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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 ...
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      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 ...
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      votes
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      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)...
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      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. ...
      5
      votes
      2answers
      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 ...
      5
      votes
      1answer
      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 ...
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      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 ...
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      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, ...
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      2answers
      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 ...
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      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 $M$ ...
      3
      votes
      1answer
      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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      101 views

      Mysterious identity in cosimplicial $R$-module with Lie brackets

      I have a cosimplicial $R$-module $\mathscr{A}:=(A_n)_{n\ge 0}$ and on each $A_n$ a Lie bracket $[-,-]_n:A_n\otimes A_n\to A_n$. Denote the cofaces by $d_i:A_n\to A_{n+1}$ and the codegeneracies by $...
      4
      votes
      0answers
      174 views

      Are vertex operator algebras ever conspiratorial?

      I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
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      1answer
      99 views

      infinite fold tensor product of universal enveloping algebra

      Let $\mathfrak a$ be a Lie algebra graded by the abelian semigroup $S$, then the universal enveloping algebra $U(\mathfrak a)$ of $\mathfrak a$ is $S \sqcup \{0\}$ graded. I have the following ...
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      1answer
      122 views

      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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