7
votes
3answers
478 views

nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
5
votes
1answer
259 views

The surjectivity of the exponential map for the isometry group

Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective. Let $M$ be a noncompact connected Riemann manifold, and ...
3
votes
1answer
163 views

Lie group GL(4) representation decomposition

Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$. Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation. My question: How does $$V\otimes ...
1
vote
0answers
61 views

Representations of $\mathfrak{so}(3)$ ($\mathfrak{so}(2,1)$) and $SO(3)$ ($SO(2,1)$)

(Apologies if this question is too basic!) I have explicit 5-dimensional real representations of $\mathfrak{so}(3)$ and $\mathfrak{so}(2,1)$, and I want to know whether it's necessarily true that the ...
2
votes
0answers
34 views

Locus maximizing the holomorphic sectional curvature in a non-compact Hermitian symmetric space

Is there a quick way to prove the following statement, if possible without resorting to the classification of simple Lie groups? Let $G$ be a simple Lie group of non-compact Hermitian type of rank ...
1
vote
0answers
54 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
2
votes
1answer
169 views

R-linear representations of sl(2,C)

Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$? Equivalently, what ...
1
vote
1answer
58 views

A canonical G_m (or G) action on the Slodowy slice

Question By Slodowy slice I mean a transverse slice at a subregular nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ (in particular I am not intersecting with the nilpotent cone). Consider the ...
3
votes
0answers
296 views

Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$? Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
4
votes
3answers
554 views

How many three dimensional real Lie algebras are there?

The main point of the question is: I would like to know whether there are only finitely many, countable infinitely many or even uncountable many isomorphism classes of $3$-dimensional real lie ...
3
votes
1answer
92 views

necessary and sufficient conditions for littlewood richardson coefficients to be non zero

Is there any necessary and sufficient conditions for $V(\tau)$ to be an irreducible component of the tensor product of two irreducible representations $V(\lambda)$ and $V(\mu)$ of a simple lie algebra ...
1
vote
1answer
85 views

tensor product of two irreducibles having same maximal weight

Is there any explicit decomposition of tensor product of two finite dimensional irreducible modules of simple lie algebras whose highest weights are same?
2
votes
0answers
74 views

Automorphisms of Nilmanifolds

Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...
2
votes
2answers
153 views

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. ...
1
vote
2answers
184 views

Are two distinct Weyl chambers always disjoint?

Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian ...
1
vote
0answers
87 views

Subgroups of $GL(n,\mathbb{R})$ which are $Aut(L)$ for some Lie structure [closed]

What is a sufficient condition for a lie subgroup $G$ of $GL(n,\mathbb{R})$ to be the automorphism group of a Lie structure on $\mathbb{R}^{n}$. In particular does $O(n)$ satisfies this property?
1
vote
1answer
152 views

The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$

Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...
5
votes
1answer
292 views

Root space decomposition

What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form $\left( \begin{array}{cc} X & Y \\ \overline{Y}^t & Z ...
4
votes
1answer
997 views

Goin' with the flow with Kummer and Pascal: Combinatorics and geometry underlying the logarithm of the derivative operator

In a MO-Q111165 and associated MSE-Q125343, I present a pair of raising / lowering (creation / annihilation) operators $R_x = log(D)$ and $L_x = -x·D$ with $D=d/dx$ (for a sequence of functions ...
3
votes
1answer
104 views

Reference to definition of matrix log with domain SO(3) which is Borel measurable

I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...
2
votes
0answers
99 views

On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?

Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...
1
vote
1answer
188 views

labeling state vectors in representation space of a simple lie algebra

Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation ...
1
vote
0answers
200 views

Mathematica package for Lie algebra computations?

I am interested in performing Lie algebra computations in Mathematica. I did a bit of searching and found several packages (LieART, KILLING, SuperLie, maybe more), and wondered if anyone would ...
1
vote
0answers
40 views

Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group

Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...
4
votes
0answers
80 views

Adjoint orbit of two vectors

Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector ...
3
votes
0answers
153 views

What is a pure algebraic interpretation for this dynamical property?

According to comments of Yves Cornulier to the previous version of this question I revise the question as follows: To what extent the following types of Lie algebras $A$ are classified? And what is ...
1
vote
1answer
189 views

Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$

Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...
1
vote
1answer
135 views

Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure (In sense of ...
6
votes
0answers
241 views

Injectivity of Lie group exponential function

If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and ...
5
votes
2answers
235 views

Summary of Lie-Algebra integration tactics

If this is in the scope of MO, I would like to gather here the known tactics of Lie algebra integration, since it appear surprisingly hard to find such a compendium, library or any other kind of ...
0
votes
2answers
164 views

A question on Lie algebras

To what extent, the following types of Lie algebras are classified : Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.
6
votes
1answer
319 views

Origin of symbols used for half-sum of positive roots in Lie theory?

The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here ...
0
votes
0answers
106 views

A question on lie groups( Lie algebras)

What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property: There are two non zero vector fields $X, Y \in ...
1
vote
1answer
64 views

Hermitian symmetric structure on a homogeneous subspace

Let $G$ be a semisimple group over $Q$ and $K$ a maximal compact subgroup of $G(R)^+$. I am assuming that $G(R)^+/K$ has a structure of a non-compact Hermitian symmetric domain. Let $g= p + k$ be ...
3
votes
1answer
416 views

How was this Lie algebra found?

In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself. ...
6
votes
1answer
378 views

Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...
4
votes
1answer
216 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
votes
1answer
218 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
votes
2answers
325 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 ...
4
votes
3answers
275 views

Lie groups vs Lie monoids

Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie grousp but with the axiomatic existence ...
2
votes
1answer
141 views

Right Invariant Randers metrics

I'm hoping to determine the geodesic equation for a right invariant Randers metric $F(x) = \sqrt{a(x,x)} + b(x)$ on $SU(N)$. In my special case the navigation data $(h,W)$ for the Randers metric are ...
2
votes
1answer
155 views

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 ...
4
votes
3answers
553 views

Reg the motivation behind Lusztig-Vogan bijection

Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$. Now, ...
5
votes
1answer
334 views

History of Jordan Canonical Form?

Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in: When and how was it first stated? (I understand it was independently stated ...
1
vote
2answers
140 views

real representation of real semi simple Lie algebra

Let $\mathfrak g$ be a real simple noncompact Lie algebra. Are there any correspondence between irreducible real representations of $\mathfrak g$ and the highest weight correspond to some positive ...
7
votes
2answers
245 views

Realizing a subgroup of a Lie group as a stabilizer subgroup

Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to ...
5
votes
1answer
202 views

Determining the Lie algebra elements exponentiating to the center of a Lie group

For a semi-simple compact Lie group $G$ with center $Z(G)$, one can characterize the preimage of $Z(G)$ in the Cartan subalgebra under the exponential map as the nodes of the Stiefel diagram (see for ...
0
votes
1answer
276 views

Symplectic structure on $Sym^kG^{\mathbb{C}} $

Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification. I am looking for a symplectic structure (without use of coordinates) on $$ Sym^kG^{\mathbb{C}}, $$ PS:Here ...
5
votes
1answer
368 views

Geometric structure of flag manifolds, Borel -Weil-Bott theorem

I want to know if there is proof of Borel Weil Bott theorem, that is as geometric as it can be. Let $G$ be a semisimple compact Lie group and $T$ be a maximal torus. We know that $G/T$ is a ...
3
votes
2answers
172 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 ...