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,...
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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 $\...
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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 ...
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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 $\...
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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 ...
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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 ...
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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
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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 ...
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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.
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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 ...
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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{...
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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 $\mathfrak{g}_\Theta$ ...
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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}...
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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 $V \...
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Abstract Jordan decomposition maybe not exist
An abstract Jordan decomposition of an element of a Lie algebra L is a decomposition
of the form a = a$_{s}$ + a$_{n}$, where
(a) ad a$_{s}$ is a diagonalizable (equivalently semisimple) endomorphism ...
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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 ...
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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 (...
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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 ...
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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(...
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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[...
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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 ...
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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 ...
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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(\...
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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 $L(\eta)$ ...
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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 ...
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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 ...
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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 $\mathfrak{g}$?
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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 ...
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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)...
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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 ...
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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 ...
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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 $G$ (...
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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 ...
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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^{\...
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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 ...
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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 ...
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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 ...
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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 $S(V)$...
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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 ...
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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 ...
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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 ...
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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($N$) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 \...
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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 ...