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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.
5
votes
Accepted
History of the study of Verma modules in terms of Kazhdan Lusztig Theory
It's probably too soon to expect a good historical overview, but for example Steve Kleiman has already written a scholarly article (The development of intersection homology theory) emphasizing the ori …
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5
votes
Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ ...
The answer to the question here is yes. (More generallly, if $M$ and $N$ lie in $\mathcal(O)$ and are both infinite dimensional, then $M \otimes N \notin \mathcal{O}$.)
The proof is eaiest to writ …
- 52.9k
8
votes
Weyl's theorem and Representations
Maybe it would clarify matters if I gave a little more background, in community wiki format.
The basic idea of this algebraic proof goes back to a short paper by Richard Brauer (1936) in German in Ma …
3
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0
answers
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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 …
- 52.9k
1
vote
Reflection reverses a root string
Apart from notation, the abstract root system argument is given at the end of section 9.4 in my 1972 textbook, Springer GTM 9. (See also section 8.4 for the origin in semisimple Lie algebras. Tog …
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3
votes
Nilpotent elements of Lie algebra and unipotent groups
YCor's question about your definition of "nilpotent" is definitely in order, because Borel and Springer already defined this term in general for affine (=linear) algebraic groups over an arbitrary inf …
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1
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Weight spaces of representations of finite dimensional simple Lie algebras
EDIT: I misunderstood at first what your basic question is but now understand it better. One cautionary case comes from older work of Richard Block here, which includes the rank 1 simple Lie algeb …
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0
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Cyclic vectors in irreducible representations of simple Lie algebras
In an irreducible representation (finite or infinite dimensional), every nonzero vector is cyclic. This has nothing to do with Lie algebras as such, as it is true over an arbitrary ring.
Thus for e …
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4
votes
Accepted
Distance between Verma modules in certain "strongly" standard filtrations
It's worthwhile to explain something of the background, since Patrick Delorme's preprint never got published in full. It's a 23 page typed double-spaced document with symbols inserted by hand, dis …
- 52.9k
5
votes
Existence of a weight of a representation in the fundamental Weyl chamber
The problem with your highlighted formulation is that it's wrong as stated, unless for example you require that an "irreducible" representation be finite dimensional or have an integral highest weight …
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4
votes
Indecomposable, non-simple, modules of quantum groups at roots of unity
In the case of the rank 1 simple Lie algebra, your references give a good account of what is known. But in general, it's wise to keep in mind that many of the indecomposable $U_q(\mathfrak{g})$-mod …
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5
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Reference Request: Structure constants for G2
Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No speci …
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Extensions of modules over universal enveloping algebra with fixed central action
The question is indeed somewhat too vague, since a knowledge of certain Ext$^1_\chi$ would be enough to prove the Kazhdan-Lusztig Conjecture: take $M_1$ to be a Verma module and $M_2$ to be a simple h …
- 52.9k
1
vote
Accepted
Definition of the weight lattice for a nonreduced root system
Bourbaki has the most detailed treatment, but they tend not to deal with weight lattices (or co-weight lattices) so explicitly outside their account of some of the representation theory. Thus you ca …
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Accepted
Generalizing Polar Decomposition of Matrices
The basic answer to the question here is "Yes, there is a strong analogy via the Iwasawa decomposition for a semisimple Lie group". If I were trying to study this kind of question, I'd probably st …
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