# Tagged Questions

**-3**

votes

**1**answer

254 views

### Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...

**1**

vote

**1**answer

181 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 ...

**8**

votes

**0**answers

242 views

### Lie algebras vs. graph complexes

A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...

**1**

vote

**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 ...

**3**

votes

**1**answer

106 views

### Linear independence in (graded) Lie algebras

I asked a mixed-up version of this question earlier.
The Lie algebras I have in mind are the homotopy Lie algebras of wedges of finitely many spheres (in dimensions greater than $1$). Thus each ...

**2**

votes

**0**answers

96 views

### Maurer-Cartan elements of the extension of an $L_{\infty}$-algebra

Let $g$ be a nilpotent $L_{\infty}$-algebra. For every commutative differential graded algebra $A$, one can form the extension $g\otimes A$ and endow it with a nilpotent $L_{\infty}$-algebra ...

**8**

votes

**3**answers

686 views

### HIgher Homotopy Groups and Representation Theory

Let $G$ be a compact Lie group, and $g$ its associated Lie algebra.
In what ways do the higher homotopy groups $\pi_{n}(G)$ with $n>1$ appear in the representation theory of $G$?
As an example, ...

**2**

votes

**2**answers

219 views

### Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (ie. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...

**14**

votes

**1**answer

337 views

### Proof for which primes H*G has torsion

In 1960 Borel proved a beautiful result:
Theorem. Let G be a simple, simply connected Lie group. Suppose that p is a prime that does not divide any of the coefficients of the highest root (expressed ...

**6**

votes

**6**answers

1k views

### Weight lattice and the first fundamental group

Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of ...

**10**

votes

**1**answer

626 views

### Torsion for Lie algebras and Lie groups

This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...

**2**

votes

**0**answers

283 views

### Cohomologies associated to residually torsion-free nilpotent groups

This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...

**11**

votes

**1**answer

501 views

### 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, ...

**9**

votes

**5**answers

2k views

### Does a finite-dimensional Lie algebra always exponentiate into a universal covering group

Hi,
I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly.
Consider a finite-dimensional Lie algebra, A, spanned by its d generators, ...

**3**

votes

**1**answer

522 views

### Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = ...

**16**

votes

**1**answer

2k views

### Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...