Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
2,938
questions
4
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2
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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 ...
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 _”?
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 ...
1
vote
0
answers
78
views
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 ...
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 ...
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 ...
3
votes
1
answer
213
views
Semidirect product of two linear groups
Let $G$ and $H$ two connected linear Lie groups. Is $G\ltimes H$ also linear?
3
votes
1
answer
189
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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)....
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 ...
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 ...
1
vote
0
answers
69
views
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 ...
3
votes
0
answers
81
views
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 ...
0
votes
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 \...
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 = [\...
3
votes
1
answer
187
views
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 ...
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 ...
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}\\ ...
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{...
0
votes
0
answers
164
views
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 ...
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. ...
1
vote
1
answer
123
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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 ...
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?
1
vote
1
answer
257
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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 ...
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 ...
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 $...
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 ...
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,\...
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 ...
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 ...
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 \...
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(...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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$...
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 $\...
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 ...
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 (...
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $...
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 ...