All Questions
Tagged with lie-groups reference-request
298 questions
17
votes
2
answers
2k
views
Where did Sophus Lie write the group commutator for two one parameter groups
If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the ...
- 25.3k
9
votes
2
answers
1k
views
Levi decomposition in disconnected linear algebraic group (characteristic 0)?
For algebraic groups or Lie groups, the subject of Levi decompositions tends to be surrounded by some mystery in the literature (and in an older question raised here). While I postpone further my ...
- 52.9k
9
votes
1
answer
639
views
Borel's Paris Lectures
I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or ...
- 105
4
votes
1
answer
713
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 ...
- 52.9k
4
votes
4
answers
284
views
Stratifications and Cohomology Computations
I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...
- 4,920
6
votes
2
answers
1k
views
Parallel forms and cohomology of symmetric spaces
Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then
$$
(\alpha \text{ is induced by an $\...
- 3,244
2
votes
2
answers
311
views
Lie groups and NSS+LC group
Let $G$ be a locally compact group without small subgroups. Is $G$ a "finite" dimensional Lie group? (i.e, $G$ is not infinite dimensional Lie group.)
Are Lie groups precisely the locally Euclidean ...
5
votes
2
answers
452
views
"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 2,201
6
votes
3
answers
1k
views
Good book on representation theory of GL(n)
I am interested in a recommendation for a good book which discuses representation theory of GL(n)(say over field of complex numbers).
I know only a basic representation theory.
The question I am ...
4
votes
3
answers
2k
views
Lie subgroups of SU(4)
Other than subgroups of SU(3), what are the Lie subgroups of SU(4)? Assume that the subgroup is closed but not necessarily connected.
Additionally, which of these subgroups admit four dimensional ...
- 141
1
vote
2
answers
341
views
Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
2
votes
0
answers
165
views
Reference request: injective homomorphisms between unitary groups
Let $U(n)$ be the group of unitary $n\times n$ matrices over $\mathbb{C}$. Is there a classification of the continuous, injective group homomorphisms $U(m)\to U(n)$? If so, is there a modern account ...
- 1,336
7
votes
2
answers
595
views
Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 327
2
votes
1
answer
319
views
Reference request: Calculation in exceptional Lie groups
Let $G$ be a compact connected simple exceptional Lie group. Let $G$ be contained in a unitary group ${\rm U}(n)$ by some standard (low dimensional) unitary representation. For example in the case of $...
- 804
4
votes
2
answers
505
views
comprehensive presentation of the unitary dual of $SO_0(n,1)$
The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case $SO_0(...
- 2,446
16
votes
5
answers
2k
views
About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
It may be a easy question for experts.
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$...
- 9,019
5
votes
1
answer
393
views
Spin and SO groups associated to a degenerate symmetric bilinear form
In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But the rest of the book ...
- 4,312
3
votes
2
answers
337
views
Does every embedding of one unipotent group (over R) in another extend to an embedding of the respective upper triangular matrix groups?
Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.
Does every (...
- 1,089
5
votes
1
answer
653
views
Reference request for the list of maximal subgroups of SU(3,1)
Is there a reference with the list of maximal subgroups of SU(p,q) for "small" values of p and q? (such as SU(3,1) as suggested in the title of the question)
- 1,675
6
votes
1
answer
2k
views
A request for suggestions of advanced topics in representation theory
Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast ...
33
votes
8
answers
9k
views
"Modern" proof for the Baker-Campbell-Hausdorff formula
Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula?
All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and
are not at all geometric (...
- 2,074
8
votes
0
answers
388
views
Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
- 431
6
votes
3
answers
466
views
Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator
Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
- 3,244
6
votes
1
answer
1k
views
Decomposition of semisimple Lie group into almost simple factors
Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk ...
- 511
19
votes
3
answers
1k
views
Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
- 965
5
votes
2
answers
1k
views
Borel–Weil theorem - reference request
I am asking about good references (both books and papers) for the well-known Borel–Weil theorem. Thank you very much!
- 1,219
6
votes
2
answers
921
views
Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved)
I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference ...
- 799
1
vote
0
answers
155
views
A Weyl invariance constructed from Clebsch-Gordan Coefficients.
Let $V$ and $\tilde{V}$ be irreducible representations of SU(N) with tensor decomposition:
\begin{equation}
V \otimes \tilde{V} = \bigoplus_i U_i
\end{equation}
\noindent were $U_i$ are also irreps ...
- 161
17
votes
4
answers
5k
views
Exceptional isomorphisms of Lie groups
It is known that in low dimensions certain exceptional isomorphisms arise between Lie groups. I have read about some of them in some papers, but I have not been able to find a "systematic" treatment ...
- 171
2
votes
0
answers
115
views
Special class of bi-hamiltonian systems
A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$.
I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $...
- 513
7
votes
2
answers
1k
views
Lattices in SOL
Consider a semi-direct product $\mathbb{Z}^2\rtimes_A\mathbb{Z}$, where $A\in SL_2(\mathbb{Z})$ and $|Tr(A)|>2$. It is clear that it is isomorphic to a lattice in the 3-dimensional solvable Lie ...
- 11.1k
4
votes
1
answer
710
views
Are there explicit formulas for spherical functions on oriented real grassmannians?
Let $p$ and $q$ be integers. The group $K=SO(p) \times SO(q)$ can be naturally seen as a subgroup of $G=SO(p+q)$. The quotient space $G/K$ is identified with the space of oriented $p$-dimensional ...
- 9,486
10
votes
2
answers
3k
views
References on Lie groups and dynamical systems
I'm interested in Lie theory and its connections to dynamical systems theory. I am starting my studies and would like references to articles on the subject.
7
votes
1
answer
360
views
Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
- 2,210
8
votes
2
answers
572
views
reference containing the list of irreducible finite dimensional representation of real general linear group
It seems that it is not easy to find a reference containing a classification and construction of finite dimensional irreducible representations of $GL_n(\mathbb{R})$. One way to look at it is via $(\...
- 2,709
2
votes
1
answer
660
views
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
3
votes
3
answers
2k
views
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
5
votes
2
answers
1k
views
Complex root systems
This question is twofold.
1) What is the best reference on root systems?
2) Do complex root systems exist?
- 475
4
votes
3
answers
340
views
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 ...
- 21k
3
votes
1
answer
293
views
the relation between cohomology and Dynkin graphs of lie groups
I heard it said that the cohomology rings of some Lie groups and Grassmannians can be read from the Dynkin graph. Can someone give me any reference?
- 139
20
votes
6
answers
4k
views
Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
- 2,160
21
votes
2
answers
1k
views
Geometric interpretation of exceptional symmetric spaces
Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
- 3,022
17
votes
2
answers
1k
views
What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?
Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
- 8,167
6
votes
1
answer
2k
views
How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?
By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
- 3,541
3
votes
1
answer
152
views
Defining a family of rotations with certain properties
Let $d \ge 2$, and consider the sphere $S^{d-1}$ embedded in $\mathbb R^d$. Does there exist a family of rotations $\{\mathcal O_v\}_{v \in S^{d-1}}$ which satisfies:
$\mathcal O_v e_1 = v$, and
$\...
- 8,512
2
votes
2
answers
1k
views
Reductive Lie algebra of a Lie group
In the answer of my question:
On the full reducibility of representations of reductive Lie algebras
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in ...
- 671
14
votes
1
answer
503
views
Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?
I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group.
Is it true? Where can I find a proof?
A counterexample for open subsets of $...
- 2,479
8
votes
3
answers
2k
views
Cohomology rings of $ GL_n(C)$, $SL_n(C)$
Can anyone provide me with the reference for the following fact
(idea of the proof will be appreciated too):
Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what ...
- 1,783