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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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        Gram matrix determinant in dimension 4 and $E_8$

        Consider a determinant of a Gram matrix in dimension 4. $$ \begin{vmatrix} 1 & -cos(\alpha_1) & -cos(\alpha_2) & -cos(\alpha_3)\\ -cos(\alpha_1) & 1 & -cos(\alpha_6)& -cos(\...
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        Why is Jacobi Identity equivalent to holonomy of system? [migrated]

        Or equivalently, why is jacobi identity equivalent to integrability of system? How do I understand it intuitively? Thanks.
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        Basic notation question involving Lie Groups and Lie algebras

        I just started reading "On the functional equations satisfied by Eisentstein series" by Langlands http://publications.ias.edu/sites/default/files/Eisenstein-ps.pdf . I wasn't sure of some notation/...
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        References for representations of Heisenberg Lie algebra

        Please suggest some reference material for the representations of the infinite dimensional Heisenberg Lie Algebra or the oscillator algebra. I already looked at Kac and Rainas book, any other ...
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        Compact image of adjoint action

        Let $G$ be a connected Lie group and $H$ a connected Lie-subgroup. Suppose that for every compact subset $K\subset G/H$ the $H$-orbit $H.K$ is relatively compact in $G/H$. Is it true that the image ...
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        Description of real roots of Kac—Moody algebra

        Let $\Delta$ be a root system associated to a generalized Cartan matrix, $\alpha_1,\ldots,\alpha_n$ its simple roots. It is known that if $\Delta$ is of finite, affine or hyperbolic type, $\alpha=\...
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        Examples of three-dimensional non-nilpotent Leibniz or Lie algebras

        Can one give me some examples of three dimensional Non Nilpotent Leibniz algebras? Any references to the classification of three dimensional Non Nilpotent Leibniz or Lie algebras will also be helpful.
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        Simple modules for direct sum of simple Lie algebras

        I think that the following statement is true, but I do not know how to prove it. Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
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        Openly available software to work with Demazure modules

        Does someone know of any sort of software openly available online which can be used to compute various characteristics of Demazure modules for semisimple Lie algebras? Specifically, I'm interested in ...
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        Definition of a Dirac operator

        So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as: $D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
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        Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra

        Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the Poincaré-Birkhoff-Witt (PBW) isomorphism, inverse to that natural map from the ...
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        1answer
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        Branching to Levi subgroups in SAGE and the circle action

        In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup: http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...
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        2answers
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        GAP versus SageMath for branching to Lie subgroups

        Which computer package is better, GAP or SageMath, for decomposing an irreducible representation of a (simple) Lie group $G$ into representations of a Lie subgroup. I am most interested when ...
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        Lagrangian subgroup of a nonabelian Lie group

        My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory. See a previous post for other background ...
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        1answer
        348 views

        Looking for access to original paper for Category O

        It is well-known that the BGG category $\mathcal{O}$ was introduced in the early 1970s by Joseph Bernstein, Israel Gelfand and Sergei Gelfand. I google for a while but I cannot find out the original ...

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        山西福彩快乐十分钟
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