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

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Classification of (compact) Lie groups

I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your ...
4
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1answer
206 views

The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
4
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1answer
190 views

Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
3
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2answers
297 views

Complete classification of six dimensional non-semi simple Lie algebra

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$). In this paper, in page 7, it ...
2
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1answer
92 views

Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's “Classical Groups”

First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but ...
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17answers
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Learning about Lie groups

Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
10
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3answers
892 views

Representations of \pi_1, G-bundles, Classifying Spaces

This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces). Here he says that for a Riemann surface $\Sigma$ the first ...
22
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3answers
724 views

Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...
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8answers
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Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up. In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
13
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4answers
923 views

Group Structure on CP^infinty

I was inspired by the following algebraic topology orals question: "Is $S^1$ the loop space of another space?" This is easy to see if you recognize that $S^1$ is a $K(\mathbb{Z},1)$, and the loop ...
12
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3answers
639 views

There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?

How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) ...
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2answers
809 views

If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie groups?

Let $G_{1}$ and $G_{2}$ be compact connected Lie groups. If $G_{1}$ and $G_{2}$ are homeomorphic as topological spaces, are they isomorphic as Lie groups?
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3answers
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Matrix expression for elements of $SO(3)$

Hi all. Is there any explicit matrix expression for a general element of the special orthogonal group $SO(3)$? I have been searching texts and net both, but could not find it. Kindly provide any ...
7
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3answers
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Curvature of a Lie group

Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold has some scalar curvature R. Is there a nice formula which relates the lie algebra of the ...
16
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2answers
927 views

Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?

Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ...
10
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3answers
1k views

What is the outer automorphism group of SU(n)?

All the automorphisms of $SU(2)$ seem to be inner, which would mean that $\mathrm{Out}$ $SU(2)$ is trivial. Is that correct? Is this true in general $SU(n)$? I can't quite see -- any thoughts would be ...
7
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3answers
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Definitions of Reductive and Semisimple Groups

I'm a graduate student. I've been reading Knapp's two books Representation Theory of Semisimple Groups and Lie Groups Beyond an Introduction. He seems to give wildly different definitions for the ...
7
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1answer
634 views

unitary irreps of O(p,q)

I am interested in the irreducible unitary representations of the orthogonal groups $O(p,q)$. By $O(p,q)$ I mean the real Lie groups which preserve the quadratic form of signature $(p,q)$ in ...
6
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2answers
489 views

Ricci curvature of the symplectic group

Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$? For $O(n)$ and $U(n)$ I know many references which state such a ...
7
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2answers
630 views

Cohomology of lattice subgroups

I am trying to find a reference for lower cohomology groups $H^i(G, \mathbb{Z}),$ for $i=1, 2, 3$ for lattices in higher rank (for example, $SL(n, \mathbb{Z}), Sp(2n, \mathbb{Z}),$ and possibly ...
5
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1answer
555 views

Can we realize Weyl group as a subgroup?

Given a semisimple Lie group G, let T be a maximal torus, W be the Weyl group defined as the quotient N(T)/C(T), where N(T) denotes the normalizer of T and C(T) denotes the centralizer. Two ...
2
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1answer
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A class of Lie groups with $f^{abc} \neq -f^{acb}$ (not fully anti-symmetrized) or $f^{abc} \neq f^{bca}$ (not-cyclic)

With the motivation to understand the Lie group structure constraint on a non-Abelian Chern-Simons theory, could some experts give a class of Lie groups with structure constants cannot fully ...
22
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2answers
655 views

In any Lie group with finitely many connected components, does there exist a finite subgroup which meets every component?

This question concerns a statement in a short paper by S. P. Wang titled “A note on free subgroups in linear groups" from 1981. The main result of this paper is the following theorem. Theorem (Wang, ...
15
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2answers
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Borel set plus a closed set = Borel

Hi, Let $R$ be equipped with the usual Borel structure. Let $F$ be a Borel subset and $E$ be a closed subset of $R$. Then $F+E=(f+e: f\in F, e \in E \)$ is Borel? If yes, is it true for any locally ...
7
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1answer
469 views

A question on eigenvalues

Let $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$ be linearly independent Hermitian matrices in the the space of $6$ by $6$ Hermitian matrices as a vector space over $\mathbb{R}$. Does there always ...
6
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1answer
201 views

Are infinite dimensional Lie algebras related to unique Lie groups?

For every finite dimensional Lie algebra $g$, there is a unique simply-connected Lie group $G$ whose Lie algebra is $g$. Is this true in the infinite dimensional case?
4
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2answers
670 views

Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$

I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them. It can be shown that $H^d[U(1), Z]$ is $Z$ for ...
2
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1answer
216 views

Characterization of the weight orbit in the projective space via second order Casimir.

This is the spin-off of the question I previously asked. First, let me remind you some notation from that question: $G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
3
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3answers
473 views

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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3answers
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On the Weyl character formula

So let $G$ be a compact real Lie group. Let $\rho:G\rightarrow GL_n(\mathbb{C})$ be an irreducible representation of $G$ and let $\chi_{\rho}$ be the character associated to $\rho$. Let ...
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orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices. QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not necessarily ...
8
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2answers
484 views

Isomorphism between Spin(3,2) and Sp(4, R)

I've been using the fact that Spin(3,2) is isomorphic to Sp(4, R) for a while, but I've never seen a proof. Can anyone point me in the direction of a good reference?
5
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3answers
378 views

What is a good introduction to branching rules in representation theory?

I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups. When a Lie group has a set of irreducible representations, I'd like to know ...
2
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2answers
104 views

A property of bi-invarient Finsler metrics on SU(N)

Does the set of bi-invariant Finsler metrics on $SU(N)$ exactly coincide the set of Finsler metrics with the one-parameter subgroups as their geodesics through the identity? I know that being ...
1
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1answer
247 views

The (-)-Connection on a Lie Group

Is the geodesic exponential map for a Lie group with the (-)-connection a diffeomorphism? This connection is one of two flat connections introduced by Cartan and Schouten on a Lie group and has ...
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2answers
310 views

Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?

Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups? I know that ...
0
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1answer
182 views

Subgroups with trivial Centralizers

Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups? ...