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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6 votes
0 answers
185 views

Eigenvalues of spherical function on $\mathrm{SL}(2,\mathbb{R})$

Lie algebraically, the eigenvalue of the spherical function \begin{align*} \phi_{\lambda}(g)=\int_{K} e^{(i \lambda+\rho)(A(k g))} \mathrm{d} k \quad (g \in G,\,\lambda\in\mathfrak{a}^*) \end{align*} ...
2 votes
0 answers
59 views

4-dimensional Lie subalgebras of $o(4,\mathbb{C})$ and of $o(3,\mathbb{C})\ltimes\mathbb{C}^3$

I would like to know all complex 4-dimensional Lie subalgebras of $o(4,\mathbb{C})$ and of $o(3,\mathbb{C})\ltimes\mathbb{C}^3$.
8 votes
1 answer
530 views

Are invariant forms on homogeneous spaces necessarily closed?

Take a compact homogeneous space $G/K$, and a left $G$-invariant differential $k$-form $\omega \in \Omega^k(G/K)$. Will $\omega$ necessarily be closed? Might it even be harmonic when $G/K$ is endowed ...
4 votes
0 answers
79 views

Technics for computing weights of kernel, image or cokernel in category $\mathcal{O}$

As in the title I looking for technics to compute weights of kernels, images or cokernels in category $\mathcal{O}$ besides checking everything directly by hand. To be more concrete, consider the ...
3 votes
0 answers
89 views

Semisimple subgroup of Euclidean group

Let $G$ be a closed and connected semisimple subgroup of the Euclidean group $E(n)$ (the group of isometries of $\mathbb R^n$). Can we prove that $G$ is conjugate to a subgroup of $O(n)$?
3 votes
0 answers
119 views

Jacobson-style Galois theory on perfect closure

Promoted from stack.exchange since I received no response: Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he ...
4 votes
1 answer
134 views

coset of affine Lie algebra

In many books about conformal field theory, when we talk about a coset $\mathfrak{g}_k/\mathfrak{h}_{k'}$, we would talk about how the modules of $\mathfrak{g}_k$ are decomposed into those of $\...
3 votes
1 answer
195 views

GKO (or coset) construction - all possible highest weights $h$

I am reading the famous paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. From a compact simple Lie algebra $\mathfrak{g}$ and a Lie subalgebra $\...
6 votes
0 answers
166 views

Characteristic polynomials of Cartan matrices of Lie algebras

Let $L$ be a simple Lie algebra over $\mathbb{C}$ with Cartan matrix $C_L$ (as in https://de.wikipedia.org/wiki/Cartan-Matrix ) Question 1: Is is true that the characteristic polynomial $f$ of $C_L$ ...
6 votes
0 answers
218 views

Is $\mathrm{End}-\{0\}=\mathrm{Aut}$ for derivation Lie algebra?

Is it true that every nonzero endomorphism of Lie $\mathbb{C}$-algebra $\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\partial_{x_n}$ is an automorphism? As I ...
2 votes
0 answers
53 views

Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras

I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on. What are some ...
11 votes
0 answers
187 views

Quiver and relations for blocks of category $\mathcal{O}$

In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ . ...
10 votes
0 answers
96 views

Non-linear version of the Chevalley–Eilenberg complex

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. In small degrees, the differentials of the Chevalley–Eilenberg complex $C^\bullet(\mathfrak{g}, \mathfrak{g})$ with values in the adjoint ...
2 votes
1 answer
157 views

Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras?

Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\...
0 votes
1 answer
259 views

Curvature collineation and the Killing identity

The Lie derivative of a general covariant $4$-tensor is given by $$\mathcal{L}_{K}R_{abcd} = X^{e}\nabla_{e}R_{abcd} + R_{ebcd}\nabla_{a}X^{e} + R_{aecd}\nabla_{b}X^{e} + R_{abed}\nabla_{c}X^{e} + R_{...
3 votes
1 answer
425 views

Solvable Lie algebra whose nilradical is not characteristic

Say that an ideal in a Lie algebra is characteristic if it is invariant under every derivation of the algebra. It is well known that the nilradical of a finite-dimensional Lie algebra over a field ...
3 votes
0 answers
211 views

On the Gelfand-Kirillov Conjecture

The base field $k$ is of zero characteristic. Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...
2 votes
1 answer
114 views

An inequality for weights of affine Lie algebras, level, and dual Coxeter number

Suppose $\mathfrak{g}$ is an (untwisted) affine Lie algebra with the normalized invariant form $(\cdot | \cdot)$. Let $\lambda \in \mathfrak{h}^\ast$ be a dominant integral weight such that $\lambda(d)...
3 votes
1 answer
390 views

A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$

Edit: According to the comment by @LSpice we realise the existing link to the main motivation of the question is not available. Then we search for the paper we found the following version: https://www....
1 vote
0 answers
63 views

Abelian ideals of $\mathfrak{n}_+$

Let $g$ be a finite dimensional simple Lie algebra. Then we have the root space decomposition $$ g=\mathfrak{h}\oplus \mathfrak{n}_+ \oplus \mathfrak{n}_-. $$ My question: how to classify the abelian ...
3 votes
3 answers
498 views

3-dimensional Riemannian manifolds with 4-dimensional isometry group

Is there a list of all 3-dimensional, connected Riemannian manifolds with 4-dimensional isometry group?
3 votes
1 answer
353 views

When is the exterior derivation $d$ a Lie algebra morphism?

In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We ...
5 votes
1 answer
327 views

Equivariant cohomology of a semisimple Lie algebra

Suppose $\mathfrak{g}$ is a real Lie algebra integrating to the connected Lie group $G$. One may consider the $G$-equivariant cohomology of $\mathfrak{g}$ ($\mathfrak{g}^*$) where the $G$-action is ...
7 votes
2 answers
655 views

Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914

Question 1. Does Élie Cartan's paper Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355 contain a classification of $\Bbb C$-linear involutions of simple ...
2 votes
0 answers
159 views

Jordan normal form in a reductive group

Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
3 votes
0 answers
54 views

Natural appeareances of (commutative) algebras in $\mathfrak g$-modules

$\newcommand{\g}{\mathfrak g}$ Let $\g$ be a Lie algebra, and observe that since $U(\g)$ is a cocommutative Hopf algebra, it makes sense to look for (naturally arising and perhaps commutative?) ...
1 vote
0 answers
95 views

From restricted root space to root space

I am reading Knapp's book "Lie Groups beyond Introduction" https://link.springer.com/book/10.1007/978-1-4757-2453-0. I do not understand the following argument in Page 377 (2nd edition). I ...
1 vote
1 answer
84 views

Iwasawa decompostion and simply connected subgroups [closed]

Let $G$ be a semisimple Lie group, i.e. $G$ is connected and Lie algebra of $G$ is semisimple. We know by Iwasawa decomposition, there are connected subgroups $K$, $A$ and $N$ of $G$ such that the ...
5 votes
1 answer
280 views

Why aren't $B_n$ and $C_n$ the other way around?

In the classification of complex simple Lie algebras/groups, I have always been vaguely puzzled why $B_n$ and $C_n$ are labeled the way they are. I always instinctively want to put the special ...
4 votes
2 answers
458 views

Applications of the PBW theorem on enveloping algebras

What are some nice corollaries or applications of the Poincaré Birkhoff Witt theorem? There's this immediate corollary that a Lie algebra sits inside the universal enveloping algebra so in particular, ...
15 votes
0 answers
256 views

Lie theoretic meaning to $e^{\text{cycle}} = \text{permutation}$?

It is well known that exponentiating the EGF(exponential generating function) for cycles gives the EGF for permutations: link here. Usually summarized under the catchy slogan ...
2 votes
1 answer
209 views

Explicit Normalizer of SU(3) Cartan subalgebra

The normalizer $N(\mathfrak{h})$ of the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{su}(3)$ is defined as $$N=\left\{ x \in SU(3)\;|\; x^\dagger\mathfrak{h}x \in \mathfrak{h}\right\}$$ I would like ...
4 votes
0 answers
184 views

Books on integration on semisimple Lie groups

Can anyone suggest me some good books where I can find integration theory on semisimple Lie groups (using KAK, KAN and other type of decompositions)? I have read Knapp's book "Lie groups beyond ...
3 votes
0 answers
232 views

Reductive Lie groups and existence of maximal compact subgroup

I am reading Knapp's book "Lie groups beyond an introduction" (2nd edition). I am struggling to understand the following point. Recall that $G$ is a reductive Lie group. If the Lie algebra $\...
6 votes
0 answers
195 views

Integral Milnor-Moore theorem

Given a field K of char. zero the theorem of Milnor Moore states that taking the enveloping hopf algebra defines an embedding $\mathcal{U} $ from Lie algebras over K into hopf algebras over K. Taking ...
4 votes
1 answer
351 views

Generalization of Killing form

I am reading Knapp's book "Lie groups beyond introduction". On page 369, he has described the following. Let $\mathfrak g$ be a real semisimple Lie algebra. Suppose $\theta\colon\mathfrak g\...
1 vote
0 answers
46 views

What is the maximal weight submodule of $\text{Hom}_{\mathfrak{g}}(M,N)$?

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. For ...
13 votes
1 answer
672 views

Why the name O for category O?

What is the motivation behind naming the category O appearing in the theory of Lie algebras? Does O stand for something? Here is a question Why the BGG category O? that further confuses me. It seems ...
3 votes
1 answer
113 views

Linear Lie algebra generated by $\mathbb{R}$-diagonalizable matrices

If $\mathfrak{gl}_n(\mathbb{R})$ denotes the Lie algebra of real $(n \times n)$-matrices, then the $\mathbb{R}$-diagonalizable matrices generate $\mathfrak{gl}_n(\mathbb{R})$ as a Lie algebra. If $\...
3 votes
0 answers
67 views

Family of Lie algebras parametrized by a discrete valuation ring

I have a family of Lie algebras parametrized by a discrete valuation ring, whose generic fiber is reductive and whose special fiber is nilpotent. I'd like to learn about the relationship between the ...
3 votes
0 answers
95 views

Decomposing a compact connected Lie group

I want to prove the following. Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ and center $Z_G.$ It is not hard to prove that $\mathfrak g$ is reductive. Therefore, we can ...
3 votes
0 answers
118 views

Nakajima reflection functors and the flavour/framing group action

Nakajima has constructed so-called reflection functors that are isomorphisms between different quiver varieties that have the same framing $\mathbf{w}:$ $$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
1 vote
1 answer
123 views

Irreducible non-Abelian subgroup of $\mathrm{U}_n(\mathbb{C})$, containing diagonal matrices

Consider an irreducible non-Abelian subgroup $\mathrm{H}$ of group of unitary matrices $\mathrm{U}_n(\mathbb{C})$, that contains the subgroup of diagonal matrices. Does there exist any result ...
8 votes
2 answers
1k views

ad-nilpotent degree of a nilpotent Lie Algebra

Let $\mathfrak{g}$ be a Lie Algebra (finite dimensional, over $\mathbb{C}$). Engel's theorem tells us that if there exists a $m\in \mathbb{N}$ such that $\mathrm{ad}(x)^m = 0$, $\forall x\in \mathfrak{...
3 votes
1 answer
128 views

Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...
53 votes
5 answers
9k views

Motivating the Casimir element

Weyl's theorem states that any finite-dimensional representation of a finite-dimensional semisimple Lie algebra is completely reducible. In my mind, the "natural" way to prove this result is by way ...
2 votes
1 answer
138 views

Checking axiom of Category $\mathcal{O}$

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G$ be a split connected reductive algebraic group over $K$ with Borel $B$. We have the associated Lie algebras $\mathfrak{g}=$Lie$(G)$ and $\...
7 votes
0 answers
174 views

Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?

For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
4 votes
1 answer
247 views

BGG Category $\mathcal{O}$ is not closed under extension

What is the reason for the BBG category $\mathcal{O}$ failing to be closed under extensions i.e which of the 3 axioms of $\mathcal{O}$ does not hold under taking extensions? Is there a prototype of ...
6 votes
1 answer
318 views

An extension of symplectomorphism group

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$. We consider the ...

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