# Questions tagged [lie-algebras]

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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### Semi-direct products and associated graded Lie algebras

Let $G$ and $H$ be groups and $G\ltimes H$ their semi-direct product given by $f\colon G\to \operatorname{Aut}(H)$ satisfying $f(g)=\operatorname{id}_H$ in $H/[H,H]\,$ for all $g\in G$. In this ...
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### When do there exist $n$ commuting, derivations which locally generate $T_{A/k}$ as an $A$-module?

Let $\operatorname{Spec}(A)$ be a non-singular, $n$-dimensional, affine variety over a field $k$ of arbitrary characteristic. For the definition of "variety" I use Hartshorne's ...
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### Difference between two definitions of affine Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, we have the notion of affinization of $\mathfrak{g}$, which is the central extension of the corresponding loop algebra. ...
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### Modules over the unipotent subalgebra as direct summands of modules over a semisimple Lie algebra

Let $\mathfrak g$ be a semisimple finite-dimensional Lie algebra over the field of complex numbers $\mathbb C$. Let $\mathfrak n\subset\mathfrak g$ be the maximal unipotent subalgebra of $\mathfrak g$...
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### Lie algebras for which all one-dimensional extensions split

I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will ...
1 vote
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### Carleman approximation for functions from $\mathbb R$ to (closed convex subset of) a Lie algebra

I am looking for an approximation result dealing with continuous functions of a real parameter with values in (some subset of) the unitary algebra. However, I wouldn't be surprised if the following ...
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### About certain elements in the zero weight space of an irreducible representation of the complex simple Lie algebra of type G$_2$

$\newcommand{\fg}{\mathfrak g}\newcommand{\ee}{\varepsilon}$Let $\fg$ be the complex simple Lie algebra of type G$_2$. We consider its root system as follows (though it is probably not necessary to ...
1 vote
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### Problem in understanding a fact about Belavin-Drinfeld triple

A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental ...
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### Infinite dimensional representations of $\frak{sl}_2$

The finite-dimensional representations of a complex semisimple Lie algebra $\frak{g}$ are well known to be classifiable by their highest weight vectors, giving a convenient countable indexing set. I ...
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### Methods for calculating (one-parameter subgroup) actions

For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form \begin{equation} \mathrm{e}^{t L(z)} f(z) \end{equation} ...
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### Classification of 3-dimensional simple Lie algebras over $\mathbb{Z}$

What is the classification of 3-dimensional simple Lie algebras over the integers?
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### Is there a classical version of Yetter-Drinfeld modules?

One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$. If we think ...
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### Question regarding orthogonality of a map

I am reading the materials discussed in lecture $5$ from the lecture notes on quantum groups about Belavin-Drinfeld classification theorem written by Pavel Etingof and Oliver Schiffmann. In the first ...
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Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as $$V(\pi_1) \... 5 votes 4 answers 357 views ### The dimension of the Grassmannian cohomology ring H^*(\mathrm{Gr}_{n,d}) and the fundamental \frak{sl}_n-representation V_{\pi_d} The vector space dimension of the cohomology group of the 2-plane Grassmannian \mathrm{Gr}_{2,n} is given by the number of tuples (\lambda_1,\lambda_2) satisfying$$ n - 2 \geq \lambda_1 \geq \...
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Suppose I have an extension of fields $L/K$, a group scheme $G_K$ over $\operatorname {Spec} K$. Let $G_L$ denote the pullback of $G_K$ to $\operatorname{Spec} L$. Then, under what conditions on the ...
Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible ...