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Results tagged with lie-algebras
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user 11877
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.
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Matrix representations of Lie groups of type $B_n$
For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^ …
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How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic r...
In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $J_{6, 11}$ with the following Dynkin diagram:
\ …
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Reference about the root systems $E_{n}$, $n \ge 10$
I am trying to understand the root systems $E_{n}$, $n \ge 10$. In particular, I would like to find some references which describe the number of real roots and imaginary roots of a given degree.
Cons …
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Imaginary roots in $\widetilde{E}_8$
Consider the root system of 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 associated with $n$.
Th …
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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 as …
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How to understand extremal vector?
Extremal vectors are defined in Kashiwara's paper. The definition is as follows.
Simple reflections in the Weyl group of $\mathfrak{g}$ acts on the crystal basis of integrable $U_q(\mathfrak{g})$-mo …
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Some questions about $\rho^{\vee}$ in Lie theory
Let $\mathfrak{g}$ be a semisimple Lie algebra and $I$ its vertices of Dynkin diagram. The weight $\rho$ is defined by $\rho = \sum_{i \in I} \omega_i = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, w …
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Lie algebra action obtained from Lie group action [closed]
Suppose that $G, H$ are Lie groups and $\mathfrak{g}$ the Lie algebra of $G$. Suppose that there is a Lie group action $G \times H \to H$. Is there a natural $\mathfrak{g}$ action on $C^{\infty}(H)$? …
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Decompose elements in $SL_2$ as a pair of elements in $SL_2^*$.
I have a question about decomposing elements in $SL_2$ as a pair of elements in $SL_2^*$. Here $SL_2^*$ is the dual Poisson Lie group of $SL_2$ which is defined as follows.
Let $G$ be a Poisson-Lie …
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How to show that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} ...
Let $(\mathcal{G}, \mathcal{G}^*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^*$ and $\mathcal{G}$ are
\begin{align}
& [t^a, t^b]_* = C_c^{ab} t_c, \\
& [t_a, t_b] …
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answer
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What is the trace of this map?
Let $g$ be a Lie algebra and $Q^+$ the set of dominant weights. For every $\lambda \in Q^+$, there is an irreducible $g$-module $V_{\lambda}$ with highest weight $\lambda$. Let $\lambda, \mu \in Q^+$, …
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What is the Cartan matrix for a dihedral group?
Dihedral groups are Coxeter groups of type $I_m$, $m \geq 3$. The Coxeter matrix of $I_m$ is
\begin{align}
\left( \begin{matrix} 1 & m \\ m & 1 \end{matrix} \right).
\end{align}
When $m=3,4,6$, $I_m$ …
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How to write down the map $T(V)_n \to S(Lie(V))_n$ explicitly?
Let $V$ be a vector space with a basis $v_1, v_2, \ldots, v_n$. Let $T(V)$ be the tensor algebra of $V$. Let $S(Lie(V))$ be the symmetric algebra of the free Lie algebra of $V$. I think that $T(V)$ is …
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Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie al...
Let $V$ be a finite dimensional vector space. Let $T(V)$ be the tensor algebra over $V$.
Do we have $T(V) \cong S(Lie(V))$ as a graded vector space? Here $S(Lie(V))$ is the symmetric algebra of the …
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How to show that the graded dual of the universal enveloping algebra of a free Lie algebra o...
In the article, the universal enveloping algebra of a free Lie algebra on a set X is defined to be the free associative algebra generated by X.
It is said that the graded dual of the universal envel …
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