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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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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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1
vote
1
answer
549
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How to draw a Littelmann path?
Littelmann path is a combinatorial tool to compute multiplicity. I have some questions about the definition of Littelmann path. It is said that a Littelmann path is a piecewise-linear mapping
$$\pi …
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1
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1
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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1
answer
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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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vote
0
answers
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Integrable modules and comodules
Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? T …
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1
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0
answers
138
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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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134
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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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3
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1
answer
383
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Modules which are direct sum of weight spaces.
For a semisimple Lie algebra $\mathfrak{g}$, a highest weight module $V(\lambda)$ with highest weight weight $\lambda$ has the property that every submodule $W$ of $V(\lambda)$ is a direct sum of the …
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1
vote
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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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Compute formal character of semisimple Lie algebras.
Let $\mathfrak{g}$ be a semisimple Lie algebra and $V_{\lambda}$ be the irreducible $\mathfrak{g}$-module with highest weight $\lambda$. Are there some softwares which can compute the formal character …
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3
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Reference request: representation of type G2 Lie algebras.
Let $\mathfrak{g}$ be an Lie algebra of type G2. Are there some combinatorial ways to describe a basis of a $\mathfrak{g}$-module? For classical types, there is a method used tableaux. Thank you very …
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answer
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PBW basis and canonical basis
Consider the example of $\mathfrak{g} = sl_3$. Then
$$
\mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-},
$$
where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, $\mat …
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Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \...
Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) \ot …
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answer
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Faithful linear representation of a nilpotent Lie algebra
Let
\begin{align}
\mathfrak{g} = Span_{\mathbb{C}}\{ e_1, e_2, e_3, e_4, e_5: \text{ non-zero brackets are } [e_1, e_i]=e_{i+1}, i=2,3,4, [e_2, e_3]=e_5 \}
\end{align}
be a $5$-dimensional Lie algebr …
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0
votes
1
answer
284
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Whitehead's lemma (Lie algebras) for reductive Lie algebras [closed]
Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: $\mathfrak{g} \to V$ a linear …
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