# Tagged Questions

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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### Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with $2$,...
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### Lie functor preserves “surjections” in synthetic differential geometry?

In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism. As pointed out ...
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### Affine analog of the theory of sheets

In the study of adjoint orbits in a complex semi-simple lie algebra, there is a well known object known as a "sheet". These are the irreducible components of the union of orbits of the same dimension. ...
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### The parity of the full automorphism group order of finite non-abelian groups of prime exponent

Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
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### Tensor products of simple modules over algebras [closed]

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...
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### del Pezzo surfaces and exceptional algebras

It is well-known that $H^2(dP_n, Z)$ for a del Pezzo surface of degree $9-n$ which is $\mathbb{P}^2$ blown up at $n$ generic points, is encoded by the exceptional Lie algebra $E_n$. However, the Mori ...
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### The existence of zero-divisors in the universal enveloping algebra of an infinite-dimensional Lie algebra

The intuition for this problem comes from $\S$17 Exercise 1 Humphreys' Introduction to Lie Algebras and Representation Theory which essentially asks us to use PBW in order to prove that if a Lie ...
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### Differential of the adjoint quotient map

My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...
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### 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 ...
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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 ...
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### 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 ...
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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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### 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, ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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?
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### 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 $\mathfrak{sl}_3(\mathbb{C})$-...
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### 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 ...
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### 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 $SL_n(\mathbb{R})$. ...
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### 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 $3$ ...
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### 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 ...
Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
### $(L, \nabla)$ 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 $A$ ...