**0**

votes

**0**answers

116 views

### A solvable Lie algebra

Consider the Lie algebra $\mathfrak{g}$ span by $x_1,\ldots,x_n,y_1,\ldots,y_n$ with the commutator
$\left[y_i,x_i\right]=y_i, \left[x_i,x_{i+1}\right]=y_i, \left[y_i,x_{i+1}\right]=-y_i,\quad 1\leq ...

**2**

votes

**0**answers

118 views

### $l$-weights and $l$-character of finite-dimensional highest $l$-weight representation of $L\mathfrak{g}$

I am trying to solve the following problem, which is related to relatively recent results, but I am not sure how to do it.
Problem
In this problem, $\mathfrak{g}=\mathfrak{sl}_{2}$. We study ...

**0**

votes

**1**answer

132 views

### Equivalence of Lie subalgebras, within a (irreducible) representation

Lie subalgebras inside simple Lie algebras (of type ABCDEFG) have been classified up to equivalence, and linear equivalence (by Dynkin et al). How does one classify embeddings of a Lie algebra h ...

**0**

votes

**1**answer

113 views

### Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices.
Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...

**2**

votes

**3**answers

508 views

### Root in positive Weyl chamber

Let $\mathfrak{g}$ be a complex simple Lie algebra. We fix a Cartan subalgebra $\mathfrak{t}\subset \mathfrak{g} $. Let $R\subset \mathfrak{t}^*$ the set of roots. We fix $\Pi\subset R$ the set of ...

**6**

votes

**2**answers

238 views

### Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and ...

**3**

votes

**1**answer

141 views

### Prescribed spherical representations, symplectic group $Sp(n)$

An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by ...

**18**

votes

**1**answer

297 views

### On a drawing in Dixmier's Enveloping Algebras

This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are ...

**4**

votes

**1**answer

223 views

### When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras?

Let $V$ be a vector space over some number field $k$. (I'm fine with $\mathbb{Q}$.)
Let $\phi \colon V \to k$ be a non-degenerate quadratic form. Associated with $\phi$ is the orthogonal group ...

**1**

vote

**0**answers

152 views

### How the exceptional simple Lie groups/ algebras were first discovered and by whom?

I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...

**2**

votes

**0**answers

135 views

### Kernel of the Weil homomorphism for compact symmetric spaces

Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...

**2**

votes

**3**answers

408 views

### First Explicit Irreducible Representations

Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...

**0**

votes

**0**answers

113 views

### Reference about a formula of coroot in an affine root system

Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that
$$
(\alpha + p\delta)^{\vee} = \alpha^{\vee} + ...

**0**

votes

**0**answers

70 views

### Cayley graphs for finitely presented Lie algebras

I have seen that an important tool of finitely presented groups consists in writing down its Cayley graph with respect to a given set of generators, and then try to extract data like the coarse ...

**2**

votes

**1**answer

110 views

### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...

**6**

votes

**0**answers

135 views

### vanishing of Lie algebra cohomology with coefficients in an infinite-dimensional module

Let $G$ be a real semisimple Lie group, $K$ its maximal compact subgroup, $\mathfrak g, \mathfrak k$ the corresponding Lie algebras. Let $V$ be a locally convex, Hausdorff vector space, which is a ...

**2**

votes

**0**answers

113 views

### Exploiting a finite presentation of a Lie algebra

In "my youth" I computed a finite presenation of the Poisson algebra on $S^2$ (finite presentation as a Lie algebra).
In what ways might this be useful? Does this allow you to extract information ...

**0**

votes

**1**answer

231 views

### generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$.
Let $\mathfrak p$ be a
...

**3**

votes

**1**answer

220 views

### The Jordan Plane and Enveloping Algebras

Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ...

**4**

votes

**1**answer

645 views

### 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:
...

**1**

vote

**0**answers

62 views

### How to define the determinant of a morphism between graded Lie algebras?

I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...

**5**

votes

**0**answers

104 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**-1**

votes

**1**answer

175 views

### Reductive space & Reductive Lie algebra

If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, ...

**2**

votes

**0**answers

161 views

### How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?

**2**

votes

**1**answer

164 views

### Explicit generators of $Z(U(\mathfrak{g}))$

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field. By Harish-Chandra, the center of its universal enveloping algebra $Z(U(\mathfrak{g}))$ is a polynomial ring and the ...

**2**

votes

**0**answers

159 views

### Free-field representations: how to study highest-weight submodules of the Fock module?

Suppose we have a representation of some affine Lie algebra $\mathfrak{g}=\mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$ on a Fock space $V$. The module $V$ will contain a lot of ...

**3**

votes

**1**answer

371 views

### Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? Open problem

I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer.
Is the manifold
$$M=\frac{E_{7(7)}}{SU(7)}\times ...

**3**

votes

**0**answers

92 views

### Homology of derivations of Differential Graded Lie algebra

Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule).
On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading ...

**6**

votes

**1**answer

339 views

### Torsion in profinite groups

Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ?
Can ...

**1**

vote

**0**answers

181 views

### rational representation of semisimple algebraic group

Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$?
...

**0**

votes

**0**answers

137 views

### List of irreducible representations whose weights are in a single Weyl group orbit

Let $\mathfrak g$ be a (finite dimensional) simple (nonabelian) Lie algebra
over $\mathbb C$. I need a complete list of irreducible (nontrivial) representations $V$
of $\mathfrak g$ such that the Weyl ...

**3**

votes

**3**answers

564 views

### 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 ...

**5**

votes

**0**answers

126 views

### LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...

**9**

votes

**1**answer

189 views

### Comparing a Chevalley basis with the canonical basis of the adjoint module?

First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} ...

**4**

votes

**0**answers

198 views

### 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 $Q$ ...

**0**

votes

**0**answers

47 views

### format of grading Witt Lie Algebra

Let $W(n,m)$ be generalized Jacobson-Witt algebra over a field of characteristic p>3, according to the grading of $W(n,m)$ , we know that it inherit the grading from $A(n,m)$ as follows: ...

**1**

vote

**0**answers

200 views

### A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...

**3**

votes

**1**answer

238 views

### Reference for exercises with solutions for affine Lie algebras

I am doing self-study of affine Lie algebras, and while I have all recommended reference material, I find it hard to prepare for the exam.
It was quite easy to study finite-dimensional simple Lie ...

**5**

votes

**1**answer

189 views

### Invariant Laurent polynomials under cyclic group action

Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...

**-1**

votes

**1**answer

221 views

### cartan killing metric [closed]

I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...

**4**

votes

**2**answers

229 views

### An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
the first one is the adjoint ...

**2**

votes

**0**answers

74 views

### Computing maximal ideals of a Lie algebra

Would you know an algorithm (or an automatic method) that computes all maximal ideals $J$ of a given Lie algebra?
Or an algorithm that computes all maximal ideals $J$ containing a given minimal ideal ...

**2**

votes

**0**answers

255 views

### The geometry of the holomorph of a Lie group

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left ...

**2**

votes

**3**answers

471 views

### What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...

**2**

votes

**1**answer

128 views

### Commutators of Schur polynomials of Lie algebra elements

Question:
Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...

**5**

votes

**2**answers

309 views

### Convention about “long” roots for simple Lie algebras of types ADE?

The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in ...

**2**

votes

**1**answer

339 views

### highest weight representations inside tensor product

Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus.
Let $\lambda,\mu,\nu$ be dominant characters of $T$.
...

**5**

votes

**0**answers

289 views

### A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...

**6**

votes

**1**answer

258 views

### Closure order on nilpotent orbits in exceptional Lie algebras

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent ...

**5**

votes

**0**answers

152 views

### Finite-codimension subalgebras of generalized Kac-Moody lie algebras

Do generalized Kac-Moody lie algebras of infinite dimension contain subalgebras of finite codimension? If so, is there a classification?