Skip to main content
Search type Search syntax
Tags [tag]
Exact "words here"
Author user:1234
user:me (yours)
Score score:3 (3+)
score:0 (none)
Answers answers:3 (3+)
answers:0 (none)
isaccepted:yes
hasaccepted:no
inquestion:1234
Views views:250
Code code:"if (foo != bar)"
Sections title:apples
body:"apples oranges"
URL url:"*.example.com"
Saves in:saves
Status closed:yes
duplicate:no
migrated:no
wiki:no
Types is:question
is:answer
Exclude -[tag]
-apples
For more details on advanced search visit our help page
Results tagged with
Search options questions only not deleted user 24965

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.

16 votes
5 answers
2k views

About the intrinsic definition of the Weyl group of complex semisimple Lie algebras

It may be a easy question for experts. The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h} …
Zhaoting Wei's user avatar
  • 9,009
10 votes
3 answers
632 views

Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ ...

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra (according to a comment of Victor Ostrik, we need to further require that $\mathfrak{g}$ is simple) and we can consider its BG …
Zhaoting Wei's user avatar
  • 9,009
8 votes
3 answers
694 views

What is the categorical significance of the trivial $\mathfrak{g}$-module in the category of...

This question may be trivial for experts. Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and consider the category $\mathfrak{g}$-mod of $\mathfrak{g}$ modules. We can add suitable conditions, …
Zhaoting Wei's user avatar
  • 9,009
7 votes
2 answers
417 views

About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$

Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) …
Zhaoting Wei's user avatar
  • 9,009
7 votes
0 answers
166 views

How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in th...

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be t …
Zhaoting Wei's user avatar
  • 9,009
6 votes
0 answers
207 views

The meaning of a "subcomplex" of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex $(\wedge^{\cdot} \mathfrak{g}^* ,\text{d}_{C …
Zhaoting Wei's user avatar
  • 9,009
6 votes
1 answer
400 views

Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the di...

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define …
Zhaoting Wei's user avatar
  • 9,009
5 votes
1 answer
419 views

Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping al...

Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as $\text{Mat}_ …
Zhaoting Wei's user avatar
  • 9,009
5 votes
2 answers
915 views

Could we define the semi-direct product of two universal enveloping algebras?

If we have two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ over a field $k$, and if we have a Lie algebra homomorphism $\mathfrak{g}\rightarrow \text{Der}_k(\mathfrak{h})$, then we can define the s …
Zhaoting Wei's user avatar
  • 9,009
4 votes
0 answers
154 views

Is one of the hyperplane partitions of a irreducible root system always generate the whole W...

Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots. We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\prim …
Zhaoting Wei's user avatar
  • 9,009
4 votes
1 answer
323 views

About the term "tangential derivation" on a free Lie algebra.

Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, …
Zhaoting Wei's user avatar
  • 9,009
4 votes
1 answer
197 views

Can we have a nontrivial division of a irreducible root system as the union of two closed su...

The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the st …
Zhaoting Wei's user avatar
  • 9,009
4 votes
2 answers
787 views

What are the "tensor-closed" object of the BGG category $\mathcal{O}$ of a semisimple Lie al...

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its BGG category $\mathcal{O}$. It is well-known that $\mathcal{O}$ is not closed under tensor product, i. …
Zhaoting Wei's user avatar
  • 9,009
2 votes
1 answer
255 views

The real group orbits on the flag variety always contains the holomorphic directions?

Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of $\mathf …
Zhaoting Wei's user avatar
  • 9,009
2 votes
2 answers
715 views

Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural sym...

Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$). My questions is: it is always true that we have …
Zhaoting Wei's user avatar
  • 9,009

15 30 50 per page