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Results tagged with lie-algebras
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user 9970
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.
2
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A remark in Jantzen's 'Lectures on Quantum Groups'
In Jantzen's AMS text 'Lectures on Quantum Groups' he makes the following remark (p.187, preface to Chapter 9):
"For general (complex semisimple f.d. Lie algebra) $\frak{g}$ we can consider for each …
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11
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5
answers
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Applications of Chevalley Restriction Theorem
Let $G$ be a simple linear algebraic group (over $\mathbb{C}$, say) and $\mathfrak{g}$ be its Lie algebra, $\mathfrak{t}\subset \mathfrak{g}$ the Lie algebra of a maximal torus in $G$ and $W$ the corr …
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Accepted
Nilpotent Lie Algebras
Whenever $ad_{\xi}$ is an endomorphism of $\mathfrak{g}$ whose corresponding partition $\pi: 1^{s_{1}}2^{s_{2}} \cdots \;$ of $\dim \mathfrak{g}$ is such that $s_{1} =0$, then we have $im\; ad_{\xi} \ …
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5
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1
answer
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Endomorphisms in Category O and Schubert Classes
Let $\mathfrak{g}\supset \mathfrak{b}\supset \mathfrak{h}$ be a complex semisimple Lie algebra, with choice of Borel and Cartan subalgebras, $W$ the Weyl group.
W. Soergel's 'Endomorphismensatz' all …
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3
votes
1
answer
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'Generalised' coinvariant algebras
Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, whe …
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4
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About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
Yes: this is the approach to defining the 'abstract Weyl group' introduced in "Representation Theory and Complex Geometry" by Chriss/Ginzburg on p. 135 (2nd Edition, Birkhauser).
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Connectedness of Springer Fibers
Yes: this is discussed in Chriss & Ginzburg 'Representation Theory and Complex Geometry', p.161 Remark 3.3.26. In short, the nilpotent cone is normal and we can apply Zariski's Main Theorem to deduce …
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