4
votes
2answers
121 views

Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...
3
votes
1answer
104 views

Reference to definition of matrix log with domain SO(3) which is Borel measurable

I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...
0
votes
2answers
158 views

A question on Lie algebras

To what extent, the following types of Lie algebras are classified : Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.
3
votes
3answers
218 views

Computation of restricted Lie algebra (co)homology

My question is the following: Is there a small complex, perhaps analogous to the Chevalley-Eilenberg complex, computing the (co)homology of a restricted Lie algebra over a field of characteristic ...
5
votes
1answer
274 views

Geometric structure of flag manifolds, Borel -Weil-Bott theorem

I want to know if there is proof of Borel Weil Bott theorem, that is as geometric as it can be. Let $G$ be a semisimple compact Lie group and $T$ be a maximal torus. We know that $G/T$ is a ...
1
vote
0answers
119 views

About the Lie algebra of polyvector fields

I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
2
votes
1answer
108 views

Casimir of a three dimensional solvable lie algebra

Good morning everyone. I've encountered recently during my computations the following lie algebra $$\mathfrak g=\text{span}(f_0,f_1,f_2),$$ with $$\begin{eqnarray}[f_2,f_1]&=&f_0+a f_2,\\ ...
6
votes
1answer
203 views

Easy argument for “connected simple real rank zero Lie groups are compact”?

Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact. Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has ...
9
votes
0answers
192 views

differentiating positive energy LG reps

Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...
1
vote
2answers
138 views

References request: representations of Heisenberg algebra.

Let $p_1, p_2, \ldots$, be the power sum symmetric functions. Let $p_n^* = n \frac{\partial}{\partial p_n}$. Then $$ p_n^* p_m - p_m p_n^* = \delta_{m, n} 1. $$ Where could I find this result in some ...
3
votes
3answers
354 views

Decomposition into irreducibles of symmetric powers of irreps.

Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lambda$, indexed by its ...
11
votes
0answers
331 views

Source of a formula for tensor product multiplicities?

This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other ...
2
votes
1answer
273 views

Reference request - localisation de g-modules

Does anyone have a link to a copy of Beilinson-Bernstein's "Localisation de g-modules", in which they prove the Beilinson-Bernstein theorem? I can't find it anywhere.
2
votes
1answer
152 views

Criterion for nilradical of a maximal parabolic subalgebra to be abelian?

This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which ...
4
votes
2answers
259 views

BGG-like resolutions and translations

This question originated from my confusion after I read the following paragraph (page 31, section 4.8) in Resolutions and Hilbert series of the unitary highest weight modules of the exceptional ...
1
vote
1answer
187 views

translation functors in parabolic category $\mathcal{O}$

I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$. I am mainly interested in the ...
4
votes
2answers
539 views

Clifford Lie Algebras

I'm studying the "Clifford Lie Algebra" (see http://arxiv.org/pdf/1007.2481.pdf page 30). It's basically a way to look at Clifford algebras and their properties in a Lie algebraic setting (which I ...
12
votes
2answers
542 views

Deligne's 1996 note on exceptional Lie groups

This is about Deligne's "La série exceptionnelle de groupes de Lie, C.R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 321–326". When this came out, that was quite something! People were often ...
1
vote
2answers
175 views

Copies of ax+b inside the AN part of an Iwasawa decomposition?

As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
7
votes
0answers
254 views

Dual versions of “folding” symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams ...
2
votes
0answers
100 views

Borel (parabolic) subalgebras of twisted affine Lie algebras.

The notion of Verma-type modules for affine Lie algebras is related to the concept of Borel subalgebras. The literature is extensive when the affine algebra is untwisted and all constructions come ...
10
votes
1answer
453 views

Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`

The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently ...
10
votes
5answers
644 views

About the intrinsic definition of the Weyl group of complex semisimple Lie algebras

It may be a easy question for experts. The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra ...
0
votes
1answer
106 views

Reference request: Tensor products of modules for reductive Lie algebras

I am looking for a reference that describes how to decompose a tensor product of two finite dimensional simple modules for a reductive Lie algebra over $\mathbb{C}$. In particular, I would like a ...
9
votes
2answers
379 views

Lie algebras over non-algebraically closed fields

I am independently studying Lie algebras (in preparation for grad school) from James Humphrey's text "Introduction to Lie Algebras and Representation Theory. Fairly early on in the development ...
23
votes
8answers
3k views

“Modern” proof for the Baker-Campbell-Hausdorff formula

Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula? All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and are not at all geometric ...
0
votes
0answers
157 views

Generalizing groups via the Hall-Witt identity

In studying the integrability problem for Lie algebra representations, I have been led to wonder whether generalizing the notion of group by dropping associativity, while keeping the Hall-Witt ...
1
vote
1answer
365 views

Schur Weyl duality for sl_n representations

Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
11
votes
3answers
437 views

Rational forms of simple Lie algebras

I am more or less familiar with the classification of real forms of complex semisimple Lie algebras. But as soon as I wander off into the domain of very-non-algebraically closed fields, things seem to ...
4
votes
1answer
168 views

Equivariant Levi subalgebras.

Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$. Does there necessarily exist a Levi ...
4
votes
4answers
453 views

Structure of $[S(\mathfrak{g})\otimes S(\mathfrak{g})]^G$ for semisimple $\mathfrak{g}$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. $S(\mathfrak{g})^G$ is a polynomial algebra with rank $\mathfrak{g}$ generators. Call them $c_i(x)$, where $x\in \mathfrak{g}$ and ...
10
votes
4answers
682 views

Decompose tensor product of type $G_2$ Lie algebras.

Let $G$ be a semisimple Lie algebra over $\mathbb{C}$. Let $V(\lambda)$ be the irreducible highest weight module for $G$ with highest weight $\lambda$. If $G$ is of type A, we can decompose ...
4
votes
3answers
476 views

Reference request: representation of type G2 Lie algebras.

Let $\mathfrak{g}$ be an Lie algebra of type G2. Are there some combinatorial ways to describe a basis of a $\mathfrak{g}$-module? For classical types, there is a method used tableaux. Thank you very ...
11
votes
4answers
896 views

Three-dimensional simple Lie algebras over the rationals

I ran into this question on math.stackexchange.com "How many 3 dimensional simple Lie algebras are there over the rationals?" The question has been sitting idle for a long time. I thought it was ...
5
votes
3answers
404 views

twisted affine algebras

Let $g$ a finite-dimensional complex simple Lie algebra and $\sigma$ a finite order Dynikin diagram automorphism of $g$. Consider $\tilde g$ as the loop algebra associated to $g$, and $\tilde ...
5
votes
0answers
263 views

Reference for the Thick Affine Grassmanian

Let $G$ be a reductive group and $LG$ be the algebraic loop group of $G$; i.e. $LG(k) = G( k((t)) )$. There is a fair amount of literature on the affine Grassmanian $LG(k)/G(k[[t]])$ and its Picard ...
5
votes
0answers
299 views

The Hochschild-Serre spectral sequence relative to an ideal containing the derived subalgebra

Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak ...
6
votes
0answers
371 views

Uniform proof of dimension formula for minimal special nilpotent orbit?

Given a simple Lie algebra over an algebraically closed field of good characteristic such as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...
9
votes
5answers
1k views

How does the group algebra look as a Lie algebra

It's probably a well known question, so it is just a reference question. Let $G$ be a finite group and let $C[G]$ be a group algebra. Then we can define a bracket on $C[G]$ by $[f,h]=f*h-h*f$. What ...
6
votes
2answers
750 views

References on Lie Groups and Dynamical systems

Hi everybody! I'm interested in Lie Theory and its connections to Dynamical Systems theory. I am starting my studies and would like references to articles on the subject.
4
votes
1answer
366 views

Commutator formulas in a universal enveloping algebra

I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum ...
2
votes
1answer
296 views

Character formulas for non-integrable modules?

Let $\mathfrak{g}$ be a Kac-Moody Lie algebra (actually, I'd already be happy with an answer addressing the case where $\mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$). 1st ?: I'm wondering ...
2
votes
2answers
732 views

Complex root systems

This question is twofold. 1) What is the best reference on root systems? 2) Do complex root systems exist?
1
vote
0answers
214 views

Is it possible to construct category $\mathcal{O}^{\mathfrak{p}}$ with non-standard parabolic subalgebra

The usual definition of the parabolic category $\mathcal{O}^{\mathfrak{p}}$ is the following. We consider Lie algebra $\mathfrak{g}$ of the rank $r$ with the root system $\Delta$ and the set of ...
4
votes
3answers
271 views

Invariant symmetric bilinear forms and H^4 of BG

I am reading this paper of Teleman and Woodward. On page 4, they say that $H^4(BG;\mathbb{R})$ can be identified with the space of invariant symmetric bilinear forms on $\mathfrak{g}_k$. Why is this ...
9
votes
6answers
1k views

Polynomial invariants of the exceptional Weyl groups

Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let ...
2
votes
3answers
485 views

Computing the index of a Lie algebra: what is known beyond the reductive case?

Recall that an index of a Lie algebra $\mathfrak{g}$ is $\mathrm{ind}\ \mathfrak{g} := \min\limits_{\xi \in \mathfrak{g}^*} \dim \mathrm{Ann}_{\xi}$ where $\mathrm{Ann}_{\xi}=\{h\in\mathfrak{g}| ...
7
votes
2answers
791 views

Inverse of Baker-Campbell-Hausdorff

This should be quite simple to answer. I have a situation in which I must have an explicit expression for the inverse of Baker-Campbell-Hausdorff. More precisely: I have two power series $P_1(X,Y)$, ...
5
votes
1answer
1k views

How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?

By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
6
votes
2answers
420 views

Characterisation of parabolic subalgebras: reference sought

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{p}$ a subalgebra. As we all know, $\mathfrak{p}$ is parabolic if it contains a Borel (thus maximal solvable) subalgebra. In this ...