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# Questions tagged [lie-algebras]

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. See also the [Wiki page](http://en.wikipedia.org/wiki/Lie_algebra).

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### Lie algebra elements commuting with a principal nilpotent element

Let $\mathfrak{g}$ be a semisimple complex Lie algebra, $A \in \mathfrak{g}$ a principal nilpotent element (i.e. its centralizer is of dimension equal to the rank of $\mathfrak{g}$). I wish to ...
94 views

### Lie algebras with unique invariant bilinear symmetric form

A complex, finite dimensional Lie algebra admitting a (up to multiplication by a non-zero scalar) unique, invariant, symmetric, non-degenerate bilinear form is 1-dimensional or simple. Cf 'Lie ...
101 views

### Weyl Group Action on Littelmann Paths

In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via  \tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi)...
56 views

### Weight spaces of modules over Lie algebras

I know that an irreducible infinite-dimensional weight module over the Virasoro algebra in which it has a non-zero finite-dimensional weight space, then all its weight spaces have finite dimension. ...
230 views

### Hopf structure on the universal enveloping of a super Hopf algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
283 views

### Identification problem: Does this group have a name?

I've encounter a group with properties that are very familiar, but I cannot say what group is it. Consider the variables $(t,x,y,z)$, an affine transformation $M \in A(3)$ on the last three variables ...
92 views

### Cyclic version of Lie algebra cohomology

Lie algebra cochains have a differential $d$ where $d^2 =0$ because of the Jacobi identity, which can be written in the cyclic form or the Leibniz form. $L_\infty$ algebra cochains have a ...
127 views

### Describing compact Lie groups in purely topological terms

Compact Lie groups are a very special type of compact group, namely those which admit a differentiable structure. Is it possible to describe compact Lie groups in purely topological terms, that is, ...
203 views

### Cyclic vectors in irreducible representations of simple Lie algebras

Is there a notion of "cyclic element" in a simple Lie algebra? In particular, is it independent of the irreducible representation chosen? Explanation. An endomorphism A is called cyclic if there is ...
151 views

### Re asking:On the proof by Chu-Kobayashi that transformation groups are Lie groups

I have similar questions asOn the proof by Chu-Kobayashi that transformation groups are Lie groups and even more, how can $Y\in\mathfrak{g}^{*}$ generate 1-parameter global transformation group of $M$ ...
64 views

### Does the Weyl group preserve coprimality in Kac-Moody algebras?

Let $\mathfrak g$ be a Kac-Moody algebra (symmetric, or hyperbolic, or whatever other assumptions you need) with simple roots $\alpha_i$. For $\alpha$ a root, write $\alpha$ in the basis of simple ...
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### Knizhnik-Zamolodchikov equation is a connection on “affine slice”

The question is - what is the precise meaning of the phrase in the title? I heard it from Andrey Okounkov during one of his lectures. The problem is that he didn't really specified which slice is ...

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