Questions tagged [lie-groups]

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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Reference for group-algebra/exp-log like identites in combinatorics

I've encountered several identities in combinatorics that resemble inversion formulas, as shown below, Here, $f_i, g_k$, $\forall i,k \in \mathbb{N}$, are coefficients of some formal power series. I ...
total dependent random choice's user avatar
3 votes
0 answers
108 views

Names for split Lie groups

Do any of the simply connected simple Lie groups of the split real classical Lie algebras have names other than “the universal cover of _”?
Daniel Sebald's user avatar
3 votes
1 answer
165 views

A filtration on Drinfeld-Jimbo quantum enveloping algebras

For the universal enveloping algebra $U(\frak{g})$ of a Lie algebra $\frak{g}$, one can define in a natural way an increasing $\mathbb{N}_{0}$-filtration. By the Poincaré-Birkhoff–Witt theorem, the ...
Lorenzo Del Vecchiopontopolos's user avatar
1 vote
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Questions on the differential of the Lie logarithm

Let $G$ be a Lie group. Recall the Lie logarithm is well-defined about a neighborhood $U \subset G$ of the identity: $\log:U\to \mathfrak{g}$. I am dealing with a research problem that concerns the ...
Spencer Kraisler's user avatar
2 votes
0 answers
51 views

Normal form of this group action?

Let $d\in\mathbb{N}$. We consider the vector space $V=\mathbb{C}^2\otimes\mathbb{C}[x_0,x_1]_d$ where $\mathbb{C}[x_0,x_1]_d$ is the space of homogeneous binary forms of degree $d$. We have a natural ...
Hans's user avatar
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8 votes
1 answer
744 views

What are the important open problems about Lie groups?

I know that the theory of Lie groups is a very old subject, and the literature is incredibly vast, so I am wondering what contemporary research on Lie groups is about. What open problems are there in ...
cgb5436's user avatar
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3 votes
1 answer
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Semidirect product of two linear groups

Let $G$ and $H$ two connected linear Lie groups. Is $G\ltimes H$ also linear?
M. Han's user avatar
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3 votes
1 answer
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Eigenforms of the Laplacian on Lie groups

I am a bit rusty in my differential geometry and I would like to confirm that my reasoning below holds, and I have some related questions (and all references to related concepts are of interest to me)....
Daniel Robert-Nicoud's user avatar
2 votes
0 answers
90 views

Embeddings of symplectic group into the orthogonal group

Let $\mathfrak{sp}$ denote the complex symplectic Lie algebra and $\mathfrak{so}$ the complex orthogonal one. Do we have an embedding $$ \mathfrak{sp}_{2n-2} \hookrightarrow \mathfrak{so}_{2n}? $$ In ...
Dr. Evil's user avatar
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0 votes
1 answer
198 views

Adjoint action on the universal enveloping algebra and the PBW theorem

Let $\frak{g}$ be a semisimple Lie algebra and $U(\frak{g})$ its universal enveloping algebra. The adjoint action of $\frak{g}$ on itself extends to an action of $\frak{g}$ on $U(\frak{g})$. How does ...
Béla Fürdőház 's user avatar
1 vote
0 answers
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Minimal $K$-orbit on $\mathfrak{g}$

Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra with Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the ...
Hebe's user avatar
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3 votes
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Explicit computation of the transfer in the representation ring for unitary groups

For a compact Lie group $G$ we let $R(G)$ be the ring of finite dimensional complex $G$-representations studied by Segal in http://www.numdam.org/item/PMIHES_1968__34__113_0.pdf. This comes with extra ...
MLV's user avatar
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0 answers
20 views

How to find condition on a linear operator $b: g \rightarrow g$ so that $b$ induces a twisted homomorphism on its simply connected Lie group

Let $G$ be a simply connected Lie group with Lie algebra $g$. Let $b: g \rightarrow g$ is a linear operator. Suppose I induce a map $B: G \rightarrow G$ from $b$ in following manner: Let $e : g \...
NIshant Rathee's user avatar
1 vote
0 answers
44 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 = [\...
kindasorta's user avatar
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3 votes
1 answer
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Lie's third theorem via graded geometry

Lie's third theorem : Given any finite dimensional Lie algebra $\mathfrak{g}$, there exists a Lie group $G$ whose Lie algebra is equal to $\mathfrak{g}$. In one of the talks, speaker mentions that ...
Praphulla Koushik's user avatar
2 votes
0 answers
63 views

Are the integer points of a simple linear algebraic group 2-generated?

Set Up: Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
Ian Gershon Teixeira's user avatar
4 votes
0 answers
89 views

Lie bracket of general unipotent matrices

Let $k$ be a field (of characteristic $0$). Let $$ X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
Li Guanyu's user avatar
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14 votes
2 answers
557 views

Existence of a regular semisimple element over $\mathbb{F}_{q}$

This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help. Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{...
D. Dona's user avatar
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0 answers
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A Lie group whose Lie algebra is the (Lie algebra?) of all functions with fibrewise polynomial growth

Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket ...
Ali Taghavi's user avatar
6 votes
1 answer
354 views

An alternative form of the Kazhdan-Lusztig conjecture

Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. ...
Estwald's user avatar
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1 vote
1 answer
123 views

Commuting time dependent vector fields and pullback invariance

Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields. Is there some analogue of the following fact in finite ...
Theo Diamantakis's user avatar
2 votes
1 answer
163 views

About finitely generated lattices in Lie groups

Let $G$ be a connected Lie group. Let $\Gamma$ a lattice in $G$ not necessarily uniform (cocompact). Is it true that $\Gamma$ is finitely generated?
M. Han's user avatar
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1 vote
1 answer
257 views

Uniqueness of spinor representation

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$I asked a similar question on math stack exchange here, but I wonder if it may be better received here. Let $n$ be ...
Chris's user avatar
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4 votes
1 answer
173 views

What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?

Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...
Sami Douba's user avatar
1 vote
0 answers
113 views

A compact Lie group $G$ acting on a compact Lie group $K$ transitively. Is there a $C$ such that $d(gx,gy)\leq Cd(x,y)$?

Let $G$ be a compact connected Lie group acting transitively and smoothly on another compact Lie group $K$. Let $d$ be the distance in $K$ that is not $G$-invariant. Is there a constant $C$ such that $...
André Gomes's user avatar
4 votes
0 answers
96 views

Frobenius norm bounds on exponentials of anti-Hermitian matrices

Suppose $X$ and $Y$ are two anti-Hermitian matrices satisifying $\|X\|, \|Y\| \leq \pi$, where $\|\cdot\|$ is the spectral norm. I'm trying to prove the following bounds on the Frobenius norm of the ...
Haimeng Zhao's user avatar
2 votes
0 answers
343 views

Analogue of Margulis height function in non lattice subgroups

I have been reading this paper https://link.springer.com/article/10.1007/s11854-017-0033-4 on singular system of linear forms and non escape of mass in homogeneous spaces $G/\Gamma$ where $ G=SL(m+n,\...
User1723's user avatar
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5 votes
0 answers
135 views

When is a unitary group over a ring of integers dense?

Let $ SU_n(O_d) $ denote an integral unitary group of $ n \times n $ matrices over a totally real number field $ K_d:=\mathbb{Q}(\cos(\frac{2\pi }{d})) $ where $ O_d $ is the ring of integers of $ K_d ...
Ian Gershon Teixeira's user avatar
4 votes
1 answer
200 views

Laplace beltrami eigenspaces of compact Lie groups

For a Riemannian manifold $\mathbb M$, let $0=\lambda_0<\lambda_1<\cdots$ be the eigenvalues of (negative of) its Laplace-Beltrami $-\Delta_{\mathbb M}$, with corresponding eigenspaces $\mathcal ...
bosch_et_tu's user avatar
1 vote
0 answers
56 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 \...
Koundinya Vajjha's user avatar
1 vote
0 answers
64 views

Classical groups generated by tensor products of subgroups

Let $ G $ denote a classical group. Question: Is it the case that $$ \langle G_n \otimes G_m,G_m \otimes G_n\rangle=G_{nm} $$ as long as $ n \neq m $? For example, if $ G $ is the classical group $ GL(...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
91 views

Properties of smooth vectors for Banach representations of Lie groups

I think the following should be known. However, I was not able to find an answer. Suppose $ V $ is an infinite dimensional Banach space representation of a Lie group $ G $. In case the answer depends ...
JaSch's user avatar
  • 183
5 votes
1 answer
360 views

Geodesic distance on $\mathrm{SO}(n)$

$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
Math_Newbie's user avatar
1 vote
0 answers
114 views

Question on Artin's Gamma function on $\operatorname{SO}(2,0)(\mathbb R)$

$\DeclareMathOperator\SO{SO}$Let $G=\SO(2,0)(\mathbb{R})$, a quasi-split group with signature $(2,0)$. Let $e$ be an element in $O(2,0)(\mathbb{R}) \setminus \SO(2,0)(\mathbb{R})$. Let $\pi$ be an ...
Andrew's user avatar
  • 875
6 votes
0 answers
227 views

What can lattices tell us about lattices?

A general group-theoretic lattice is usually defined as something like A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
Mark Schultz-Wu's user avatar
5 votes
0 answers
110 views

What do the Carnot groups act on?

My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive. A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
Theo Diamantakis's user avatar
3 votes
0 answers
71 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 ...
Giovanni Moreno's user avatar
3 votes
2 answers
167 views

The orbits of an algebraic action of a semidirect product of a unipotent group and a compact group are closed?

We consider real algebraic groups and real algebraic varieties. It is known that the orbits of an algebraic action of a unipotent algebraic group $U$ on an affine variety are closed. The orbits of an ...
Vladimir47 's user avatar
2 votes
1 answer
116 views

Polar decomposition with respect to the nonstandard involution of quaternionic matrices?

The quaternions admit infinitely many involutions. But up to isomorphism, there are only two: The standard one $t+xi+yj+zk\mapsto t-xi-yj-zk$ and the nonstandard one $\phi:t+xi+yj+zk\mapsto t-xi+yj+zk$...
wlad's user avatar
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4 votes
0 answers
174 views

What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?

$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
NIshant Rathee's user avatar
4 votes
1 answer
145 views

An analogue of Mostow-Palais equivariant embedding theorem for the group of conformal automorphisms of the 2-sphere

Is there a smooth embedding of $S^2$ into some Euclidean space that is equivariant with respect to a linear representation of $PSL(2,\mathbb C)$? A counterexample to a more general question can be ...
Igor Belegradek's user avatar
5 votes
1 answer
227 views

Single sum of squares of Clebsch–Gordan coefficients

Let $C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch–Gordan coefficients of $\operatorname{SU}(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (...
onamoonlessnight's user avatar
2 votes
0 answers
132 views

Proof of Zimmer's cocycle super-rigidity theorem

I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
John Depp's user avatar
  • 187
4 votes
1 answer
223 views

Geodesics on orthogonal matrix

Let $ O(n) $ be the manifold of orthornormal matrix, i.e. $$ O(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=I\}. $$ Then $ O(n) $ is a submanifold of $ \mathbb{R}^{n\times n} $. On $ O(n) $, there is a ...
Luis Yanka Annalisc's user avatar
0 votes
0 answers
88 views

Concrete examples of quantum duality principle

Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
yohei ohta's user avatar
0 votes
0 answers
68 views

Integrating homomorphisms of Borel subalgebras

Let $G$ be a connected simple complex Lie group and $\mathfrak{g}$ be its Lie algebra. Let us fix a root decomposition, let $\mathfrak{b}_\pm$, $\mathfrak{n}_+$ and $\mathfrak{h}$ be the corresponding ...
Grabovsky's user avatar
1 vote
0 answers
71 views

Does Poincaré duality link topological study and representation study of a given Lie group?

The Poincaré duality for an oriented n-manifold M takes the form : $$H^\star(M) \simeq H_c^{n-\star}(M)^\vee.$$ Instead of M take now a real Lie group G. We can basically study it by looking at its ...
TopGenAx's user avatar
3 votes
0 answers
282 views

The definition of a homogeneous vector bundle

For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...
Béla Fürdőház 's user avatar
5 votes
2 answers
660 views

Does every connected Lie group have a dense torsion-free subgroup?

Question: Do all connected Lie groups have dense torsion-free subgroups? Context : Let $ R_\alpha \in SO_2(\mathbb{R}) $ be a rotation by $ \alpha/2\pi $. If $ \alpha $ is irrational, then $ R_\alpha $...
Ian Gershon Teixeira's user avatar
1 vote
0 answers
37 views

Splitting of the conformal group into $PSL(2,\mathbb{R})$ and other factorizations

In 1+1 dimensions of Minkowski spacetime, the conformal group can be split into two copies of $PSL(2,\mathbb{R})$ acting on null lines. I'm curious to know if a similar split exists for the conformal ...
Gabriel Palau's user avatar

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