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Results tagged with lie-algebras
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user 97316
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.
3
votes
Does every nilpotent lie in the span of simple root vectors?
Actually I think this has to fail for $G_2$. Using fundamental coweights, we can construct a semisimple element that scales the simple root vectors $u_1, u_2$ by arbitrary nonzero constants $c_1, c_2$ …
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4
votes
3
answers
271
views
Does every nilpotent lie in the span of simple root vectors?
Let $G$ be a reductive group (for simplicity, we can work over $\mathbb{C}$). Let $N \in \operatorname{Lie} G$ be nilpotent. Does there exist a Borel pair $(B,T)$ of $G$ such that $N$ lies in the span …
- 953
7
votes
1
answer
329
views
Nilpotent orbits of a parabolic subgroup
Suppose $G$ is a reductive group over an algebraically closed field of characteristic $0$ with parabolic $P$, Levi quotient $M$, and unipotent radical $U$. We denote the nilpotent elements of $\mathrm …
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3
votes
Accepted
Nilpotent orbits of a parabolic subgroup
I thank Emile Okada for suggesting the following argument that there is a unique $P$-orbit in $q^{-1}(O) \cap p^{-1}(O_G)$ (all mistakes are due to me, however). The argument is inspired by Lemma 5.2. …
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