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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 ...

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403 views

About the definition of E8, and Rosenfeld's “Geometry of Lie groups”

I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...
3
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2answers
106 views

Generating Irreducible representations of a simple lie algebra with Schur functors

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...
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1answer
157 views

Symmetric tensor of Lie algebra of $su(N)$

I am interested in knowing the exact form of the anti-commutation of two generators of $su(N)$ lie algebra. Let us denote $T^a$ to be the generator of $su(N)$ lie algebra in the defining ...
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1answer
182 views

Book on algebraic structures

What is the most complete book on algebraic structures that deals with the complete taxonomy from magmas to Lie algebras and inner product spaces?
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1answer
63 views

A question about centralizer of a vectors in the positive Weyl chamber

Given a compact Lie group $K$ and a maximal torus $T\leq K$, and choose a positive Weyl chamber $\mathfrak t^*_+\subset\mathfrak k^*$, where we used a $K$-invariant inner product on $\mathfrak k$. ...
4
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1answer
147 views

The Ungraded Milnor-Moore Theorem

Let $k$ be a field of characteristic $0$. There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\...
5
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1answer
95 views

Can I bring the Kirillov 2-form on coadjoint orbits to adjoint orbits?

I tried asking this question on stackexchange and received no response. Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I ...
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0answers
60 views

Pairing half the sum of the roots with a simple coroot

I was calculating something with the root system $A_n$ and I think there might be a more general principle at work. Here is the example: let $G = \operatorname{GL}_5$, with maximal torus $T$ and ...
3
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1answer
178 views

Framing dependence of HOMFLY polynomial

I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...
7
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0answers
44 views

Criterion for existence of a homogeneous space associated to a Lie pair

Recall that every finite-dimensional real Lie algebra $\mathfrak{g}$ is the Lie algebra of a simply-connected Lie group, which is unique up to isomorphism. This statement generalises somewhat to ...
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0answers
58 views

How the roots and weights changed under a folding?

Let $e_1,e_2,e_3,f_1,f_2,f_3$ be the generators of the Chevalley basis of the Lie algebra $sl_4$. Let $e_1' = e_1+e_3$, $e_2'=e_2$, $f_1'=f_1+f_3$, $f_2'=f_2$. Then the subalgebra generated by $e_1', ...
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70 views

Why Lie algebra (Chevalley–Eilenberg) cohomology are graded Lie algebras but not G-algebras?

I was reading a paper related to Gerstenhaber algebra structure and came across to this- "Lie algebra (Chevalley–Eilenberg) cohomology are graded Lie algebras but not G-algebras(Gerstenhaber algebra)"....
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0answers
45 views

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 ...
2
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0answers
55 views

Lie algebra bundle associated to a Lie group bundle

I was reading something(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 gave http://www.pphmj.com/Images/...
3
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1answer
180 views

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 $k$ ...
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0answers
38 views

Dual space of polynomial one-form

Recently I read a paper "Quasi-particles models for the representations of Lie algebras and geometry of flag manifold". In section 2, author gives a fact without proof. Now I rephrase this fact as ...
7
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1answer
266 views

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{...
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2answers
208 views

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 ...
3
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1answer
125 views

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/...
4
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2answers
224 views

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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0answers
67 views

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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0answers
98 views

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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0answers
54 views

Adjoint orbits of a finite group of type $G_2$ [reference request]

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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66 views

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 ...
9
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1answer
183 views

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 ...
5
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1answer
65 views

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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1answer
197 views

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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1answer
219 views

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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1answer
111 views

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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0answers
64 views

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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0answers
17 views

How to express free Lie algebra elements in terms of the right-normed basis?

Article A right normed basis for free Lie algebras and Lyndon–Shirshov words shows that it is possible to build right-normed (right-nested) basis of a free Lie algebra. I am looking for references (...
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0answers
55 views

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\...
5
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1answer
284 views

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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1answer
90 views

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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0answers
129 views

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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1answer
128 views

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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3answers
578 views

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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0answers
60 views

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 $\rho = \sum_{i \in I} \omega_i = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, ...
3
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1answer
101 views

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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27 views

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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1answer
230 views

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 $x\in f+\mathfrak{g}_e$. ...
4
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0answers
97 views

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 ...
5
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1answer
210 views

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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54 views

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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0answers
78 views

$U(sp_2)$ 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,...
3
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1answer
122 views

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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1answer
74 views

Semi-direct product of Lie algebras [closed]

Let $A_{4,1}$ be a Lie algebra of dimension four such that non-vanishing Lie brackets on $A_{4,1}$ are given by $$[e_2,\, e_4]= e_1, \: [e_3,\, e_4]= e_2.$$ Furthermore, we observe that the Lie ...
4
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0answers
182 views

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 ...
7
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2answers
217 views

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 ...
9
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1answer
226 views

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 ...