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Results tagged with lie-algebras
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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.
3
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Lie algebras and pulled back group schemes
As the link in Erica's comment shows, you can find this in SGA3 Exp. 2, but it is not so easy to extract from the very general language. Here is a rough guide: From Definition 3.9.0, the Lie algebra …
- 12.9k
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votes
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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 …
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3
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GKO (or coset) construction - all possible highest weights $h$
The terminology is explained earlier on that page and the previous page in the paper.
On the same page, we see that they set $\mathfrak{g} = \mathfrak{su}(2) \times \mathfrak{su}(2)$, and let the suba …
- 45.7k
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Examples of simple vertex operator algebras (VOAs)
I expect there will never be a classification of simple VOAs, unless perhaps one is only sorting according to very rough criteria. This is because there are too many of them - even the rational case …
- 45.7k
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1
answer
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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 …
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The use of Schur's lemma for Lie algebras in physics (CFT)
Let $\mathfrak{g}$ be a complex Lie algebra with a distinguished nonzero central element $x$, and let $V$ be an irreducible representation of $\mathfrak{g}$. The usual proof of Schur's lemma can be a …
- 45.7k
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Physicists misuse the term "Kac Moody algebra". Does that bring problems?
I can't address all uses by all physicists, but in many contexts, they consider only representations at a fixed level that admit a well-behaved energy grading. That is, sometimes an energy grading is …
- 45.7k
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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 …
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Classification of quasi-lisse vertex algebras
I do not have a complete answer to your questions, but this is what I can say for now:
Question 1: A classification is impossible (see the response to question 3).
Question 2: Additional examples ar …
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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 …
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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 …
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Why is there a discrepancy between the normalizations of the central terms for the commutati...
As far as I can tell, a $\sigma$-model with $d$ dimensional target space will have Virasoro central charge $d$ with bosonic strings, and $3d/2$ with supersymmetric strings. I believe the normalizatio …
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2
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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 …
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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 …
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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 …
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