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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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        De Rham cohomology of homogeneous spaces

        Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a ...
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        116 views

        What subalgebras of $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ are orthogonal to the centre of $\mathfrak{gl}(n)$?

        Consider the Lie algebra inclusions $\mathfrak{gl}(n) \subset \mathfrak{so}(2n)$ and $\mathfrak{gl}(n) \subset \mathfrak{sp}(2n)$. Let $\mathfrak{c} \subset \mathfrak{gl}(n)$ denote the centre. ...
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        62 views

        Branching rules for E6 into SU(3)^3

        I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
        4
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        0answers
        179 views

        $L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

        First of all I want to apologize for the much too long post. A Lie group $G$ is acting on a smooth manifold $M$, then we define \begin{align*} T^k_G(M)= (S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
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        88 views

        Representation theory [closed]

        Let $L$ be a complex semisimple Lie algebra, then by Weyl's theorem every finite-dimensional representation of $L$ is completely reducible. My question is: can we find an example showing that in the ...
        1
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        70 views

        Classical nilpotent and solvable Lie algebras [closed]

        In this question $F$ denotes a Field. We have the set of strictly upper triangular matrices of size $n \times n$ denoted by $n(n,F)$ is nilpotent of index $n$, so it is easy to check that the algebra $...
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        43 views

        Homotopy transfer of cyclic L-infinity algebras

        Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...
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        3answers
        314 views

        Nilpotent elements of Lie algebra and unipotent groups

        Let $k$ be a field of characteristic 0 (not necessarily algebraically closed), let $G$ be a connected split reductive group over $k$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Let $X \in \...
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        27 views

        Finite dimensional Lie algebras: bracket of generalized eigenspaces of a derivation

        Let $\delta$ be a derivation of a complex Lie algebra L, and for $\lambda \in C$, let $$L_{\lambda}=\lbrace x \in L:(\delta-\lambda 1_{L})^{m}\;x=0 \mbox{ for some } m \ge 1 \rbrace$$ be the ...
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        1answer
        88 views

        On maximal closed connected subgroups of a compact connected semisimple Lie group?

        Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra. Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie ...
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        votes
        2answers
        341 views

        Weight spaces of representations of finite dimensional simple Lie algebras

        This question has probably been asked before on this website, but I could not find any solution and neither can I solve this question. So again I am asking the following question: Let $\mathfrak{g}$ ...
        5
        votes
        1answer
        116 views

        Number of real roots in type $\tilde{E}_8$

        Let $\Phi_+$ be the set of all positive roots for a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$ by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root ...
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        44 views

        Commutator space of regular nilpotent elements

        This is a follow-up to this question. Let $\mathfrak{g}$ be a semisimple complex Lie algebra, $A \in \mathfrak{g}$ a regular nilpotent element (i.e. its centralizer is of dimension equal to the rank ...
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        1answer
        102 views

        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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        votes
        0answers
        101 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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