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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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      Nice Form of Vector Field

      Let $G$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $k$ of characteristic zero. Let $X$ be a homogeneous affine $G$-variety, i.e. $X=G/K$ for ...
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      1answer
      84 views

      Indecomposable, non-simple, modules of quantum groups at roots of unity

      Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...
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      87 views

      Lusztig's completion for universal enveloping algebra

      In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
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      votes
      0answers
      37 views

      Factorization of characters

      Linked to the end of this question here and because the subject involves many deformations of shuffle, I came to the following Let $k$ be a $\mathbb{Q}$-algebra and $\mathfrak{g}$ a $k$-Lie algebra, ...
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      57 views

      Centraliser of $\Delta U$ in $U\otimes U$

      Let $U$ be a universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. What is a good reference for the centralizer of $\Delta (\mathfrak{g})$ in $U \otimes U$ ? Here $\...
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      votes
      1answer
      76 views

      What is the connection between Frechet Lie groups and Lie algebras?

      An ordinary Lie group has a differentiable manifold structure, i.e. it is locally isomorphic to a finite-dimensional Euclidean space. A Frechet Lie group, on the other hand, has a Frechet manifold ...
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      votes
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      150 views

      Homeomorphisms of Springer fibers

      Let $V$ be a complex $n$-dimensional vector space and denote by ${\cal F}$ its space of complete flags. Let $g \in Gl(V)$ be unipotent and consider the Springer fiber ${\cal F}_g$ of its fixed points ...
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      174 views

      Non-faithful irreducible representations of simple Lie groups

      For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group. ...
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      98 views

      Motivation and Difference of Category O Definition for Kac-Moody Algebras

      My first encounter with Category $\mathcal{O}$ was (perhaps unusually) learning about Kac-Moody algebras from Kac's book. Kac takes the following definition: The Category $\mathcal{O}$ has objects $\...
      2
      votes
      1answer
      88 views

      Characterisation of even nilpotent elements in $\mathfrak{sl}_n$

      Is there a ''nice'' classification of even nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, ...
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      votes
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      94 views

      Tensoring Harish-Chandra bimodules with Verma modules

      The question is about the functor $T_\lambda$ defined by Bernstein and Gelfand in the paper Tensor Products of Finite and Infinite Dimensional Representations of Semisimple Lie Algebras. Setup: Let $\...
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      votes
      1answer
      137 views

      Distinguished dominant integral weight related to a branching problem

      Let $G$ be a simple compact connected Lie group and let $K$ be a connected closed subgroup of $G$. Let $\widehat G$ and $\widehat K$ denote the corresponding unitary duals, that is, the (equivalence ...
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      Questions of the paper “PBW-pairs of varieties of linear algebras”

      I am reading this paper "PBW-pairs of varieties of linear algebras", the link is here:https://www.tandfonline.com/doi/abs/10.1080/00927872.2012.720867. At page 672, there is a definition of PBW-pair. ...
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      54 views

      The idealizer of the space of vector fields with vanishing divergence

      The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure. Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...
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      votes
      1answer
      408 views

      Has the geometry of the variety of nilpotent matrices over $\mathbb{C}$ been studied?

      Consider the complex projective variety given by $X^n = 0$, where $X\in \mathrm{M}_n(\mathbb{C})$ and, say, $n\geq 3$. Some basic properties of it are already mentioned in this question: https://...

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