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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Why is $O(n;k)$ not connected, and has four connected components? [closed]

Why is $O(n;k)$ not connected and has four connected components when $nk\ge 1$? Here $O(n;k) =\{A\in GL(n+k,\mathbb{R}) \mid A^{T}GA=G\}$ where $G=\begin{pmatrix} 1&&&&&\\ &\...
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Connected subgroups of $SL(2,C)$

Where can I find a list of all connected real Lie groups inside the 6-dimensional real Lie group $SL(2,C)$, up to conjugacy? How can one verify that a partial list is complete? I found on wikipedia a ...
Arnold Neumaier's user avatar
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Property of lattices in Lie groups

Let $\Gamma$ be a lattice in a (real or p-adic) Lie group. Is it true that for a given natural number $n$ there exists a finite index subgroup $\Sigma\subset\Gamma$ such that each $\sigma\in\Sigma$ is ...
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Cartan-Hadamard Theorem

Can someone point out the gap in this argument. Consider a simply-connected Lie group with the (-)-connection. This connection is flat and so the sectional curvatures are zero. Then, by the Cartan-...
Oliver Jones's user avatar
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Minimal non-abelian groups -> Lie groups/algebras

A group is called minimal non-abelian if it is non-abelian and all proper subgroups are abelian. Does this notion also exist with Lie groups or algebras? As an example, consider the Lie algebra ...
Hauke Reddmann's user avatar
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Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group?

I need to answer (affirmatively, I hope) the following question: In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset ...
Xin Nie's user avatar
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weyl group representations

I am looking for some references about irreducible representations of the Weyl Group over simple Lie Groups, both classical and exceptional ones. In particular I want to know the dimensions and the ...
wky's user avatar
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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 ...
XYC's user avatar
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Faithful locally free circle actions on a torus must be free?

Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free? I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$. Another related question is: ...
chan kifung's user avatar
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Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation

We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...
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How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe ...
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Relationship between Laplacian and Hessian on compact Lie groups

If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has $$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|_F^2\,d\mathbf{x}\,,$$ where $\Delta$ ...
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Finite Order Automorphisms on Complex Simple Lie Algebras

Let $L$ be a finite dimensional complex simple Lie algebra, and let $F(L)$ be the set of all finite order automorphisms on $L$. Suppose that we declare $f,h \in F(L)$ to be equivalent if there exists ...
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Proper subgroups of $\rm{SU}(d)$ that act transitively on $\rm{CP}^{d-1}$?

The special unitary group $\rm{SU}(d)$ has a canonical action on the Hilbert space of dimension $d$, and this action induces a canonical action on the projective space $\rm{CP}^{d-1}$, which is ...
Huangjun Zhu's user avatar
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SU(6) -> SU(3) branching rule

I read in at least one paper and in the wiki below http://en.wikipedia.org/wiki/Quark_model that the 56 symmetric irrep of SU(6) breaks down into 10^{3/2} + 8^{1/2} irreps of SU(3)xSU(2). Here the ...
Y Macdisi's user avatar
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Invariant symmetric bilinear forms and H^4 of BG

I am reading this paper of Teleman and Woodward. On page 4, they say that $H^4(BG;\mathbb{R})$ can be identified with the space of invariant symmetric bilinear forms on $\mathfrak{g}_k$. Why is this ...
Kevin H. Lin's user avatar
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About the conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...
Golden Wave 's user avatar
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reference help indecomposable representations of SL(2,R)

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible $(\mathfrak{g},K)$-...
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Lie group about the quantum harmonic oscillator [closed]

We konw that in quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ is annihilation and creation operator, $H$ is the Hamiltonian operator. ...
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Reference request - localisation de g-modules

Does anyone have a link to a copy of Beilinson-Bernstein's "Localisation de g-modules", in which they prove the Beilinson-Bernstein theorem? I can't find it anywhere.
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Kostant's theorem on principal 3-dimensional subalgebras

I have a few questions concerning Kostant's work on principal three-dimensional subalgebras (TDS). Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and $\mathfrak{a}\subseteq\...
Peter Crooks's user avatar
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Semi-Simple Kahler Groups?

We say that a Kahler manifold is a Kahler group if it is also a Lie group. I would like to know which semi-simple Lie groups are also Kahler groups?
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Action of $ax+b$ with compact support

I wonder whether it is possible to have a smooth action of the $ax+b$ Lie group with compactly supported fundamental vector fields on $\mathbb{R}^2$ in such a way that it is non-trivial at least at ...
Stefan Waldmann's user avatar
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What does this paper have to do with Hilbert's fifth question?

Apparently, there is a paper M. Sablik, Final part of the answer to a Hilbert's question. Functional Equations - Results and Advances. Edited by Z. Daróczy and Zs. Páles, Kluwer Academic Publishers ...
Imre G.'s user avatar
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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 ...
Giovanni Moreno's user avatar
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Finite models for torsion-free lattices

Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$? I know this to be true in many instances (e.g. ...
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Orbits of tensor product $\operatorname{St}_2\otimes\operatorname{Sym}^2(\mathbb C ^3)$

Let $G_1=\operatorname{GL}_2(\mathbb C)$ act on $V_1=\mathbb C^2$ via the standard multiplication. Denote this representation by $\operatorname{St}_2$. Let $G_2=\operatorname{SL}_3(\mathbb C^3)$ act ...
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GAP versus SageMath for branching to Lie subgroups

Which computer package is better, GAP or SageMath, for decomposing an irreducible representation of a (simple) Lie group $G$ into representations of a Lie subgroup. I am most interested when ...
Nadia SUSY's user avatar
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The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators

Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$). How is the embedding $\mathfrak{g}...
Christoph Mark's user avatar
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Is equivariant oriented cobordism finite?

It is known that for $n \not\equiv 0 \mod 4$, the oriented cobordism ring $MSO_n$ is finite. That is, for oriented n-dimensional manifold $Y$, there exists $m\in \mathbb{N}$, such that $mY$ bounds. ...
Bo Liu's user avatar
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Uniform lattice in semidirect product

A uniform lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact. My question is whether a uniform lattice exists in the group $$ G={\mathbb R}^...
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What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
Axiom's user avatar
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the group of all biholomorphic group automorphisms of complex tori

My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group. Let $...
user108005's user avatar
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1 answer
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How does one calculate homotopy classes for group coset spaces?

Inspired by Witten's Wess-Zumino term arguments, I'm curious to know how one calculates homotopy classes more generally for coset spaces. In the above example the coset is $G/H=(SU(3)_L\times SU(3)_R)...
homotopyquestions's user avatar
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Center of the algebraic group $G_{\mathbb{R}}$ for a centerless $G$

This must be an easy question but I don't have a good argument for it and have not found a counterexample: Let $G$ be a connected semisimple algebraic group over $\mathbb{Q}$ such that the center of $...
Jack's user avatar
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Complexification or 'real'ization of Mapping Class group.

So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
Anant Atyam's user avatar
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1 answer
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Character determines the representation?

Consider a semisimple Lie group or a $p$ adic reductive group $G$. To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
Marc Palm's user avatar
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Is a Lie subgroup whose center is closed, a closed subgroup itself?

I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, ...
Pablo Lessa's user avatar
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Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made. In this paper, $\mathfrak{g}$ is a ...
Ji Woong Park's user avatar
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A weight generalization of root systems?

For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized ...
johhnyelgerton's user avatar
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2 answers
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Schur positivity of a polynomial

Suppose a polynomial of the form $$\prod_i^d \sum_j^p x_i^{f_j}$$ clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...
Nicolas Medina Sanchez's user avatar
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1 answer
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On maximal closed connected subgroups of a compact connected semisimple Lie group?

Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra. Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie ...
emiliocba's user avatar
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Spin groups in terms of matrices and/or linear operators

Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
Libertron's user avatar
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Submanifold of a Lie group whose tangent bundle is invariant under group (left) action

Edit: According to the interesting comment of Tobias Fritz we revise the question. Assume that $G$ is a Lie group and $M\subseteq G$ is a closed connected smooth submanifold of $G$ containing the ...
Ali Taghavi's user avatar
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1 answer
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On the isometry group of a self cartesian product of a Riemannian space

Let $X$ be a complete Riemannian space. Let us denote by $Iso(X)$ the group of isometries of $X$. It is a well-known fact that the group $Iso(X)$, when endowed with the compact-open topology, is a Lie ...
Hugo Chapdelaine's user avatar
4 votes
1 answer
276 views

If $N_G(S)/Z_G(S)$ is a reflection group, is it a Weyl group?

Let $G$ be a compact, connected Lie group and $S$ a torus in $G$ not assumed maximal. Then conjugation in $G$ induces a faithful representation of $N = N_G(S)/Z_G(S)$ in the Lie algebra $\mathfrak s$ ...
jdc's user avatar
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Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...
Hebe's user avatar
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2 answers
368 views

Moving Between Weight Spaces in Highest-Weight Representations

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$ and let $\Delta\subseteq Hom(T,\mathbb{C}^*)$ be the ...
Peter Crooks's user avatar
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4 votes
1 answer
669 views

Criterion for nilradical of a maximal parabolic subalgebra to be abelian?

This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which ...
Jim Humphreys's user avatar
4 votes
1 answer
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Transitive action on the sphere

Hello! From the book "Einstein manifolds" by Arthur L. Besse (at section 7.B), Lie groups $Sp(n)$, $Sp(n)\cdot U(1)$, $SU(2n)$ and $U(2n)$ constitute the complete list of Lie subgroups of $U(2n)$ ...
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