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Results tagged with lie-algebras
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user 4231
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.
13
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Is there a reasonable way to define "reductive Lie algebra" in prime characteristic?
Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's tre …
CommunityBot
- 1
19
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2
answers
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Dual versions of "folding" symmetric ADE Dynkin diagrams?
Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams admi …
- 4,713
7
votes
Chevalley Groups over an arbitrary ring.
A couple of further clarifications, to supplement the extensive answer by marguax and the many comments:
Chevalley's influential 1955 paper was mainly concerned with finding a uniform approach to mos …
- 35.5k
3
votes
Computation of restricted Lie algebra (co)homology
I'm not sure how best to answer the question formulated here, but I can comment further on references. As Dietrich says, there is a large literature. Ever since the foundational work by Jacobson an …
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3
votes
Accepted
Stabilizers for nilpotent adjoint orbits of semisimple groups
To supplement what Francois Ziegler says, I'd point out that the structure of semisimple complex Lie groups has been developed piecemeal over a century or so. The basic results on nilpotent elements …
- 12.9k
4
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Good even grading and principal Levi type
Short answer: The two conditions on $e$ are logically unrelated. For example, a long root vector in type $B_2$ (generating the minimal nonzero nilpotent orbit) satisfies 1) but not 2), while an elem …
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Computing the index of a Lie algebra: what is known beyond the reductive case?
There is quite a bit of literature by now, in the classical characteristic 0 setting of finite dimensional Lie algebras. Looking up some of the papers listed below on arXiv (usually under math.RT) a …
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2
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Accepted
Reg the motivation behind Lusztig-Vogan bijection
Like Jay, I don't see any reasonable way to address all parts of your wide-ranging question. You are looking at the intersection of numerous lines of research, motivated in different ways for differ …
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41
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2
answers
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Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?
In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031),
Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corr …
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17
votes
2
answers
926
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Are the unipotent and nilpotent varieties isomorphic in bad characteristics?
In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its L …
- 12.9k
9
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2
answers
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Examples of Richardson orbit closures not having a symplectic resolution?
This is a follow-up to a recent question asked by Peter Crooks here. The answer by Ben Webster includes a helpful link to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper Symplect …
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Accepted
Centralizers of nilpotent elements in semisimple Lie algebras
This determination of component groups goes back to Elashvili and Alexeevskii, but has been improved somewhat in a 1998 IMRN paper by Eric Sommers and a later joint paper by him and George McNinch her …
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6
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Accepted
Can we count the number of simple modules for a reduced enveloping algebra?
The answers to your several closely related questions are not yet known, though many parts of the story have emerged. In particular, there is no "formula" for the number of simple $U_\chi(\mathfrak{g …
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6
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Centralizers of regular elements are abelian
EDIT: The question and some of the responses are out of focus, so it's worth clarifying a few of the issues here. First, the intended notion of "regular element" is unclear. In Kostant's early pa …
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homotopy type of connected Lie groups
I'm a bit skeptical as to how simple and low-tech a proof can be, since a fair amount of Lie group theory has to be in hand. Short of going into the full details of Iwasawa decomposition, you might fi …
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