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Results tagged with lie-algebras
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user 3696
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.
4
votes
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Name of upper triangular/lower triangular Lie Algebra decomposition
I think people use the term "triangular decomposition" or sometimes "polarization"
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23
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How do you switch between representations of an algebraic group and its Lie algebra?
If $G$ is semisimple simply connected in characteristic zero, the differential at $1$ gives an equivalence of (tensor) categories $Rep(G)\to Rep({\mathfrak g})$. If $G$ is not semisimple, this is not …
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4
votes
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Dolbeault Operators for $CP^1$ as $\mathfrak{su}(2)$ Actions.
Let $K$ be a compact connected Lie group, and $L$ a Levi subgroup of $K$ (the centralizer of an element of the Lie algebra). Then $X=K/L$ is a complex manifold (a coadjoint orbit).
So one can ask abo …
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7
votes
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Comparing two similar procedures for quantizing a Casimir Lie algebra
The second construction (Lie bialgebra quantization) in fact also uses a Drinfeld associator. The braided tensor categories obtained in these two ways are equivalent, since the quasitriangular QUE alg …
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8
votes
Accepted
Is there a canonical Hopf structure on the center of a universal enveloping algebra?
In the semisimple case, one has the Harish-Chandra isomorphism between the center ${\mathcal Z}(\mathfrak g)$ and $(S{\mathfrak h})^W$, where ${\mathfrak h}$ is a Cartan subalgebra of ${\mathfrak g}$ …
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9
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Cohomology of Lie groups and Lie algebras
In question 1, it seems that the subspace $H^*(g(Q))$ in $H^*(g(C))$ indeed depends on the rational form of g. Consider the example of $sl(3)$ and two rational forms, $su(3,Q)$ and $sl(3,Q)$, and look …
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