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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Normalizers over centralizers of Levi subalgebras

Let $\mathfrak g$ be a simple Lie algebra and $e$ a nilpotent element of $\mathfrak g$. Using Jacobson-Morozov theorem, there exist a nilpotent element $f$ and a semisimple element $h$ such that $\{e,...
1 vote
0 answers
153 views

A linear map on $\chi^{\infty}(\mathbb{R}^2)$ arising from the Cauchy integral formula

The space of smooth vector fields on $\mathbb{R}^2$ and open unit disc $\mathbb{D}$ are denoted by $\chi^{\infty} (\mathbb{R}^2)$ and $\chi^{\infty}(\mathbb{D})$, respectively. A vector field on $\...
6 votes
1 answer
473 views

Derivations of universal enveloping algebra of Lie algebras

We know a lot about derivations of Lie algebra. However, for the universal enveloping algebra of Lie algebra, we have few references about it. My question: describing the derivations of enveloping ...
2 votes
1 answer
163 views

The sum of the weights of an irreducible simple Lie algebra module

Let $\frak{g}$ be a simple Lie algebra (over $\mathbb{R}$ or $\mathbb{C}$) and $V_{\lambda_i}$ a fundamental representation. What happens if I take the sum, in the dual of the/a Cartan subalgebra $\...
1 vote
0 answers
91 views

Application of cup-product for Leibniz cohomology

In 1995, Loday introduced a cup-product operation on the graded cohomology of Leibniz algebra and showed that the cup-product operation satisfies the graded Zinbiel relation. My question is how this ...
2 votes
0 answers
185 views

Reference: Subalgebras of compact lie algberas

If $\mathfrak{g}$ is a simple, compact Lie algebra, plenty is known about its subalgebras: They're all reductive. Any simple subalgebra is also compact, and these are in 1 to 1 correspondence with ...
1 vote
0 answers
62 views

References: Lie derivations of Full matrix algebra [closed]

I want, if possible a list of references that trait Lie derivations of Full matrix algebras. Other than Lie derivations of generalized matrix algebras . Thanks
2 votes
1 answer
209 views

Lie algebra (co)homology of the Lie algebra of differential operators

Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case ...
1 vote
0 answers
57 views

Representations of Chevalley algebras over arbitrary fields

In professor Humphrey’s “Introduction to Lie Algebras and Representation theory” it is explained how we can reduce a semisimple complex Lie algebra (and its representations) to an arbitrary field. ...
2 votes
0 answers
70 views

Does $\mathfrak{g}^*$ split off from the augmentation ideal

(Note: I had this posted on MSE for a while but didn't get much of a response... so I'm posting it here now.) Let $G$ be an affine algebraic group over an algebraically closed field $k$ of ...
1 vote
0 answers
113 views

Equivalence of categories between the loop algebra of $sl_{n+1}$ and the affine Weyl group of $GL_\ell(C)$

In this paper here, Theorem 4.9 page 18, Charri and Pressley are claiming that there exists an equivalence of Categories between certain categories of the Lie algebra $\tilde{ \mathfrak g}=\mathfrak{...
7 votes
3 answers
404 views

When is the Lie algebra of automorphisms of a geometrical structure finite-dimensional?

Let $M$ be an $n$-dimensional smooth manifold and $\Theta$ some tensor field on $M$, so a smooth section of $TM^{\otimes r} \otimes T^*M^{\otimes s}$ for some $(r,s)$. Let $\mathfrak{g}_\Theta$ ...
5 votes
1 answer
539 views

Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative functions

Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
2 votes
2 answers
235 views

Tensoring $\frak{g}$-modules by fundamental representations

Given a fundamental representation $V(\nu_k)$ of a semisimple Lie algebra $\frak{g}$, and a general irreducible finite-dimensional representation $V$, is it ever possible that the tensor product $V \...
6 votes
1 answer
478 views

Abstract Jordan decomposition maybe not exist

An abstract Jordan decomposition of an element of a Lie algebra L is a decomposition of the form a = a$_{s}$ + a$_{n}$, where (a) ad a$_{s}$ is a diagonalizable (equivalently semisimple) endomorphism ...
4 votes
2 answers
335 views

Double centralizer in special linear algebra

It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence: $$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$ where $Z(A)$ is the ...
3 votes
0 answers
318 views

Maximal symmetry at the speed of light

Are there examples of 1 + 3 dimensional pseudo-Riemannian manifolds with 6 dimensional isometry group whose orbits are light-like (i.e., the metric restricted to each orbit is degenerate)? Here is a (...
6 votes
2 answers
309 views

Duals of the spinor representations of $\frak{so}_{2n}$

For the $D_n$-series simple Lie algebra $\frak{so}_{2n}$ a curious phenomenon occurs for the fundamental representations corresponding to the spinor nodes of the Dynkin diagram, which is to say the ...
4 votes
0 answers
145 views

Division in the universal enveloping algebra

Let $\mathfrak g$ be a (semisimple) Lie algebra, $\mathfrak b\subset \mathfrak g$ a Borel and $\mathfrak n = [\mathfrak b,\mathfrak b]$. Then I am interested in solving certain division problems in $U(...
5 votes
1 answer
136 views

Construction of non-split extension of simple modules of Lie algebras using linear differential operators

Consider the natural action of $W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}_2$-tuple ($\left[...
0 votes
1 answer
176 views

Can we write $sl(4,\mathbb{C})$ as the vector space sum of two copies of $sl(3,\mathbb{C})$? [closed]

We know $sl(4,\mathbb{C})$ has dimension $15$ and $sl(3,\mathbb{C})$ has dimension $8$. Is it possible to write $sl(4,\mathbb{C})$ as the vector space sum of two Lie subalgebras that are isomorphic to ...
3 votes
1 answer
124 views

Euler characteristic of a holomorphic homogeneous vector bundle

Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic ...
7 votes
1 answer
379 views

Unicity of the BGG complex

A friend and I are writing a paper that uses the BGG resolution of $L(\lambda)$ (where $\mathfrak g$ is a semisimple complex Lie algebra, $\lambda \in P^+$ is a dominant integral weight, and $L(\...
8 votes
1 answer
546 views

History of the study of Verma modules in terms of Kazhdan Lusztig Theory

Let $\mathfrak{g}$ be a complex finite dimensional semisimple Lie algbera, $W$ be the Weyl group, $\rho$ be the half sum of positive roots, $M(\eta)$ be the Verma module of weight $\eta$ and $L(\eta)$ ...
2 votes
1 answer
440 views

Classifying ample line bundles over the flag manifold $G/B$

For a complex Lie group $G$, with $B$ a choice of Borel subgroup. The line bundles over the flag manifold $G/B$ are indexed by elements of the weight lattice of $\frak{g}$. Which of these line bundles ...
1 vote
2 answers
894 views

Witt Lie algebras

For Witt Lie Algebras over field of characterestic $p>3$ we know that $\operatorname{dim}W(n;m):=np^{|m|}$ , such that $|m|=m_1+⋯+m_n$ . I would like to know what is the dimension of Witt ...
4 votes
0 answers
147 views

Differential graded Lie algebra over an ordinary Lie algebra

Given a dg (differential graded) Lie algebra $L$ and an ordinary Lie algebra $\mathfrak{g}$, is there written somewhere a formal definition of $L$ as a dg Lie algebra over $\mathfrak{g}$?
6 votes
1 answer
416 views

Analog of the Lie Product formula for commutators

Let $X, Y$ be elements of a Lie algebra. Consider the group $G$ generated by (limits of) arbitrary products of the elements $$ G = \langle{e^{tX},e^{sY}\rangle}$$ for all $t,s$. The Lie product ...
1 vote
0 answers
50 views

Decompose $D_n$-module $L(\lambda)$ as $B_{(n-1)}$-module

The Lie algebra of type $B_{(n-1)}$ can be got by taking the fixed point subalgebra of the diagram automorphism of $D_n$. So we can restrict the highest weight module $L(\lambda)$ of $D_n$ to $B_{(n-1)...
11 votes
2 answers
1k views

Realizing a subgroup of a Lie group as a stabilizer subgroup

Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to ...
3 votes
2 answers
398 views

Universal central extension of Lie algebras

In the literature, there is the notion of the universal central extension of a Lie algebra. My question is: is there also a notion of universal extensions that are not central? If yes, can you provide ...
1 vote
1 answer
315 views

Connectedness of stabilizer of regular element

Let $\mathfrak{g}$ be a complex simple Lie algebra and $x \in \mathfrak{g}$ be a regular element, i.e. its centralizer is of minimal dimension. Consider the adjoint action of the adjoint group $G$ (...
3 votes
0 answers
93 views

A question to the derived length in modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...
4 votes
0 answers
151 views

Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?

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^{\...
4 votes
1 answer
177 views

Can we have a nontrivial division of a irreducible root system as the union of two closed sub-root systems?

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 ...
1 vote
1 answer
227 views

Can we have a nontrivial division of a irreducible root system as $\Phi=\Phi_{[\lambda]}\cup \Phi_{[\mu]}$?

Let $(\mathfrak{g},\mathfrak{h},\Phi)$ be a root system of a complex simple Lie algebra, where $\Phi$ is the set of all roots. For each $\alpha\in \Phi$, let $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$ be ...
10 votes
3 answers
1k views

Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
4 votes
2 answers
379 views

Invariants in the symmetric algebra of a module

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$...
14 votes
0 answers
355 views

Gel'fand and Fuks' "Globalizing" of cohomology of formal vector fields

I apologize for the length of this question. If anybody already spent some time with cohomology of (formal) vector fields and the results of Gel'fand and Fuks, I imagine a lot can be skipped. I do ...
5 votes
3 answers
297 views

Tensoring irreducible $B$-series representations/ Type B Littlewood-Richardson

When tensoring finite dimensional representations of the Lie algebra ${\frak sl}_n$, we have an explicit algorithm given in terms of Young diagrams. See Section 4 of this paper. Do there exist ...
2 votes
0 answers
95 views

About geometric quantisation and application to real system

Quantisation is a important step to properly define a quantum system from a classical one. In a nutshell : On a symplectic manifold $(M,\omega)$ and an algebra of function $f$ on $M$, one defines an ...
1 vote
0 answers
74 views

Classically compute ahead time if Lie Algebra is either polynomial finite or geometrically closed?

Given a set of $N$ operators $\mathcal{O}$ with a known set of Lie Algebra group multiplication rules $\mathcal{G}$ that can be programmed into a classical computer, is there a classical poly($N$) ...
10 votes
3 answers
601 views

Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ of a semisimple Lie algebra always not in $\mathcal{O}$?

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 ...
4 votes
2 answers
993 views

Weyl's theorem and Representations

Let $L$ be a semisimple Lie algebra and let $(V,\varphi)$ be a finite-dimensional $L$-module representation. Our main goal is to prove that $\varphi$ is completely reducible. Consider an $L$-submodule ...
4 votes
0 answers
284 views

Number of connected components of the centre of a Levi subgroup

Let $G$ be a connected complex semisimple algebraic group and $T\subset B\subset G$ a choice of maximal torus and Borel subgroup. Let $\Phi$ be the root system and $\Pi\subset\Phi$ the set of simple ...
2 votes
1 answer
133 views

Characterization of all $w$ in the Weyl group satisfying $w \geq w_l w_{l, \theta}$

Let $W$ be the Weyl group of a root system $\Phi$ with base $\Delta$ and system of positive roots $\Phi^+$. Let $S = \{ w_{\alpha} : \alpha \in \Delta \}$ be the set of simple reflections ...
4 votes
1 answer
295 views

Morphism of Lie algebras giving an action of Lie group on manifolds

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $ M$ be a manifold and $\mathfrak{X}(M)$ be its Lie algebra of vector fields on $M$. Let $G\times M\rightarrow M$ be an action of ...
5 votes
1 answer
291 views

Bruhat order and positive roots made negative

Let $(\Phi, V)$ be a reduced root system with base $\Delta$ and Weyl group $W$. Let $\ell$ be the length function of $W$ with respect to the set of simple reflections $S = \{s_{\alpha} : \alpha \in \...
3 votes
0 answers
154 views

Lie algebra cohomology of loop algebra

Let $G$ be a simple algebraic group over the complex numbers. Is it true that the Lie algebra cohomology $H^*(L\mathfrak{g}, \mathbb{C})$ of the loop Lie algebra $L\mathfrak{g}=\mathfrak{g} \otimes \...
3 votes
0 answers
156 views

Origin of the standard result on convex hull of weights of an irreducible finite dimensional representation?

What is the earliest published statement and proof of the well-known result: for a simple Lie algebra over $\mathbb{C}$ or other algebraically closed field of characteristic 0, the convex hull (in the ...

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