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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Can discrete groups be Lie groups? Are all finite groups Lie groups? [closed]
I was trained as a physicist, rather than a mathematician. So, I apologize if my question is naive.
As a physicist, I think of Lie groups as being continuous groups. I have begun studying the second ...
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Counting fixed points on flag variety and Deligne-Lusztig functors
Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
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Reference Request: Structure constants for G2
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
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Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?
Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
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Maximal geodesic spheres in the "octooctonic projective plane"
Boris Rosenfeld claimed that the 128-dimensional compact Riemannian symmetric space on which $\mathrm{E}_8$ acts as isometries could be seen as the "octooctonionic projective plane", $(\...
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Question about an example in symplectic geometry
Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, ...
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When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus?
This is related to Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups. There I was told that in general, the symplectic group $\text{Sp}_{2n}(R)$ is not generated by its ...
user341790
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Bounding the dimensions of faithful representations of a quotient group
For $G$ a compact Lie group, let $\operatorname{mdfr}(G)$ be the minimum dimension of a faithful complex representation of $G$. Is there a bound on $\operatorname{mdfr}(N(H)/H)$ for $H$ a subgroup of ...
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What is the situation with Hilbert's Fifth Problem?
The common knowledge in this regard seems to be that Hilbert's Fifth Problem was completely solved by Gleason, Montgomery, and Zippin. However, such wisdom was contested by Peter Olver:
Olver, Peter ...
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Closedness of subgroup corresponding to semi-simple real Lie subalgebra
I have an impression (but could be wrong) that I heard that for any semi-simple (real) Lie subalgebra $\mathfrak{k}$ of $\mathfrak{gl}(n,\mathbb{R})$ there exists a connected closed Lie subgroup $K\...
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Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$
Edit: Thoughts updated (22/3/2021).
I've come across with the following problem.
Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
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Does $SU(N)$ have pseudo-real representation?
For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).
A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\...
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Centralizers of regular elements are abelian
Let $G$ be a complex semisimple Lie group with Lie algebra $\mathfrak{g}$. In a particular paper, the following statement is made:
If $X\in\mathfrak{g}$ is regular (i.e. has centralizer of minimal ...
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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 ...
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Principal bundles with no trivializable extensions
Let $Q \to M$ be a principal $G$-bundle. Given a homomorphism $\phi: G \to H$, we can ‘extend the structure group’ of $Q$ to $H$, by defining an associated principal $H$-bundle: $Q_{H} := (Q \times H)/...
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Finite covolume of uniform lattice in quotient group
Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact.
Suppose ...
user215380
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Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$
Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\...
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Positive genus Fuchsian groups
Let $G$ be a lattice in $SL(2,\mathbb{R})$. Is it always true that there exists a finite index subgroup $F$ of $G$ such that the quotient surface $\mathbb{H}/F$ has positive genus? Is the statement ...
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Automorphism group of the special unitary group $SU(N)$
Let us consider the automorphism group of the special unitary group $G=SU(N)$.
We know there is an exact sequence:
$$
0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.
$$
For $G=SU(2)...
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Decomposition about splitting of symmetric spaces of compact type
I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
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Are the odd dimensional spheres Poisson homogeneous spaces?
Are the odd dimensional spheres $S^{2n+1}$, for $n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld?
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Estimate word-metric length in free nilpotent groups
I would like to estimate the length of a word in a free nilpotent group. As the first example, I would like to estimate the word metric in the Heisenberg group $H_3$. This is the group of upper ...
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Irreducibility of root-height generating polynomial
The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of ...
- 2,753
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Models for Eilenberg-MacLane space K(Z,3)
Denote by $U(H)$ and $PU(H)$ the unitary and projective unitary groups on an infinite-dimensional Hilbert space $H$. Recall that $U(H)$ is contractible by Kuiper's theorem and that $PU(H)$ is a $K(Z,2)...
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Lie group flows [closed]
I recently came across a way to think of an ordinary differential equation on a smooth manifold $M$ is as a Lie group homomorphism $\phi : (\mathbb{R}, +) \rightarrow \operatorname{Diff}(M)$ where $\...
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Normalizer of SU(2) x SU(2) in SU(4) [closed]
What is the normalizer of SU(2) x SU(2) in SU(4) or how would I find it?
Reason for the question: with 2 qubits, if I was interested in conjugation of 2-qubit gates with generic SU(2) elements, ...
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Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: $...
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Positive roots and the longest element of the Weyl group
Take $\frak{g}$ a complex semisimple Lie algebra and its Weyl group $W$. Is it true that the number of positive roots of $\frak{g}$ is equal to the length of the longest element of $W$?
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When representations of reductive Lie group in a Banach space and in its Garding space have the same length?
Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...
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Can we find a holomorphic representative in an equivalence class?
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSU{PSU}$Let $U\subseteq \mathbb C^2$ be a given open set, $A$ be the set composed by maps (not necessarily continuous) $f:U\to \PSL(2,\mathbb C)$. ...
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Orthonormal frame on $\mathbb{S}^3$ orthogonal to foliations [duplicate]
Does there exist a smooth orthonormal frame $X_1,X_2,X_3$ on $\mathbb{S}^3$ such that the distribution spanned by $X_i$ and $X_j$ is integrable for all $1\leq i,j\leq 3$?
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Intuition for symplectic groups
My question essentially breaks down to
How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
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Cohomology of $\mathrm{BPSO}(2d)$ with $Z_2$ coefficients
$\DeclareMathOperator\BPSO{BPSO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}$The projective group is $\PSO(2d)=\SO(2d)/Z_2$.
$\BPSO(2d)$ is the classifying ...
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Relationship between Harish-Chandra Schwartz space and more generic Schwartz spaces
If $G$ is a connected semisimple Lie group with finite center, Harish-Chandra defined a Schwartz space of rapidly decreasing functions on $G$ as the space of $\mathrm{C}^\infty$ functions defined by ...
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Compact connected iff semi-simple for Lie Groups?
Are compact & connected Lie Groups in correspondence with semi-simple Lie groups? I think there is a condition on the center (discrete?) but I'm not sure.
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Specify the embedding of special unitary group in a Spin group via their representation map
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
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Existence of commuting Chevalley involution
Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra, and let $\theta$ be a complex linear involution on $\mathfrak{g}$. Let $\mathfrak{a}$ be a Cartan subspace, and choose a $\theta$...
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Is there a Morse theory proof of the Bruhat decomposition?
Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
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Does every irreducible representation of a compact group occur in tensor products of a faithful representation and its dual?
(Previously posted on math.SE with no answers.)
Let $G$ be a compact Lie group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. Is it true that every (complex, ...
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Nascent formal group law
$\DeclareMathOperator\FGL{FGL}$The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal ...
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A question about unitary and anti-unitary matrices
The question is the following: Let $U:\mathbf{C}^n\to \mathbf{C}^n$ be a unitary operator; let $\tilde{U}:\mathbf{C}^n\to\mathbf{C}^n$ be an antiunitary operator.
Can one deform $U$ to $\tilde{U}$ ...
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Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$
Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
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Probability of satisfying a word in a compact group
This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. ...
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Malcev's paper "On a class of homogeneous spaces" in English
I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
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Holonomy of a triangle in an affine symmetric space
Let $G/H$ be an affine symmetric space with involution $\sigma$, and $\mathfrak{g}=\mathfrak{m}\oplus \mathfrak{h}$ the Cartan decomposition of its Lie algebra. We can identify $G/H$ and $\exp(\...
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Universal property of induced representation
Let $H$ be a closed subgroup of the compact Lie group $G$. Let $E$ be a continuous representation of $H$. In the book "Representations of compact Lie groups" by Bröcker and Dieck the induced ...
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Invariants of symmetric forms with respect to the symplectic group
Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
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Fundamental group of a Lie group
Is it true that the class of isomorphism classes of fundamental groups of Lie groups coincides with the class of isomorphism classes of finitely generated abelian groups?
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Does a compact Lie group have finitely many conjugacy classes of maximal Abelian Lie subgroups?
Let $G$ be a compact Lie group. An Abelian Lie subgroup $A \leq G$ is a maximal Abelian Lie subgroup if, for any Abelian Lie subgroup $A'$ such that $A \leq A' \leq G$, then $A' = A$.
Of course any ...
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Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$
Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial?
The case where $G$ ...
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