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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 ...

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Centralizers in Jacobson-Witt Lie algebras

Recall the (Jacobson-)Witt Lie algebras in positive characteristic: $W(n,1)$ is the Lie algebra of derivations of $\Bbbk[X_1,\dots,X_n]/(X_1^p,\dots,X_n^p)$. (For simplicity; more generally, I'm ...
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Invariants of the group $SO(2)$

Let $V_d$ be the complex vector space of binary forms of degree $d$ endowed with the natural action of the special orthogonal group $SO(2).$ Consider the corresponding action of the group $SO(2)$ on ...
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1answer
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Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...
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1answer
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Some confusion about weights and roots in parabolic root systems

I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...
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The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps $$ L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
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2answers
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Definition of the weight lattice for a nonreduced root system

Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha_1, ... , \alpha_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha_1^{\vee}...
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Finite-dimensional graded Lie algebras with $2$ generators

Does anyone know of a classification of those (complex) Lie algebras which are: generated by two elements $\mathbb{Z}$-graded Lie algebras finite dimensional
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1answer
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Tilting Objects in BGG Categories $\mathcal{O}$

Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...
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221 views

The de Rham complex of the octonionic projective spaces

The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
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1answer
134 views

About Hom and weight space of nilpotent Lie algebra cohomology

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $\Phi$ the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ ...
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About isomorphism of Extension groups between Category $\mathcal{O}^\mathfrak{p}$ and $\mathcal{O}$

In the paper: Kostant modules in blocks of category $\mathcal{O}^\mathfrak{p}$ In section 8.2 (p.22), I use the notation in Humphrey's Category $\mathcal{O}$ book. Then the passage said the ...
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129 views

Which representations of the Lie algebra of a Lie group come from representations of the group itself?

I think this is very classic mathematics, but I can't find a complete answer in the literature. Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
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1answer
233 views

“Almost-ideals” in the (simple) Lie algebra of an algebraic group?

Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple. Is it necessarily the case that ...
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About Extension group in Category $\mathcal{O}^\mathfrak{p}$

Let $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$, $L(\lambda)$ be the simple highest weight module with highest weight $\lambda$. Suppose $\mu\le \lambda\le \nu$, does ...
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Question about Ext group in $\mathcal{O}^\mathfrak{p}$?

Let $W$ be a Weyl group, let $\mu$ be an antidominant weight. Let $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$. Denote ${}^IW$ the set of minimal length coset ...
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Geometry of Affine Kac-Moody Algebras

I recently asked this question on phys.SE and it was suggested to me to ask it here. One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric ...
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1answer
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Is there an accessible exposition of Gelfand-Tsetlin theory?

I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...
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Deformations of nilpotent parts of superalgebras

I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598 After ...
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109 views

action of Weyl group element on Weyl vector

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...
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1answer
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Splitting the injection that you get from the Poincaré-Birkhoff-Witt theorem

Let $\mathfrak g$ be a Lie algebra over a field of characteristic zero, with universal enveloping algebra $U\mathfrak g$. By the Poincaré-Birkhoff-Witt theorem one knows that $i:\mathfrak g \to U\...
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1answer
178 views

Lie algebra action on complex algebraic variety

Suppose that $X$ is a complex algebraic variety, i.e. it is integral, separated, and of finite-type over $\mathbb{C}$. Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra. Suppose that $\...
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1answer
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finite dimensional modules are highest weight modules [closed]

Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector. My feeling is, since $e_i$'s ...
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2answers
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Fundamental representations and weight space dimension

For the Lie algebra $\frak{sl}_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all ...
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It there a nice way to describe the structure of Malcev-complete groups?

Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...
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Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra: The space of ...
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Chern-Simons theory with non-compact gauge groups G

This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
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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 ...
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Openness of regular mappings and the conjugacy of Cartan subalgebras

In the book Lie Algebras of finite and affine type by Roger Carter, in chapter 3, the conjugacy of Cartan subalgebras of a finite-dimensional Lie algebra over $\mathbb{C}$ is established via a ...
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how the borel subalgebras of p(n) look like except the standard one?

I am reading the book Musson: Lie superalgebras. In Chapter 3, on page 62, Lemma 3.6.8 tells about support of Borel subalgebra. I am confused about this. I am trying to do the proof of the Proposition ...
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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 ...
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about weight decomposition of U(sl3) [closed]

the universal enveloping algebra is $\mathbb{Z}^2$-graded algebra. $U(sl_3)= \bigoplus_{(i_1,i_2) \in \mathbb{Z}^2 } U_{i_1\alpha_1+i_2\alpha_2}$ $($the weight decomposition$)$ where $\alpha_1$ and $...
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A lie group which is sat in its Lie algebra

Motivated by this question we ask the follwing question: Assume that a Lie subgroup $G$ of $Gl(n,\mathbb{R})$ is contained in a subvector space $F$ of $M_n(\mathbb{R})$ such that $dim G=...
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Universal Enveloping algebra of a L$_\infty$ algebra

In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras ...
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Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...
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Casimir operator of a given Lie algebra and relation with its matrix representation

I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
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2answers
148 views

Generalizing Polar Decomposition of Matrices

I am trying to find a certain proof of polar decomposition of complex matrices which I think should exist more generally for a certain class of Lie groups. Recall that the polar decomposition of a ...
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1answer
167 views

Simple Lie algebras: making subspaces 'very transversal'

Let $G$ be a Lie group or group of Lie type whose Lie algebra $\mathfrak{g}$ is simple. Because the Lie algebra is simple, for any proper subspace $V\subset \mathfrak{g}$, there is a $g\in G$ such ...
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A convergence condition on tempered representation

Assume $\pi$ is a tempered representation of $GL_n(\mathbb{Q}_p)$. $N_n$ is a maximal unipotent subgroup of $GL_n$, and $\xi$ is a non-degenerate character of $N_n(\mathbb{Q}_p)$. Let $\Pi$ be the ...
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A (familly) of Lie brackets associated to a Lie algebra

Let $L$ be a Lie algebra whose Lie bracket is denoted by $[.,.]$. For a given vector $V\in L$,does the following 2- linear map always define a new Lie bracket on $L$? $$[X,Y]_V=(adXadY -adYadX)(V)$$...
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Prime ideals being finitely-generated implies coherence?

Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$ $(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...
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1answer
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The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators

Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$). How is the embedding $\mathfrak{g}...
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2answers
173 views

About block $\mathcal{O}_\lambda$ of Category $\mathcal{O}$

The blocks of $\mathcal{O}$ are precisely the subcategories consisting of modules whose composition factors all have highest weights linked by $W_{[\lambda]}$ to an antidominant weight $\lambda$. I ...
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Induced $(\mathfrak{g},K)$-modules

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
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1answer
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About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$

It follows from Exercise 1.13 in Humphreys' Category $\mathcal{O}$ book that $M\in\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$ has a direct sum decomposition $M=\oplus M_i$ such that all weights of each $...
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About weight of $M\in\mathcal{O}_{\chi_\lambda}$ in the Category $\mathcal{O}$

Given $M\in\mathcal{O}_{\chi_\lambda}$, how does the weights of $M$ relate to $\lambda$? Is it something about $W\cdot\lambda$?
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2answers
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About block of category $\mathcal{O}$

In the book "Representation of Semisimple Lie Algebra in the BGG Category $\mathcal{O}$". Exercise 1.13. Suppose $\lambda\not\in\Lambda$, so the linkage class $W\cdot\lambda$ is the disjoint union of ...
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1answer
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higher Casimirs for $\mathfrak{sl}$

The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
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1answer
593 views

The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

Original question (without additional information from Wendy): Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way: Taking the E8 as {128,...
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Relationship bewteen Kazhdan-Lusztig Vogan polynomial and Kazhdan-Lusztig polynomial

Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$ Let $M(\lambda)$ be the Verma module with highest weight $\...
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1answer
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How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?

In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$"). Physically this modular fusion category describes the ...