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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47 views

Checking axiom of Category $\mathcal{O}$

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G$ be a split connected reductive algebraic group over $K$ with Borel $B$. We have the associated Lie algebras $\mathfrak{g}=$Lie$(G)$ and $\...
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118 views

Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?

For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
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1answer
89 views

BGG Category $\mathcal{O}$ is not closed under extension

What is the reason for the BBG category $\mathcal{O}$ failing to be closed under extensions i.e which of the 3 axioms of $\mathcal{O}$ does not hold under taking extensions? Is there a prototype of ...
3
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0answers
49 views

Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...
6
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1answer
196 views

An extension of symplectomorphism group

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$. We consider the ...
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24 views

Proof of parametrisation of $\hat{\mathfrak{g}}$-intertwiners of induced modules of affine Lie algebras

I am not 100% sure that this question belongs here, since I think the topic does but the specific problem I have might not. Feel free to tell me so if this is the case. It concerns the proof of ...
4
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2answers
152 views

Confusion over spin representation and coordinate ring of orthogonal Grassmannian

This is a copy from MSE where the question did not attract much attention. I'm working over $\mathbb{C}$ here. Let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic ...
4
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156 views

Cohomology and higher structures

Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear $G$-action on $M$, there is an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ ...
4
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1answer
167 views

A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$

This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here. $G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie ...
6
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1answer
86 views

Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$?

For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to ...
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59 views

Hopf algebras structure and quantum affine algebras

I'm looking for some information about the Hopf algebras structure and the quantum groups. In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...
7
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1answer
194 views

Is a 8-dimensional quadratic form recognized by its Lie algebra, modulo equivalence and scalar multiplication?

Question. Let $K$ be a field of characteristic zero (large characteristic should be fine too). Let $q,q'$ be two non-degenerate quadratic forms on $K^n$ with $n=8$. Suppose that the Lie algebras $\...
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43 views

Composition of operators in $w_{1+\infty}$ and $W_{1+\infty}$

The algebra $W_{1+\infty}$ can be defined as a central extension of the lie algebra $w_{1+\infty}$ (defined as being spanned by $\left(-\partial_z \right)^m z^{-k}$ ). See for example: Alexandrov, ...
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76 views

Existence of a maximal rank CR Lie subalgebra

Let $\mathfrak{g}$ be a real Lie algebra of dimension $2n+1$ and let $\mathfrak h \subset \mathfrak g \otimes \mathbb C$ be a subalgebra of complex dimension $n+1$ satisfying $\mathfrak h + \overline{\...
3
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1answer
93 views

Complete reducibility of integrable modules over symmetrizable Kac-Moody Lie algebras

I am reading the book "Infinite-dimensional Lie algebras" by Victor G Kac. This is a long question regarding my understanding of the following theorem. In Theorem 10.7 Kac proves the complete ...
7
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2answers
392 views

Classification of symplectic resolutions

A. Okounkov said, "symplectic resolutions are Lie algebras of the 21st century." Is there a conjecture on the classification of symplectic resolutions? Do Braverman-Finkelberg-Nakajima Coulomb ...
2
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1answer
87 views

Lifting one parameter subgroup $e^{t K}$ to the universal cover of $\mathrm{Sp}(2N,\mathbb{R})$

I would like to lift an arbitrary one-parameter subgroup $e^{t K}$ with $K\in\mathfrak{sp}(2N,\mathbb{R})$ to the universal cover $\widetilde{\mathrm{Sp}}(2N,\mathbb{R})$ (or at least its two-fold ...
3
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0answers
49 views

Reference request: Category of finite dimensional representations of loop algebra is not semisimple

For $\mathfrak{g}$ a semisimple Lie algebra, we may define its (untwisted) loop algebra as $L(\mathfrak{g}) = \mathfrak{g} \otimes \mathbb{C}\lbrack t,t^{-1} \rbrack$. Let $\mathcal{F}$ be the ...
3
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1answer
62 views

Maximal quotients of the enveloping algebra of a simple Lie algebra

Let $\mathfrak{g}$ be a simple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and centre $Z(\mathfrak g)$, $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak g$, $\lambda$ a complex ...
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50 views

Is there a functorial relation of center of universal enveloping algebra?

I asked this question before but there was no proper answer so I would like to ask it again making it more precise question. (See my earlier question : Question on the center of universal enveloping ...
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1answer
73 views

Lie algebra cohomology: $H^i(R,V)=H^i(R,V^R)$ with $R$ reductive and $V$ an $R$-module

Let $R$ be a reductive, finite-dimensional Lie algebra over a field of characteristic 0, and let $V$ be a semisimple $R$-module (also finite dimensional). I have seen a reference to the fact that $H^i(...
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57 views

Relation between topological and differentiable Lie group cohomology for unitary modules

It is well known, that if $V$ is a $C^\infty$ $G$-module, then the differentiable cohomology groups $H^*_{d}(G, V)$ are isomorphic to $H^*(\mathfrak{g}, K, V)$. I am interested if the result can be ...
3
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48 views

Choosing a complementary Lie subalgebra well

I have a compact connected Lie group $G$ with Lie algebra $\mathfrak{g}$ and a codimension one closed subgroup $H$ with Lie algebra $\mathfrak{h}$. Using an inner product on $\mathfrak{g}$ one can ...
4
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4answers
356 views

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 ...
4
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73 views

Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras

Yesterday, I came across the following preliminary theorem. Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup of $\mathcal{B}^{-1}$ (the group of ...
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135 views

Question on the center of universal enveloping algebra

Let $G$ be a unitary group $U(n)$ and $G’$ is a subgroup $U(m)$. (i.e. $m\le n$) Let $\mathfrak{g},\mathfrak{g}'$ be the complexified Lie algebra of $G,G’$ and $\mathfrak{z},\mathfrak{z}’$ be the ...
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31 views

Classification of involutions on $G_{2}$-homogeneous spaces

Are you aware of a systematic classification of involutions on $G_{2}$-homogeneous spaces?
4
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1answer
155 views

Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request

Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0. We consider the adjoint representation $$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$ ...
4
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1answer
123 views

The Lie algebra of the subgroup of $GL(n)$ preserving a given variety

Let $V=k^n$ for an algebraically closed field $k$ of characteristic 0, and let $W \subseteq V$ a subspace. Let $G_W\subseteq GL(V)$ be the set of invertible linear maps that preserve $W$, i.e. $$ G_W=\...
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109 views

When a finite codimensional subalgebra contains a finite codimension ideal?

What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property: Every finite codimensional subalgebra $B$ of $A$ ...
10
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247 views

Did anybody come across this Lie algebra representation?

Let $\mathfrak{g}$ be – for the sake of simplicity – a complex Lie Algebra. Then we define the antisymmetric linear transformations as $$ \mathfrak{A(g)} =\big\{\left.\,\alpha \in \mathfrak{gl(g)}\,\...
3
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1answer
110 views

Do rational points in a split reductive group act transitively on the orbits of the Cartan subalgebra (w.r.t. automorphism group of Lie algebra)?

Let $(G,T,M)$ be a split reductive group (over say, the integers), with Lie algebra $(\mathfrak{g}, \mathfrak{t})$, and let $R$ be a commutative ring. When $R$ is an algebraically closed field, it is ...
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100 views

Explicit branching rules from $G(n+m)$ to $G(n) \times G(m)$ (where $G = \operatorname{SL}$, $\operatorname{SO}$ or $\operatorname{Sp}$)

Is there in the literature any explicit combinatorial description of the branching rules from $\operatorname{SL}(n+m)$ to $\operatorname{SL}(n) \times \operatorname{SL}(m)$, from $\operatorname{SO}(n+...
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0answers
55 views

Matrix Lie algebra: Semisimple element = semisimple matrix?

I am considering the set $$ \mathbb{L} = \lbrace X \in \mathbb{C}^{2n \times 2n} \; ; \; J^{-1}X^HJ = -X \rbrace, \quad J = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix},$$ of complex ...
3
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2answers
185 views

Nontrivial Poisson relations for affine Poisson algebras

Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, ...
1
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1answer
107 views

Ideal of the free Lie algebra L(x,y) generated by x

Let $L=L(x,y)$ be the free Lie algebra generated by letters $x,y.$ For a vector subspace $V\leq L$ we denote by $[V,L]$ the vector space spanned by brackets $[v,l],v\in V,l\in L.$ A vector subspace $V\...
2
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1answer
93 views

Weight space dimension of the fundamental representation $\pi_n$ for type $C_n$

Will the fundamental representation $\pi_n$ of type $C_n$, for $n > 3$, have weight spaces of dimension greater than $1$? Is there some online resource where weight space multiplicities can be ...
8
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0answers
104 views

Invariant complex structures for simple Lie groups

For which simple Lie groups there exists a left-invariant complex structure?
1
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0answers
51 views

Property of a semisimple Lie algebra over complex number field

Is the following property true? Let $L$ be a finite-dimensional Lie algebra over $\mathbb{C}$. Then $L$ is semisimple if and only if for every $x \in L$, there exists $y, z\in L$, such that $x=[y, z]...
3
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1answer
120 views

Examples of simple vertex operator algebras (VOAs)

A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a ...
4
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199 views

A conjecture about the barycenter of a polytope

Could someone help me with the following conjecture? Thanks a lot! Suppose I have a polytope $\Delta$ in $\mathbb R^n (n\geq 2)$ with coordinates $(x_1,x_2,\cdots,x_n)$ defined by linear ...
5
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1answer
113 views

Maximal dimension of abelian subalgebra of exceptional simple Lie algebra in positive characteristic

For complex semisimple Lie algebras, the maximal dimension of an abelian subalgebra was determined by Mal'cev in 1945. For $E_7$, for example, it is $27$, and is the radical of the $E_6$ parabolic. ...
6
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2answers
310 views

Homotopy Gerstenhaber algebras: description via operads vs derivations

There are at least a couple of definitions in the literature for an $E_2$-algebra, also known as a homotopy Gerstenhaber algebra, also known as $G_{\infty}$-algebra. Suppose $V$ is a graded vector ...
3
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67 views

Understanding the geometric fibre twisted differential operators

Let $\mathfrak{g}$ be a Complex semisimple Lie algebra with decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}$. For $\lambda \in \mathfrak{h}^*$, we let $\mathcal{D}_{\lambda}$ be ...
3
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0answers
46 views

Embedding of Verma modules in Kac-Moody Lie algebras

Let $\mathfrak{g}(A)$ be a symmetrizable Kac-Moody Lie algebra over $\mathbb{C}$ and ($\mathfrak{h}$, $\Pi, \Pi^\vee)$ be a realization of the GCM $A$. Assume that $$\mathfrak{g}(A)=\mathfrak{h} \...
9
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1answer
230 views

What are Harish-Chandra bimodules used for?

There are many recent papers on classification of Harish-Chandra bimodules for rational Cherednik algebras and, more generally, non-commutative algebras which are quantizations of symplectic ...
4
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1answer
106 views

Algorithm to construct basis for Kac-Moody algebra

Suppose I have a Kac-Moody algebra (maybe even Borcherds-Kac-Moody) $\mathfrak{g}$ with symmetric cartan matrix $A$. Let the simple roots be $e_{\alpha_i}$ for $i = 1, \ldots n$. I know there is ...
3
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0answers
79 views

Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators

Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...
4
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1answer
101 views

Quantum Hamiltonian reduction and tensor products

Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras. Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and ...
4
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0answers
150 views

Bar notation in Bourbaki’s *Lie groups*, Chap. IX

I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they ...

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