Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with lie-algebras
Search options not deleted
user 121
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
2
votes
Recovering a Lie algebra from its affine Lie algebra
The extraction of a finite-type Lie subalgebra from an abstract affine Lie algebra is not functorial, because you have lots of automorphisms. Even if you are given a presentation with a Dynkin diagra …
- 45.7k
4
votes
faithful adjoint representation
Suppose $A \in PGL_n(\mathbb{R})$ lies in the kernel of the adjoint representation. Then for any lift $\tilde{A}$ of $A$ in $GL_n(\mathbb{R})$, and any traceless matrix $B$, we have $\tilde{A}B = B\t …
- 45.7k
3
votes
Accepted
What's the name of an algebra? is it isomorphic to $w_\infty \times w_\infty$?
If I'm not mistaken, this is the associative algebra of algebraic differential operators on the torus $\mathbb{G}_{m,\mathbb{C}}^2$. It is the tensor product of two copies of $w_\infty$, not the dire …
- 45.7k
2
votes
What is significant about the half-sum of positive roots?
If you have any free abelian group with an integral bilinear form embedded in the Lorentz space $\mathbb{R}^{n,1}$, you may consider the group of automorphisms generated by roots, i.e., reflections in …
- 45.7k
4
votes
Accepted
Kac Moody algebra defintion
I'll just elaborate on my comment from last year.
A Kac-Moody algebra is defined by generators-and-relations, with starting data given by an $n \times n$ matrix $A$. It can be written as $L \rtimes …
- 45.7k
5
votes
Accepted
Several question on Affine Lie algebra
First question: Peter's (and Emerton's) argument works not only for affine algebras, but for arbitrary Kac-Moody algebras. Decompose your integrable representations into weights, and take the tensor …
- 45.7k
2
votes
Accepted
About localization theorem for affine Lie algebra?
The main problem seems to be that you think the global section functor for (twisted) D-modules on a singular variety depends on a choice of embedding into a smooth variety. This is not true - D-modul …
- 45.7k
2
votes
Invariant symmetric bilinear forms and H^4 of BG
One way to look at the invariant symmetric forms is by noting that they describe one dimensional central extensions of the loop algebra $L\mathfrak{g} = \operatorname{Maps}(S^1, \mathfrak{g})$. As a …
- 45.7k
4
votes
Accepted
Lie algabra of symmetric group
For any $n>1$, the lower central series for the symmetric group is $S_n > A_n\geq A_n \geq A_n \geq \cdots$, so the Lie ring formed by the sum of successive quotients is the group $\mathbb{Z}/2\mathbb …
- 45.7k
11
votes
Accepted
Lie group operation and tangent vectors
Here's another way to look at the problem. The derivative of a differentiable map at any point is a linear map of tangent spaces. We have five differentiable maps in play:
The "pair of paths" map …
- 45.7k
0
votes
twisted affine algebras
I don't have Kac's book next to me right now, but I was under the impression that representations of the affine algebra do not give rise to representations of the twisted algebra in any straightforwar …
- 45.7k
7
votes
Accepted
Is the centralizer of a torus in a Kac-Moody algebra always a Borcherds algebra?
The answer to your first question is "yes", and it follows from Theorem 1 in Borcherds's paper Central extensions of generalized Kac-Moody algebras, which is available online as number 11 on his paper …
- 45.7k
18
votes
About the definition of E8, and Rosenfeld's "Geometry of Lie groups"
The algebraic group $E_8$ is the group of automorphisms of the $E_8$ lattice vertex algebra, by Frenkel-Kac and Segal. This vertex algebra has a self-dual integral form, so the construction works ove …
- 45.7k
5
votes
Accepted
coset of affine Lie algebra
This is typically given by the commutant, or coset construction. You take the vector subspace of $\mathcal{R}_\text{vac}[\mathfrak{g}_k]$ spanned by vectors $v$ satisfying $Y(u,z)v \in \mathcal{R}_\t …
- 45.7k
2
votes
1
answer
175
views
Do rational points in a split reductive group act transitively on the orbits of the Cartan s...
Let $(G,T,M)$ be a split reductive group (over say, the integers), with Lie algebra $(\mathfrak{g}, \mathfrak{t})$, and let $R$ be a commutative ring. When $R$ is an algebraically closed field, it is …
- 45.7k