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.
2,170
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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
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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
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1
answer
530
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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 ...
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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
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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
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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
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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
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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
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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 $C_L$ ...
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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
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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
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answers
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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
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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
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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
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Curvature collineation and the Killing identity
The Lie derivative of a general covariant $4$-tensor is given by
$$\mathcal{L}_{K}R_{abcd} = X^{e}\nabla_{e}R_{abcd} + R_{ebcd}\nabla_{a}X^{e} + R_{aecd}\nabla_{b}X^{e} + R_{abed}\nabla_{c}X^{e} + R_{...
3
votes
1
answer
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Solvable Lie algebra whose nilradical is not characteristic
Say that an ideal in a Lie algebra is characteristic if it is invariant under every derivation of the algebra.
It is well known that the nilradical of a finite-dimensional Lie algebra over a field ...
3
votes
0
answers
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On the Gelfand-Kirillov Conjecture
The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...
2
votes
1
answer
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An inequality for weights of affine Lie algebras, level, and dual Coxeter number
Suppose $\mathfrak{g}$ is an (untwisted) affine Lie algebra with the normalized invariant form $(\cdot | \cdot)$. Let $\lambda \in \mathfrak{h}^\ast$ be a dominant integral weight such that $\lambda(d)...
3
votes
1
answer
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A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$
Edit: According to the comment by @LSpice we realise the existing link to the main motivation of the question is not available. Then we search for the paper we found the following version:
https://www....
1
vote
0
answers
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Abelian ideals of $\mathfrak{n}_+$
Let $g$ be a finite dimensional simple Lie algebra. Then we have the root space decomposition
$$
g=\mathfrak{h}\oplus \mathfrak{n}_+ \oplus \mathfrak{n}_-.
$$
My question: how to classify the abelian ...
3
votes
3
answers
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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
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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
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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
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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
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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
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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?) ...
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95
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...