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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Yangians as unique deformation

In Drinfeld's paper "Hopf algebras and the quantum Yang-Baxter equation" there is a statement (Theorem 2) that Yangian is a unique quantization of the corresponding Lie bialgebra. My question is ...
DerLoewe's user avatar
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Differential operators on $G/K$

Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal ...
jorge vargas's user avatar
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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 ...
AccidentalFourierTransform's user avatar
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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 ...
David Roberts's user avatar
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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 ...
Weicheng Ye's user avatar
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Conformal Dimension and Highest Weight States of Coset CFT

I am trying to understand the vertex operator algebras of the following form: $$\frac{U(M|N)_{k_1}}{U(L)_{k_2}}$$ Where $U(M|N)$ is the unitary supergroup, $U(L)$ is the usual unitary group, and $...
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DGLA related to the deformation of hopf Algebras

Recently I was considering Hopf algebras and Drinfeld's twists. I stumbled upon a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by ...
Niek de Kleijn's user avatar
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Is there a notion of octonionic structure on a Lie algebra? In the same way as there is one for complex and quaternionic

Is there a notion of octonionic structure on a Lie group? In the same way as there is one for complex and quaternionic
Rony Kaplan's user avatar
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Plancherel formula for $L^2(G/N)$

Let $G$ be a connected real semisimple or reductive Lie group. Let $TA$ be a Cartan subgroup, where $T$ is compact and $A$ is split. Let $MA$ be the centralizer of $A$ in $G$, and let $N$ be the ...
user97074's user avatar
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Tensor product of half-spin representations

Let us consider half-spin representations $S^+$ and $S^-$ of the orthogonal Lie algebra $\mathfrak{so}(2n, \mathbb{C})$. It is known that $\bigwedge\limits^{n-1}V$ arises as subrepresentation in $S^+\...
quantum's user avatar
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A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...
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Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth. Let's ...
Nicolas's user avatar
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What's the story with the Hopf fibration and the Jacobi identity?

I like the Hopf fibration of the 3-sphere $S^3$ enough that I found a nice way to make a physical model of it. All you need is to combine a bunch of key rings in such a way that (ii) every pair of ...
David Richter's user avatar
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Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly. For example, take $X=\mathbb{P}^n$, ...
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Polynomials invariant with respect to a nilpotent Lie algebra

Let $\mathfrak{u}$ be a nilpotent Lie algebra and let $\mathbb{C}[\mathfrak{u}]$ be the space of polynomials with the natural coadjoint action of $\mathfrak{u}$. Can one describe $\mathbb{C}[\...
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Bruhat decomposition of $G/Q$

Let $G$ be a semisimple algebraic group over $\mathbb C$, $T$ be a maximal torus and $B$ be a Borel subgroup of $G$ containing $T$. Let $R^+$ be the set of positive roots with respect to $B$. Let $Q$ ...
Ram's user avatar
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
yang's user avatar
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Adjoint orbit of two vectors

Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector ...
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Is the "Toeplitz algebra" the representation ring of a Hopf algebra related to SU(2)?

More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
Qiaochu Yuan's user avatar
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Lie-infinity structure in Lagrangian Floer theory ?

Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?
Oliver Fabert's user avatar
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Commutator Baker-Campbell-Hausdorff formula

Consider the Baker-Campbell-Hausdorff formula $\Phi(X,Y)\in\mathbb{Q}\langle\!\langle X,Y\rangle\!\rangle$ in non-commutative variables. Define $X*Y:=\Phi(X,Y)$ and $[X,Y]=(-X)*(-Y)*X*Y$, and then (as ...
Diego Sulca's user avatar
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Classification of generalized Cartan matrices (GCMs)

A GCM is square matrix $A = (a_{ij})$ satisfying: (1) $a_{ij} \in \mathbf{Z}$ (2) $a_{ii} = 2$ for all $i$. (3) $a_{ij} \leq 0$ for $i \neq j$. (4) $a_{ij} = 0$ iff $a_{ji} = 0$. There is a standard ...
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Applications (and source) of Bourbaki exercise on root systems with two root lengths?

In Chapters 4-6 of Bourbaki's Groupes et algebres de Lie, Exercise 20 for Section VI.1 concerns irreducible (reduced) root systems with roots of two lengths: in other words, systems of types $B_\ell, ...
Jim Humphreys's user avatar
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classifying space and cohomology of integer general linear group

I have obtained that the classifying space $$ BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty) $$ is the Grassmannian. I have also obtained that the mod 2 cohomology is the polynomial ...
Shiquan Ren's user avatar
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Existence of Cartan subalgebra

I am reading Helgason's book. In Chapter 3 he proved the existence of Cartan subalgebra for a semisimple Lie algebra $\mathfrak g$ (definition: a Cartan subalgebra is a maximal abelian subalgebra ...
CEOandVIP's user avatar
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On the Weyl character formula

So let $G$ be a compact real Lie group. Let $\rho:G\rightarrow GL_n(\mathbb{C})$ be an irreducible representation of $G$ and let $\chi_{\rho}$ be the character associated to $\rho$. Let $\Lambda_{\rho}...
Hugo Chapdelaine's user avatar
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German term for "restricted Lie algebra" ?

Can anyone tell me the German term for "restricted Lie algebra" ? Many thanks in advance ! Kind regards, Stephan Kroneck.
Stephan F. Kroneck's user avatar
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about dual Verma module in BGG category O

for the Verma module $M(\lambda)$, it has a dual $M(\lambda)^{\vee}$, also as $n^{-}$ module, $M(\lambda)$ isomorphic to $U(n^{-})$ so it is very nature to ask for the dual Verma module $M(\lambda)^{\...
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Algebra of regular functions on the quadratic cone and SU(2) representations

I was reading the paper "Short Star-Products for Filtered Quantizations" by Pavel Etingof and Douglas Stryker (MSN), where in the introduction they claim that the algebra of regular functions on the ...
Simone Castellan's user avatar
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Logarithm of a matrix

I am looking for a reference to study logarithm of an invertible triangular matrix. What is a good algorithm? Are there any good reference which studies this topic both theoretically and from an ...
Turbo's user avatar
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Three dimensional real Lie groups with cocompact discrete subgroups

I would like to know what are all the real three dimensional Lie groups (simply connected) that can act transitively and locally freely on a compact three dimensional manifold? This is equivalent to ...
Kamoun's user avatar
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Homology of solvable (nilpotent) Lie algebras

Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for ...
Boris Bilich's user avatar
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3 answers
634 views

About $G$-modules versus $Lie(G)$-modules for algebraic groups

Hello, I would like to know clear references about the following facts: Let $G$ be a connected algebraic group (over alg. closed field in char. 0), $Lie(G)$ its Lie algebra, $M$ a $G$-module. I don'...
Sasha's user avatar
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Why is the trace of the Casimir on the irrep of a semisimple algebra nonzero?

A crucial step in the "purely algebraic" proof of Weyl's semisimplicity theorem is that the Casimir element $C\in U\mathfrak{g}$ acts by nonzero scalars on a nontrivial irrep $V$. However, at least ...
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What are the "tensor-closed" object of the BGG category $\mathcal{O}$ of a semisimple Lie algebra $\mathfrak{g}$?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its BGG category $\mathcal{O}$. It is well-known that $\mathcal{O}$ is not closed under tensor product, i....
Zhaoting Wei's user avatar
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Decomposition into irreducibles of symmetric powers of irreps.

Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lambda$, indexed by its ...
Moderat's user avatar
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Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$. First, let's define two matrices: ...
anderstood's user avatar
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3 answers
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Modular Forms and Root Systems

In the study of semisimple Lie groups, lattices appear all over the place. In the theory of elliptic functions and modular forms, (equivalence classes of) lattices correspond to elliptic curves and to ...
David Corwin's user avatar
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Nilpotent Lie algebras and unipotent Lie groups

$\mathbf{n}$ is nilpotent Lie algebra with $N$ being the corresponding algebraic Lie group. Now one neat feature of this setting is that you can take the exponential map to be identity. In other words ...
Najdorf's user avatar
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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 ...
Adam Hammam's user avatar
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A Manifold for which $\chi^{\infty}(M)$ is rich

Is there a manifold $M$ for which $\chi^{\infty} (M)$, the lie algebra of smooth vector fields on $M$ contains all finite dimensional Lie algebras(Up to isomorphism)? A weaker question: Is there a ...
Ali Taghavi's user avatar
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3 answers
3k views

On radicals of a lie algebra

Let $\mathbb{k}$ be a field, $\mathfrak{g}$ be a finite-dimensional Lie algebra over $\mathbb{k}$. In Bourbaki's "Lie Groups and Lie Algebras", Ch I, he defines four radical-like ideals of $\mathfrak{...
Si-Qi Liu's user avatar
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2 answers
1k views

Sum of all root lengths in simple Lie algebra

Part of one of my calculations involves (the innocent looking) expression $\sum_{\alpha\in\Sigma} (\alpha,\alpha)$ for simple Lie algebras. I have two methods of calculating it -- which don't agree. ...
Simon's user avatar
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388 views

Minimal non-abelian groups -> Lie groups/algebras

A group is called minimal non-abelian if it is non-abelian and all proper subgroups are abelian. Does this notion also exist with Lie groups or algebras? As an example, consider the Lie algebra ...
Hauke Reddmann's user avatar
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475 views

Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group?

I need to answer (affirmatively, I hope) the following question: In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset ...
Xin Nie's user avatar
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Finite Order Automorphisms on Complex Simple Lie Algebras

Let $L$ be a finite dimensional complex simple Lie algebra, and let $F(L)$ be the set of all finite order automorphisms on $L$. Suppose that we declare $f,h \in F(L)$ to be equivalent if there exists ...
sunny's user avatar
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SU(6) -> SU(3) branching rule

I read in at least one paper and in the wiki below http://en.wikipedia.org/wiki/Quark_model that the 56 symmetric irrep of SU(6) breaks down into 10^{3/2} + 8^{1/2} irreps of SU(3)xSU(2). Here the ...
Y Macdisi's user avatar
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2 answers
664 views

Compute formal character of semisimple Lie algebras.

Let $\mathfrak{g}$ be a semisimple Lie algebra and $V_{\lambda}$ be the irreducible $\mathfrak{g}$-module with highest weight $\lambda$. Are there some softwares which can compute the formal character ...
Jianrong Li's user avatar
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Is the space of polynomial functions on M_n a faithful U(gl_n)-module?

We are over some field $k$ of characteristic $0$. The general linear group $\mathrm{GL}_n$ canonically acts from the left and from the right on the space $\mathrm{M}_n$ of all $n\times n$-matrices, ...
darij grinberg's user avatar
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3 answers
327 views

Invariant symmetric bilinear forms and H^4 of BG

I am reading this paper of Teleman and Woodward. On page 4, they say that $H^4(BG;\mathbb{R})$ can be identified with the space of invariant symmetric bilinear forms on $\mathfrak{g}_k$. Why is this ...
Kevin H. Lin's user avatar
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