# Tagged Questions

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**1**answer

145 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 ...

**1**

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**1**answer

120 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

**0**answers

226 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 ...

**0**

votes

**2**answers

97 views

### Derivations of algebra of smooth $g$-valued function?

Let $M$ is a smooth n-manifold and $g$ is a $Z_2$-graded Lie algebra, we denote the algebra of smooth $g$-valued function on $M$ by $C^{\infty} (M,g)$. I wanna find all graded derivation of ...

**3**

votes

**1**answer

400 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.
...

**1**

vote

**1**answer

129 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 ...

**5**

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**1**answer

191 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 ...

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**0**answers

119 views

### About the Lie algebra of polyvector fields

I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...

**2**

votes

**1**answer

121 views

### Split real form of $SL(2,\mathbb{C})$ description of the two sphere?

If we denote the parabolic subgroup of $SL(2,\mathbb{C})$ by $P$, then we have the well known isomorphism $SL(2,\mathbb{C})/P \simeq S^2$, where $S^2$ is the two sphere. Now the compact real form of ...

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**1**answer

414 views

### Para-Complexification of Lie Groups

Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...

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**1**answer

109 views

### Casimir of a three dimensional solvable lie algebra

Good morning everyone. I've encountered recently during my computations the following lie algebra
$$\mathfrak g=\text{span}(f_0,f_1,f_2),$$
with $$\begin{eqnarray}[f_2,f_1]&=&f_0+a f_2,\\ ...

**0**

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**0**answers

55 views

### compute the determinant of a conjugacy map

Let $k$ be an algebraically closed field.
Let $F=k((\pi))$ and $\mathcal{O}$ the ring of integers, let $\gamma\in T(F)$ regular semisimple for a connected reductive group $G$.
We consider the map ...

**2**

votes

**1**answer

455 views

### Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $

My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...

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votes

**1**answer

255 views

### fiber bundle on an orbit of $\mathfrak{g}\oplus\mathfrak{g^*}$

Let $G$, be a Lie Group and $\mathfrak{g}$ be its Lie algebra ,i.e, $Lie(G)=\mathfrak{g}$. Let $\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^*}$. Here $X\in \mathfrak{g} $ and $F\in ...

**0**

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**1**answer

127 views

### A question about G-Manifolds

I am looking for a clear reason for following fact:Is there any reference ?
Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...

**2**

votes

**1**answer

260 views

### Can this Lie group written as a direct product?

Let $G=G_1.G_2$ be a Lie Subgroup of $SO(k) \times SO(2) \subset SO(k,2)$, where
$G_1=SU(k/2) \subset SO(k)$ and $G_2$ is a Lie subgroup of $SO(k) \times SO(2)$ isomorphic to $SO(2)$.
Let $G_1 \cap ...

**2**

votes

**1**answer

236 views

### symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation
$v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...

**3**

votes

**2**answers

291 views

### degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$

How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.

**3**

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**0**answers

229 views

### The normalizer a maximal compact subgroup of a semi-simple Lie group

Let $G$ be a semi-simple real Lie group such that $|\pi_0(G)|<\infty$ and let $K$ be a maximal compact subgroup of $G$.
Q1: How does one prove that $N_G(K)=K$?
So I know a nice (and low-tech) ...

**4**

votes

**2**answers

248 views

### Are maximal connected semisimple subgroups automatically closed?

(Yet another question in a series demonstrating my rather embarrassing ignorance of standard Lie theory... I hope this is not too basic for MO!)
To be a little more precise: let $G$ be a real ...

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**2**answers

175 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 ...

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**2**answers

540 views

### Maurer-Cartan structure equation derivation

Dear all.
I'm a theoretical physicist trying to understand the structure equations and their geometrical significance, this for their gravitational applications.
I know the relation between the Lie ...

**3**

votes

**1**answer

244 views

### How can one find generators of basic differential forms on homogeneous spaces?

Dear all,
In short, my problem is that I would like to have a better control of the 1-forms on a homogeneous space. Contrary to the group case, the module of differential form is not trivialisable. ...

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votes

**5**answers

2k views

### Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...

**23**

votes

**8**answers

3k 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**

votes

**1**answer

178 views

### Topologic or geometric mean of the structure constants of a semi simple lie algebra

Let $G$ be a semi simple Lie group (or real reductive), $\mathfrak{g}$ its lie algebra and $B$ its killing form. We can defined the 3-form $k$ by
$$k(X,Y,Z)=B([X,Y],Z).$$
with $X,Y,Z\in \mathfrak{g}$.
...

**1**

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**0**answers

202 views

### volume form in a symmetric space of real rank one

I want to compare the two canonical volume forms on a noncompact symmetric space fo real rank one.
The first one is the volume form induced by the Riemannian structure given by the Killing form ...

**12**

votes

**1**answer

1k views

### Isometry group of a homogeneous space

Background
Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...

**0**

votes

**1**answer

341 views

### Higher order Approximation of Lie groups [closed]

Maybe the following is trivial or folklore, but I can't find any concrete proof of
the theorem, that higher order derivatives of Lie groups don't give any new information
above what is coded in its ...

**4**

votes

**2**answers

385 views

### 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?

**3**

votes

**3**answers

1k views

### Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P?

I am an analyst struggling through some geometry used in physics.
Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection ...

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votes

**5**answers

759 views

### Lie group operation and tangent vectors

Suppose we have two differentiable paths $\alpha$ and $\beta$ thru the identity of a Lie group $G$, $\alpha(0)=\beta(0)=e$ the identity element. Denote $\alpha\beta$ the path given by ...

**2**

votes

**3**answers

674 views

### Is there any relation between deformation and extension of Lie algebras?

In a paper of A. Weinstein on the geometry of Poisson manifolds, he relates the formal linearization around a zero, p, of the Poisson bivector to extensions of the Lie algebra induced by the bivector ...

**1**

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**2**answers

271 views

### Explanation of $y = x \exp(\triangle)$ for a Lie Group

Let $M$ be a non-compact matrix Lie group and $T_e M$ its lie algebra.
Consider a point $x \in M $ and $ \triangle \in T_e M$.
To move from $x$ to a point $y \in M$ along $\triangle$, below group ...

**38**

votes

**10**answers

10k views

### why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...

**4**

votes

**1**answer

380 views

### 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 ...

**13**

votes

**4**answers

971 views

### 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 ...

**6**

votes

**4**answers

1k views

### 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 ...

**10**

votes

**1**answer

296 views

### Is there much theory of superalgebras acting on manifolds by alternating polyvector fields?

Usual story: vector fields on $M$, with their Lie bracket, form a Lie algebra. We can consider "actions" of some other Lie algebra ${\mathfrak g}$ on $M$ by looking at Lie homomorphisms ${\mathfrak ...

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votes

**2**answers

3k views

### Maurer-Cartan form

I suppose given a Lie Group (G) and its corresponding Lie Algebra (g) every element in its dual defines a Maurer-Cartan form on the whole Lie Group?
Let $\omega \in g^*$ be a Maurer-Cartan form and ...

**2**

votes

**1**answer

328 views

### How do you exponentiate a section of the adjoint bundle to get a gauge transformation?

Suppose $E$ is a vector bundle with structure group $G$ and let $P = F(E)$ be the frame bundle. Let $\mathfrak{g}_P$ denote the associated bundle to the adjoint representation of $G$ on its Lie ...

**8**

votes

**4**answers

728 views

### lists of computed cohomologies?

Is there any comprehensive list of examples for computed
1) de-Rham cohomology-groups
2) Lie-algebra-cohomology groups $H^i(\mathfrak{g},\mathbb{R})$
3) equivariant de-Rham cohomology groups ?
...

**12**

votes

**4**answers

3k views

### Formal Geometry

[edit: I posted an answer to this which summarizes one that I received verbally a few weeks after posting this question. I hope it is useful to someone.]
I am presently seeking references which ...

**27**

votes

**8**answers

5k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...