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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Is there a 'nice' interpretation of virtual representations?
Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
- 5,191
3
votes
2
answers
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Lie bracket of Invariant Vector fields
Let $G$ be a Lie group and let $\xi.\eta$ be left invariant vector fields. We can now construct right invariant vector fields $X_\xi$ and $X_\eta$ by defining $X_\xi(e)=\xi(e)$ and $X_\eta(e)=\eta(e)$....
- 1,553
3
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0
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Construction of an algebra with prescribed representation of the automorphism group.
For this discussion, $G$ is a compact semisimple Lie Group.
For many of the common representations of compact groups, there is a realization of the representation as the automorphisms of some ...
- 5,191
2
votes
1
answer
660
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The quotient of a Lie group by the Levi factor of a parabolic subgroup
I am interested in some references on the quotient spaces obtained by quotienting G, a simple Lie group, by L, the group generated by the Levi factor of a parabolic subalgebra.
Presumably the case ...
- 2,123
2
votes
1
answer
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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) ...
- 965
1
vote
2
answers
1k
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Sum relation for Clebsch-Gordan-Coefficients?
In the context of (numerically) calculating reduced density matrices in the Lipkin-Meshkov-Glick model (a model introduced to describe atomic nuclei, which has however found many other applications as ...
- 11
5
votes
1
answer
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Convexity radius of a Lie Group
Is there a nice formula/method to find the convexity radius of a matrix Lie group (the manifold can be noncompact) ?
Edited based on comments:
Definition : Convexity Radius (Berger - Panoramic View ...
- 207
4
votes
1
answer
502
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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 ...
- 8,098
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votes
4
answers
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homotopy type of connected Lie groups
Is there a simple proof (short and low-tech) of the following fact:
(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to
$K\times\mathbb{R}^n$ where $K$ is a maximal ...
- 7,609
5
votes
1
answer
988
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smooth cohomology of Lie groups
Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)=\{ f: G^n \to A|~f~\text{is smooth}\}$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of ...
- 83
2
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2
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A question about the affine Grassmanian
For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as:
$$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$
Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. ...
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6
votes
1
answer
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Does a Trivial Tangent Bundle Induce a Multiplication?
Let $M$ be a connected smooth manifold, and assume that it is parallelisable; that is, its tangent bundle is trivial. Does $M$ admit an H space structure? That is, does there exist a smooth map $\mu:...
- 208
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Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?
The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary,
the result of historical accidents. In order to avoid repetitions, the four infinite
families $A_\ell, B_\ell, C_\...
- 52.9k
4
votes
3
answers
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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 $\Lambda_{\rho}...
- 7,609
11
votes
2
answers
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views
Groups "approximately commutative" near the identity
Is the following idea something that is known?
I call a metric group[1] $G$ "approximately commutative near the identity" if there exists a $K$ such that for small enough $\epsilon$, when $d(g,id) &...
- 2,895
2
votes
1
answer
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Restriction from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$
Hi,
I am faced with a finite-dimensional representation $V$ of $\mathfrak{gl}_{2n}$, whose character I know. I know how to use this character to determine the irreducibles for $\mathfrak{gl}_{2n}$ ...
- 6,131
3
votes
1
answer
559
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unitary representation of semisimple lie groups in view of Moore's ergodicity thm
Let $G=G_1\times\ldots\times G_n$ be a product of (connected) simple Lie groups and $(H, \pi)$ be a unitary representation of $G$. In a proof of Moore's ergodicity thm it uses the following fact $$\pi=...
- 853
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votes
3
answers
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analytic structure on lie groups
I need a reference for a result I have heard only very vaguely "A lie group (smooth) has a compatible analytic manifold structure".
(Would even appreciate a concise way to refer to the result..)
I ...
- 73
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votes
4
answers
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Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
- 3,541
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answer
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Highest weight orbit characterization (reformulated and extended)
Edit 1: I think that the question was not stated clearly enough so modified it a little.
Edit 2:
I thought over the physics that lies behind this question which led me to reformulation of the original ...
6
votes
1
answer
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Words in two infinitismal rotations
I asked this as subquestion in a comment pursuant to my Banach-Tarski
question. I think it is worth promoting here to a question in its own right.
Consider these two matrices over ${\Bbb R}[[\...
- 17.6k
8
votes
2
answers
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Killing form vs its counterpart in a given represenation
Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate ...
- 965
6
votes
2
answers
815
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Classification of real forms up to inner automorphisms
I hope to know the classification of real forms of complex simple Lie algebras of types $A$, $D$, $E$ up to inner automorphisms.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be real forms of a complex ...
- 63
14
votes
7
answers
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Cheap, non-constructive, free group generating rotations for Banach-Tarski
Stan Wagon's exposition of Banach-Tarski (for example) includes a beautiful explicit construction of two 2-sphere rotations which generate a free subgroup of the rotation group.
For teaching purposes ...
- 17.6k
35
votes
5
answers
4k
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$G_2$ and Geometry
In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie ...
- 3,726
5
votes
2
answers
1k
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Complex root systems
This question is twofold.
1) What is the best reference on root systems?
2) Do complex root systems exist?
- 475
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2
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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 ...
17
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1
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Is a smooth action of a semi-simple Lie group linearizable near a stationary point?
Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...
- 15.3k
2
votes
2
answers
503
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Lie Algebras and Simple Connectivity for general algebraic groups
In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
- 15.4k
9
votes
5
answers
677
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Occurrence of semi-spin groups
In the classification of simple Lie algebras one has the familiar picture of 4 families, $A_n$, $B_n$, $C_n$ and $D_n$, and 5 exceptional groups, $F_4,$ $G_2,$ $E_6$, $E_7$ and $E_8$. The $D_n$ ...
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0
answers
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Are the extra vertices in Nakajima's doubling of a quiver related to Langlands duality?
To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$
(vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching
an extra vertex to every old vertex in $Q_0$. Then ...
- 27.8k
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votes
3
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Which groups have only real and quaternionic irreducible representations?
Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:
1) it's not isomorphic to its dual (...
- 22.3k
6
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2
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Representations of Lorentz group
Questions:
What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?
Why should one read Bargmann's paper on irred. ...
- 361
3
votes
1
answer
594
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Analogies between orthogonal/unitary groups and their indefinite counterparts
Suppose I have $A\in U(n)$ such that $A^t=A$ (which is a bit un-natural, as usually you'd consider the hermitian transpose, not the transpose).
Well, then $A=X+iY$ say, for $X$ and $Y$ real matrices. ...
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2
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3
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is the subgroup generated by one-parameter unipotent subgroups a Lie subgroup?
Let $G$ be a Lie group and $H$ be a subgroup generated by some
one parameter unipotent subgroups (in group sense). Is it true that
$H$ has a Lie group structure which makes it a Lie subgroup of $G$?
...
- 853
5
votes
4
answers
3k
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Parametrization of O(3)
Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?
- 95
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2
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Is there a good definition of the universal cover for non-connected Lie groups?
It is well-known that the universal cover $\tilde G$ of a connected Lie group $G$ has a Lie group structure such that the covering projection $\tilde G\to G$ is a Lie group morphism. Of course $\tilde ...
- 2,140
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0
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reductive lie group without $R$-characters compact?
Let $G$ be the real points of an affine algebraic group
defined over $R$. If there is no non-trivial characters
$G\to R^*$, does it imply $G$ is a compact lie group?
I guess the paper of Borel and ...
- 1
1
vote
1
answer
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Explicit Coquasi-Triangular Quantised Coordinate Algebra of a Complex Semi-Simple Lie Group?
Let $SL_q(N)$ be usual quantised coordinate algebra of the special linear group. As is well-known, this is co-quasi-triangular algebra with coquasi-triangular structure given by
$$
R(u^i_j \otimes u^...
- 1,462
9
votes
1
answer
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Cohomology of the unitary group
The de-Rham cohomology ring of U(n) is the exterior algebra generated by the odd-dimensional classes x_1, x_3, ..., x_(2n-1). Moreover, on a Lie group every cohomology class is represented by a unique ...
- 1,232
2
votes
0
answers
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Type I unimodular groups in physics
Many of the physical symmetry groups are type I and unimodular. The unitary representations of type I second countable groups in separable Hilbert spaces can be given in a direct integral form which ...
- 21
22
votes
1
answer
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Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
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23
votes
1
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Monotone functions are differentiable a.e. and Hilbert's Fifth Problem: what's the connection?
In Andrew Gleason's interview for More Mathematical People, there is the following exchange concerning Gleason's work on Hilbert's fifth problem on whether every locally Euclidean topological group is ...
- 53.3k
8
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2
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Lie algebras to classify Lie groups
What does the classification of Complex Semi-simple Lie algebras buy us in terms of classifying Lie groups? Certainly it classifies complex semi-simple lie groups but can we get any better? I know we ...
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1
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Baker–Campbell–Hausdorff formula: prime divisors of denominators
Consider the Baker–Campbell–Hausdorff formula (Wikipedia page):
$$Z(X,Y) := X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \dotsb$$
Many sources, including the Wikipedia ...
- 7,320
12
votes
1
answer
796
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Lie's third theorem via differential graded algebras?
Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following.
Let $\mathfrak{g}$ be a (finite-dimensional, real)...
- 20.9k
1
vote
1
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558
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Understanding manifold GL+(3,R)/SO(3) ?
I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation ...
- 157
21
votes
6
answers
2k
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How do I stop worrying about root systems and decomposition theorems (for reductive groups)?
I apologize for this being a very very vague question.
Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall ...
- 229
12
votes
2
answers
2k
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A decomposition of the "spin representation" of SL(2)
Let us take an N-dimensional (N odd) irreducible representation V of SL(2,R).
It is known that (e.g., Lie groups and Lie algebras III by Vinberg and Onischik, 1994 p. 94) in V there is an invariant ...
- 1,765
4
votes
3
answers
340
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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 ...
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