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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Comments/references on an obscure category of "rudimentary representations"
Consider a finite toset (cool word) $A=\{a_1<\ldots<a_N\}$. Let an ordered monomial be a finite formal product $b_1\ldots b_M$ with $b_i\in A$ and $b_i\le b_{i+1}$.
Consider the following ...
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3
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Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
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When is this map of Hopf algebras Surjective?
I'm reading Akhil Mathew's blog post on Formal Lie Theory in Characteristic Zero.
Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ ...
user30211
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Exponential map of a Formal Group Scheme
Let $k$ be a field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$. $\mathfrak{g}$ corresponds to a formal group scheme $\mathcal{G} = \text{Spf} (U(\mathfrak{...
user30211
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Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?
I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
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Holonomy groups of compact Riemannian symmetric spaces
Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
https://en.wikipedia.org/wiki/...
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Weyl's Branching Rule for $SU(N)$-Setting
On the Wikipedia page for restricted representations
https://en.wikipedia.org/wiki/Restricted_representation
there is presented a number of explicit "branching rules". In particular, there is the ...
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Good range and fair range
Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
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Which kind of functors preserve the bar-construction?
Let C, D be monoidal infinity categories that admit geometric realizations.
Let $F: C \to D$ be a monoidal functor and A an augmented associative algebra of C.
Denote $Bar(A)= \mathbb{1} \otimes_A \...
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Adjoint orbits of a finite group of type $G_2$
Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
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Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
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How many facets does the convex hull of all the roots of a root system have?
Let $V$ be an $n$-dimensional Euclidean vector space with inner product $\langle\cdot,\cdot\rangle$ and $\Phi$ an irreducible crystallographic root system in $(V,\langle\cdot,\cdot\rangle)$.
Question ...
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Restricted Lie algebras with no nonzero proper restricted subalgebras
Let $L\neq 0$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. If $F$ is algebraically closed, then it is known that $L$ has no nontrivial restricted subalgebras if and only ...
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Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does ...
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A Lie algebra associated to a symplectic manifold
Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ with the volume form $\omega^n.$
In this question we associate a Lie algebra $L(M,\omega)$ to $(M,\omega)$. Then we are interested ...
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Abstracting the properties of the category $\frak{g}$-modules
Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^*$, we know that the decomposition of $V \otimes V^*$ ...
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A property of the Weyl vector of an irreducible root system
Let $R$ be one of the $A,B,C,D,E,F$ root systems, let $\Lambda$ be the lattice generated by the roots, $R^+$ a system of poitive roots and $\rho$ and $\alpha$ be the Weyl vector and the highest root ...
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Weyl's formula and Cartan decompositiom of semisimple lie algebras
I'm working on the article of Benkart and Osborn " Flexible Lie-admissible algebras", specially, i'm working on the lemma 3.1, this lemma represent the dimension of L-module homomorphisms of $L\...
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Formula for Goldman Lie bracket of surface
Let $\Sigma_{g}$ be a closed oriented surface of genus $g$, Goldman defined a Lie algebra structure on the free module generated by the free homotopy
classes of loops on $\Sigma_{g}$. Roughly speaking,...
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Diagonalisation of invariant hermitian forms and irreducible representations of tori actions
here is my question:
Suppose that the torus $T^n = (S^1)^n$ acts on $\mathbb{C}^n$ by linear transformations $$(e^{i \theta_1},...,e^{i \theta_n}).(z_1,...,z_n) = (e^{i \theta_1}.z_1,...,e^{i \...
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Tensor product of irreducible ''anti-dominant'' representations
Let us consider the algebra $\mathfrak{gl}_{\infty}$ (or $\mathfrak{gl}_n$, or just any finite-dimensional semisimple Lie algebra; howerer, I am primarily concerned with the case of $\mathfrak{gl}_{\...
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Torus actions on $Sp(n)$-spheres
In this old question of mine
https://math.stackexchange.com/questions/1651906/spheres-as-symplectic-homogeneous-spaces
the presentation of spheres as symplectic group homogeneous spaces was ...
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About enveloping algebras of direct sums
This question is imported from MSE. It is linked to this one in the case of semi-direct products.
My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) ...
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Some questions about $\rho^{\vee}$ in Lie theory
Let $\mathfrak{g}$ be a semisimple Lie algebra and $I$ its vertices of Dynkin diagram. The weight $\rho$ is defined by $\rho = \sum_{i \in I} \omega_i = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, ...
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Poisson vertex algebra
Suppose $vir_{c}= \operatorname{span}\langle L_{-2}v_{c},L_{-3}v_{c},....\rangle$ is a vector space spanned by Virasoro algebra. Then we have a symmetric algebra $Sym(vir_{c})$. For this symmetric ...
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Classifications of the indefinite generalized Cartan matrix
I want to know that the present results about classifications of generalized indefinite Cartan matrices.
I only have known that the classifications of hyperbolic matrces.
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Is it true that $\mathfrak{g}=\mathfrak{g}_e\oplus[x,\mathfrak{g}]$?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $(e,f,h)$ a principal $\mathfrak{sl}_2$-triple (see below). Let $\mathfrak{g}_e$ be the centralizer of $e$ and let $x\in f+\mathfrak{g}_e$. ...
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Second symmetric square of the adjoint representation
I have just come across the following experimental fact. Let ${\mathfrak g}$ be a simple complex Lie algebra.
Fact: ${\mathfrak g}$ is a constituent of $S^2{\mathfrak g}$ if and only if ${\mathfrak ...
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Symmetric Powers for Lie Algebras
Let $V_\lambda$ be the irreducible representation of $sl_{n}(\mathfrak{C})$ with highest weight $\lambda$. There are well known formulas for the decomposition of $V_\lambda^{\otimes^k}= V_\lambda\...
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Lie Algebra Module Decomposition in GAP
Let $\mathfrak{g}$ be a complex finite-dimensional Lie algebra and let $V$ be a finite-dimensional $\mathfrak{g}$-module. Is there a way for me to check in GAP or some other software package whether $...
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$U(sp_2)$ subalgebra of $U(sl_4)$?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. It is well known that there is a universal enveloping algebra $U(\mathfrak{g})$ over $\mathbb{C}$ generated by generators $e_1, \dotsc, e_n, f_1,...
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Representation of a Lie algebra from a representation of Lie group
Let $\pi$ be a unitary representation of a Lie group $G$ on a Hilbert space $H_{\pi}$. Thus $\pi:G \to B(H_{\pi})$. One can extend $\pi$ to a representation of $L^1(G)$ via the formula
$\pi(f)=\int_{...
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Isn't the quantomorphism group really just the "WKB-quantomorphism" group?
Introduction
In his second-most upvoted post, called "Why quantum mechanics?" (second only to his post on fibre bundles & gauge theory) in the physics SE community, Urs Schreiber, in the setting ...
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The actual Satake diagram EIV
In table 9 of "Lie groups and algebraic groups" (1990) [OV], Onishchik and Vinberg present the Satake diagrams. The diagram corresponding to EIV has the "orthogonal" node blackened.
In table 4 of "Lie ...
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Free graded Lie algebras
Let $R$ be a graded commutative unital ring and $M$ a graded $R$-module (all gradings are over $\mathbb{Z}$). I'm looking for a reference for the following statement:
If $M$ is $R$-free, then the ...
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How to write down the connection morphism in the long exact sequence in Čech cohomology explicitly in this specific case?
Fix an integer $k$. Let $X=G/P$ be a complex rational homogeneous variety. I assume here $G$ is a simply connected semi simple complex Lie group and $P=P_k$ is a maximal parabolic subgroup defined by ...
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Is the Adjoint Action self dual over finite fields?
Given a finite group $G$, and representation $\rho: G \to H(\mathbb{F}_q)$ where $H$ is some classical algebraic group ($Gl$, $Sl$, $O$, $SO$, $SP$, $GSP$, $U$, etc), is the induced Adjoint ...
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Reference Request: Branching Rules of $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$
I have heard that the branching rules are well-known for the simple Lie algebra $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$ over fields of characteristic zero. Where can I find a ...
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Which compact (finite dimensional) Lie groups have $H^1_{DR}(G)\neq 0$
In particular, I am wondering if $H^1_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible ...
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Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)
Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows.
$$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...
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Historically, which came first: the Lie algebras or their classification?
The classification of the complex simple Lie algebras by their Dynkin diagrams gives rise to five exceptional complex simple Lie algebras: $F_4, G_2, E_6, E_7$ and $E_8$.
I am trying to find out ...
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What is the minimum number of steps for two elements of a Lie algebra to generate the whole Lie group?
Consider a compact, connected and simply connected Lie Group $G$, and two elements in the corresponding Lie algebra $X$ and $Y$. By successive action of exponential map you can get the following ...
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Branching laws for $SO(n)$
The branching laws for the $SO(n-1)$ as a subgroup of $SO(n)$ are well known and easy to find. See for example the Wikipedia article:
https://en.wikipedia.org/wiki/Restricted_representation#...
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Does $SU(2)\cong Sp(1)\subset SO(5)$ factor through $Spin(5)\cong Sp(2)$ as the standard embedding $Sp(1) \to Sp(2)$?
$SU(2)$ can be seen as a subgroup of $SO(5)$ through the following chain of subgroups
$$
SU(2) \subset SO(4) \subset SO(5).
$$
If we identify $SU(2)\cong Sp(1)$, does the inclusion $Sp(1) \to SO(5)$ ...
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Noncompact dual of $\mathrm{Spin}(2n)$ corresponding to $\mathfrak{so}^*(2n)$
Let $U=\mathrm{Spin}(2n)$, which is a simply connected compact simple Lie group, and let $\mathfrak{u}_0=\mathfrak{so}(2n)$, the Lie algebra of $U$. If $\mathfrak{g}_0$ is a noncompact dual of $\...
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Maximal abelian subalgebras of Lie algebras over $\mathbb{C}$
Let $\mathfrak g$ be the Lie algebra of a compact connected Lie group $G$. Let $\mathfrak g_{\mathbb{C}}$ be the complexification of $\mathfrak g$ and let $\mathfrak h \subset \mathfrak g_{\mathbb{C}}$...
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Universal enveloping algebra and the algebra of invariant differential operators
Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Then $\mathfrak{g}$ may be interpreted as the Lie algebra of right (equivalently left) invariant vector fields. Let $\mathcal{U}(\mathfrak{...
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2
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Difference of adjacent dominant weights is a root?
The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 ...
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When does the enveloping algebra functor lift to the category of bialgebras?
Let $\mathrm{Ass} $ denote the operad, whose algebras are associative unital algebras, considered as a dg-operad.
Denote $\mathrm{Ch} $ the category of chain complexes over a commutative ring $\...
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Lyndon words and Hall basis
I am looking for an algorithm to produce Hall basis from Lyndon words. First I will recall the definition of the Hall set following Serre's presentation.
Let $X$ be a finite set and let $M(X)$ be ...
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