Tagged Questions

1answer
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 ...
1answer
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 ...
0answers
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 ...
0answers
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 ...
1answer
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 ...
0answers
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 ...
3answers
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, ...
2answers
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 ...
1answer
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 ...
6answers
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 ...
1answer
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 ...
0answers
283 views

1answer
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 ...