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Jim gave me the answer! It is false in general! Thanks.

Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, $\Delta'\subset \Delta$ and $R(\Delta')=R\cap \mathbb Z(\Delta')$. Define $$p(\Delta')=\mathfrak h \bigop …

Representation theory of Lie groups and Lie algebras are quite close and in a suitable sense they become equivalent. However, some subjects are typically found on a Lie group theory language. For exa …

Let $\tilde{\frak g}=\frak g \otimes \mathbb C[t,t^{-1}]$ be the loop algebra associated to a finite-dimensional simple Lie algebra $\frak g$. Consider $U,V$ and $W$ universal finite-dimensional high …

I have found that a category of modules over a Lie algebra has an infinite number of (partial) tilting modules and that direct sum of these tilting modules is also an object in this category. What ar …

The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$. How to calc …

The usual action of $fg$ on $u⊗v$ , where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by $fg(u⊗v …

Let $G$ be a reductive connected algebraic group and let $B$ a Borel subgroup. One of central themes of the representation theory of $G$ is the study of the induction functor $H^0$ from $B$ representa …

Let $\frak g$ be a Lie algebra over a field of characteristic zero with a Lie subalgebra $\frak s$ and consider $M$ a $\frak s$-module. When is it possible to extend the action of $\frak s$ to $\frak …

The algebra of distribution and its relationship with the universal enveloping algebra is discussed in the Jantzen's book, as we can see a discussion in the question link (more specifically, the Jim H …

Let $\frak g$ a simple finite-dimensional complex Lie algebra. Which categories of modules has the Weyl modules for $\frak g$ (in characteristic zero or positive) as projective objects? It is an am …

What is a reference for the subject of "free resolutions for Lie algebras"? Does the term "standard resolutions" means "free resolutions"? What is a "bar resolution"? Is there only one way to talk …

Let $g$ a finite-dimensional complex simple Lie algebra and $\sigma$ a finite order Dynikin diagram automorphism of $g$. Consider $\tilde g$ as the loop algebra associated to $g$, and $\tilde g^\sig …

It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good refere …