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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What can be said about these tensor representations of $\mathrm{SL}(2)$?
Let $W = V \otimes \dots \otimes V$, the product of $n$ copies of $V = \mathbb{C}^2$. Let $G$ and $H$ be two subgroups of the symmetric group $S_n$ and let $\chi$ be a character of $G$.
Associated to $...
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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 ...
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Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...
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A compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group
Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great!
Edit: ...
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Embed exceptional non-compact simply connected simple Lie groups into classical simple Lie groups with preserving centers
It is well known that there are exactly 22 exceptional simple real Lie algebras (5 compact, 5 split, 5 complex and 7 others). To each of these algebras there corresponds a unique simply connected (...
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Discrete nonabelian free subgroups of semisimple Lie groups
I understand that the following is a theorem: If $G$ is a noncompact connected semisimple Lie group, then $G$ contains a discrete nonabelian free subgroup.
I can find proofs that such a $G$ contains a ...
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Injection of $G(k)/Z(k)$ into $(G/Z)(k)$
In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
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Why is $\operatorname{SO}(4)$ not a simple Lie group?
$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ ...
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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 ...
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Commensurability classes of subgroups of a nilpotent group
Here is a question I stumbled upon in my research.
Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many abstract commensurability classes?
Recall that two ...
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Let $G \subset \mathrm{GL}(n)$ be a matrix Lie group. Does there exist an affine connection under which the matrix and manifold exponential coincide?
Let $G\subset \operatorname{GL}(n)$ be a matrix Lie group. I am curious about curves $\gamma(t) = g \exp(tv)$, where $g \in G$, $v \in \mathfrak{g}$, and $\exp(.)$ is the matrix exponential. If we ...
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Reliable literature with the list of centers of all simply connected simple real Lie groups
Wikipedia webpage https://en.wikipedia.org/wiki/Simple_Lie_group contains a full list of all simple (centerless) real Lie groups. One of the columns in tables (therein) contains fundamental groups of ...
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Explicit representatives for Borel cohomology classes of a compact Lie group?
I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) ...
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If a discrete and faithful representation of a surface group has proximal values, does the attracting points map have a continuous extension?
For some context, I'm studying the paper Anosov Representations and Proper Actions [GGKW].
$G$ denotes a non-compact real reductive Lie group of rank greater than $1$, $\Gamma$ denotes the fundamental ...
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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 $...
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Is the inclusion of the maximal torus in a simply connected compact Lie group null-homotopic?
Let $G$ be a simply connected compact Lie group and $T$ its maximal torus with inclusion $i:T \hookrightarrow G$.
By simply connectedness of the group $G$ and asphericity of the torus $T$, the induced ...
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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 ...
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Holomorphic discrete series vs. discrete series
(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete ...
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Extension of a type A Springer fibre
Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding
partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
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Any "inherent" definition of $\mathrm{SU}(2)$ independent of any matrix representation?
$\DeclareMathOperator\SU{SU}\SU(2)$ is explained in detail here.
However, if I know right, this definition itself is known the "fundamental representation".
I wonder if there is any "...
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Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?
Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
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Spherical functions in the space of functions on real Grassmannians
Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$.
Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
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How can we construct a non-trivial central extension of a Lie group
Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
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Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?
Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
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For topological torus action, there is a subcircle whose fixed point is the same as the torus
Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1}
$ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$.
The group $G$ is said to act on a space $X$ ...
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Do the exceptional root systems arise in the real world?
I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...
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Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation
The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
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What would you want on a Lie theory cheat poster?
For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...
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Homotopy groups of Lie groups
Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
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Classification of all connected simple real Lie groups?
Is there an explicit(!) classification of all(!) connected real simple Lie groups up to isomorphism? Not just simply connected or adjoint, but all of them?
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Triple product formula on $K = \mathrm{SU}(2)$
Let $K = \mathrm{SU}(2) = \{ k[\alpha ,\beta] \mid \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with
$$ k [ \alpha , \beta ] =
\begin{pmatrix}
\alpha & \beta \\
- \...
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Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
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How are Lie groups and polynomial resolvents related?
I came across the following sentence in Stevenhagen and Lenstra's wonderful little article Chebotarëv and his density theorem:
Nikolai's interest in [polynomial] resolvents led him to study Lie ...
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Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
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What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?
Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
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Action of complex Lie group on Dolbeault cohomology
Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$.
Consider the natural representation of $G$ in (...
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$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$
Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an ...
user515519
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Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?
In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
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Problem in understanding the coadjoint action of $\mathfrak {g}^{\ast}$ on $\mathfrak {g}$
$\DeclareMathOperator\ad{ad}$Let $\mathfrak {g}$ be a Lie bialgebra. Then $\mathfrak {g}^{\ast}$ is also a Lie bialgebra which is dual to $\mathfrak {g}$. Let the brackets on $\mathfrak {g}$ and $\...
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What are the maximal closed subgroups of $ \operatorname{SU}_3 $?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$What are the maximal closed subgroups of $ \SU_3 $?
This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that ...
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distance between unitary and anti-unitary matrices
This question is related to the previous post, "A question about unitary and anti-unitary matrices". Following the suggestion of Lspice, I am posting it as a separate question, as it might ...
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Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?
$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...
- 3,477
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Lie groups generated by finitely many Lie algebra elements
Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A standard fact is that $G$ is generated by $\exp(\mathfrak{g})$, i.e. every $g \in G$ can be written as $g=\exp(x_1)\cdots\exp(x_n)$ ...
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One parameter subgroups of reductive algebraic groups
If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
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N(H)/H and the Weyl group
Let $ H $ be a connected subgroup of $ G=\mathrm{SU}(n) $ such that $ N_G(H)/H $ is finite. Is $ N_G(H)/H $ always a subgroup of the symmetric group $ \mathrm{S}_n $?
I just noticed this from the ...
- 4,081
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How are Sheffer polynomials related to Lie theory?
Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any ...
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51
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Action of Hopf algebra of identity supported distributions on a Lie group
The Hopf algebra
of identity supported distributions on a lie group is cocommutative. It is well known that it is a group object in the category of cocommutative coalgebras. Is there a canonical ...
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Signed permutations in $ \operatorname{SO}(n) $ and normalizing an extraspecial group
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \SO(n) $ of signed permutations has order $ n!2^{n-1} $. I will call ...
- 4,081
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Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?
$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
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What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
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