**2**

votes

**1**answer

54 views

### References about Hasse diagrams of root systems

This is to ask about references of Hasse diagrams of irreducible root systems. I found here and there nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical ...

**3**

votes

**0**answers

131 views

### 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 ...

**6**

votes

**1**answer

62 views

### Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions

Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends ...

**6**

votes

**1**answer

158 views

### Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$ ?

Let $n\in\mathbb N$. Let $k$ be a commutative ring. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $x_n$ over $k$. (The multiplication in ...

**8**

votes

**3**answers

830 views

### Making D-modules on affine varieties more explicit

This is a follow up to my question about D-modules supported on the nilpotent cone. I got some good answers there but now I have a more basic question.
Consider an affine algebraic variety X, a ...

**0**

votes

**0**answers

26 views

### The non-singular controls always in neighbourhood of singular controls?

Consider the case of a right invariant affine distribution: $D_{U} = \{ aU + \lambda bU | a,b \in \mathfrak{su}(n), \lambda \in \mathbb{R} \}$ on $SU(4)$.
Consider the equations:
...

**7**

votes

**1**answer

274 views

### A technical question about root systems

I'm studying root systems and coming up with an observation:
Let $\Phi$ be an irreducible root system and $\Phi^+$ be a positive root system. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ ...

**5**

votes

**0**answers

207 views

### Recurrence Formula for Zernike polynomials

I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...

**18**

votes

**3**answers

947 views

### Is the sequence of partition numbers log-concave?

Let $p(n)$ denote the number of partitions of a positive integer $n$. It seems to me that we have for all $n>25$
$$
p(n)^2>p(n-1)p(n+1).
$$
In other words, the sequence $(p(n))_{n\in ...

**42**

votes

**7**answers

7k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...

**5**

votes

**1**answer

171 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**

vote

**0**answers

32 views

### Iwasawa decomposition and Non-Abelian Centraliser of A

I'm studying Knapp's book "Representation Theory of Semisimple Groups" and am trying to understand the structure theory of non-compact groups. Namely, let $G=KAN$ be the Iwasawa decomposition, ...

**2**

votes

**1**answer

135 views

### Length of Weyl group element mapping highest root to a simple root

Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the ...

**2**

votes

**1**answer

75 views

### Seeking for abelian subalgebra of fixed dimension in finite Lie algebra

The problem is: I want to know if there is abelian subalgebra of dimension $k$ in Lie algebra of dimension $n$. My Lie algebra is given by its structure constant table. There are some algorithms ...

**3**

votes

**2**answers

101 views

### Covering derivations of a quotient algebra

Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$.
Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...

**9**

votes

**1**answer

420 views

### What happened to the fourth paper in the series “On the classification of primitive ideals for complex classical Lie algebras” by Garfinkle?

In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to ...

**4**

votes

**1**answer

82 views

### Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?

Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is ...

**3**

votes

**2**answers

201 views

### 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 ...

**3**

votes

**1**answer

109 views

### Singular curves of affine distributions on a Lie group

Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group?
Specifically the case of a right invariant affine distribution: $D_{U} = ...

**7**

votes

**1**answer

210 views

### Homotopes of simple Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For which $z\in \mathfrak{g}$ does the formula
$$
\mu(x,y)=ad (z)([x,y])=[z,[x,y]]
$$
define another Lie bracket on the same ...

**7**

votes

**0**answers

108 views

### Is $F_{f, c, \ell}$ a $G$-harmonic polynomial?

Let $G \subset \text{GL}_n(\mathbb{C})$ be a finite subgroup. The group $G$ acts naturally on $\mathbb{C}^1[\mathbb{C}^n]$ the space of degree $1$ homogeneous polynomials in $x_1, \dots, x_n$, i..e, ...

**0**

votes

**0**answers

61 views

### Conjugacy classes of involutions in Kac-Moody groups

Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix.
Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$.
...

**1**

vote

**1**answer

65 views

### A converse to Whitehead's Second Lemma (and more)

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{h}$ be a finite dimensional $k$-Lie algebra. I'm interested in knowing which (Lie algebraic) properties $\mathfrak{h}$ ...

**9**

votes

**0**answers

104 views

### Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?

Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence ...

**-2**

votes

**0**answers

33 views

### Intersection of a family of closed Lie subgroups [migrated]

If I have a Lie group $G$, and $\{H_{\alpha}\}_{\alpha\in A}$ is a family of closed Lie subgroups with Lie algebra $\{\mathfrak{h}_{\alpha}\}_{\alpha\in A}$, it's easy to see that $\bigcap_{\alpha} ...

**0**

votes

**0**answers

49 views

### Definitions of length function on a Weyl group [migrated]

Let $\Phi$ be an irreducible root system and $W$ the Weyl group of $\Phi$. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_l\}$ the corresponding base.
Can anyone give me the standard definition ...

**1**

vote

**2**answers

189 views

### Fundamental invariants for root subsystems

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$.
Now suppose $\Psi$ is ...

**2**

votes

**0**answers

78 views

### Prove that a Verma module is projective only if its highest weight is dominant?

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $\mathbb{C}$ with a fixed Cartan subalgebra $\mathfrak{h}$ and a fixed system of simple roots. It is stated in Exercise 3.11 of ...

**2**

votes

**0**answers

72 views

### Maximal possible dimension of abelian Lie subalgebra of Heisenberg Lie algebra of dimension $2n+1$?

Fix $n \in \mathbb{N}$, and let $\mathfrak{h}_n$ denote the Heisenberg Lie algebra of dimension $2n+1$ (over any given field $k$). Namely, $\mathfrak{h}_n$ is the Lie algebra with basis $x_1, \dots, ...

**31**

votes

**1**answer

2k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**4**

votes

**2**answers

185 views

### About supersolvable Lie algebras

A colleague of mine asked me the question below, and since I could not answer it, I thought I might have more luck on MO.
In Encyclopedia of Mathematics, a finite dimensional Lie algebra $L$ over a ...

**1**

vote

**0**answers

40 views

### Attainability of Global Optima In Optimal Control

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...

**8**

votes

**2**answers

329 views

### Confusion about Subcategories of Category $\mathcal{O}$

So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...

**0**

votes

**2**answers

133 views

### Classifying compact homogeneous Kähler manifolds

In this comprehensive answer to an old question, it is stated that
Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
...

**5**

votes

**1**answer

154 views

### Regular functions on nilpotent orbits and their covers

Let $G$ be a complex semisimple algebraic group with Lie algebra $\mathfrak{g}$.
In 1989 McGovern described the structure (as $G$-module) of the ring of regular functions on a finite cover of the ...

**2**

votes

**1**answer

163 views

### Where does the algebraic closure enter into Block's Theorem?

When applying Block's Theorem on the structure of differentiably simple rings to Lie algebras most authors require an algebraically closed field, but I can see no reference to algebraic closure in ...

**7**

votes

**2**answers

348 views

### Examples of Richardson orbit closures not having a symplectic resolution?

This is a follow-up to a recent question asked by Peter Crooks here. The answer by Ben Webster includes a helpful link to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper ...

**0**

votes

**1**answer

90 views

### Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$

I have the following questions:
Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra ...

**2**

votes

**0**answers

120 views

### 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 ...

**5**

votes

**1**answer

284 views

### Lie algebra and base change

I am very confused by the following and would appreciate any help.
Let $\mu_p \subset \mathbb{G}_m$ be the $p$-torsion subgroup scheme of the multiplicative group over $\mathbb{Z}_p$. I would like to ...

**3**

votes

**1**answer

93 views

### Orbits in the adjoint representation of $SU(2,1)$

How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation?

**5**

votes

**1**answer

190 views

### Plethysm of $S^3(S^2V)$ as $\mathfrak{sl}_3(\mathbb{C})$-module

I have asked this question in MSE before, but have not got any answer. So here I am asking it again with some more detail.
I believe that the following sequence of ...

**40**

votes

**15**answers

5k views

### What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty ...

**2**

votes

**1**answer

92 views

### When is the image of the adjoint representation of a real algebraic group Zariski closed?

Let $\operatorname{Ad}:\operatorname{SL}_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL_n(\mathbb{R})$. ...

**4**

votes

**1**answer

80 views

### Homogeneous Quaternionic-Kähler Structure of the Grassmannians?

Paraphrasing from Cortes' notes:
The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a
parallel subbundle $Q$ which is locally spanned by $3$
...

**2**

votes

**0**answers

47 views

### Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...

**8**

votes

**4**answers

1k views

### Constructing Affine Kac-Moody Groups

Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the ...

**10**

votes

**1**answer

222 views

### $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?

Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental methods, that if $\nabla$ is an integrable connection on a vector bundle $L$ on $A$ ...

**6**

votes

**1**answer

296 views

### A question about $O(3,1)$

Recall that $O(3,1)$ is the collection of matrices $A\in M_4(\mathbb R)$ such that
$$A\begin{pmatrix}1 ...

**4**

votes

**1**answer

129 views

### Centralizer of hermitian matrices with zero trace

In Quantum Physics one often has to deal with commutators.
Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero!
One can easily relate it to ...