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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.
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Applications (and source) of Bourbaki exercise on root systems with two root lengths?
In Chapters 4-6 of Bourbaki's Groupes et algebres de Lie, Exercise 20 for Section
VI.1 concerns irreducible (reduced) root systems with roots of two lengths: in other
words, systems of types $B_\ell, …
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answer
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Nilpotent matrices related to Lie algebras of special orthogonal groups in characteristic 0
In terms of matrix theory, the question I'm led to is the following: Start with an $n$-dimensional vector space over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, which has …
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Irreducible representations of Weyl group of F$_4$ on zero weight spaces?
This is a follow-up to a recent question here concerning the natural representation of a Weyl group $W$ on the zero weight space of an irreducible representation $L(\lambda)$ of highest weight $\lambd …
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Origin of the standard result on convex hull of weights of an irreducible finite dimensional...
What is the earliest published statement and proof of the well-known result: for a simple Lie algebra over $\mathbb{C}$ or other algebraically closed field of characteristic 0, the convex hull (in the …
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Occurrences of a simple reflection in the longest element of a Weyl group?
While looking at a preprint I've just bumped into a question about the longest element $w_0$ of a Weyl group $W$ (say irreducible of a Lie type $A$ - $G$ and of rank $n>1$, to simplify). Suppose t …
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Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?
This graph was determined in the case of simply-laced root systems by Igor Dolgachev and Norman Goldstein here. For other root systems the original question should be modified, leading to a precise c …
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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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Convention about "long" roots for simple Lie algebras of types ADE?
The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in Bourbaki …
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Comparing a Chevalley basis with the canonical basis of the adjoint module?
First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} \o …
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Source of a formula for tensor product multiplicities?
This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other algebraicall …
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What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Ka...
While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into …
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Difference of adjacent dominant weights is a root?
The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 the …
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How to decide if two surfaces occurring in Springer theory are isomorphic?
In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some …
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Uniform setting for computing orders of algebraic groups over finite quotients of the integers?
A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb …
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Origin of symbols used for half-sum of positive roots in Lie theory?
The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here an …
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