $\SL_{2}$ over a field. Why is it important to study these homologies?
I have ...
- 259
4
votes
1
answer
133
views
Universal character ring for classical groups
The universal character ring for the general linear group is well understood but I want to ask about the universal character ring for the symplectic and orthogonal groups. For the general linear group,...
- 81
3
votes
1
answer
484
views
Relationship between the representation theory of $\operatorname{Spin}(n)$ and $\operatorname{SO}(n)$
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$What is the exact relationship between the finite dimensional representations of the group $\SO(n)$ and its covering group $\Spin(n)$? More ...
- 393
4
votes
0
answers
161
views
Differential invariants and Lie symmetries
I have the following question:
Does differential invariants have the same Lie symmetries?
I want to know about the relation of differential invariants of an action and their symmetry properties. ...
- 49