**3**

votes

**3**answers

438 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 ...

**-1**

votes

**1**answer

246 views

### Heisenberg Lie algebras

Dear forum,
I would like to ask if $H(m)$ is the Heisenberg Lie algebra of dimension $2m+1$ and $M$ is an ideal of $H(m)$. Can we say that $M$ has a complement in $H(m)$?

**9**

votes

**1**answer

453 views

### Why are affine Lie algebras called affine?

Hi. I was wondering if someone could explain why we call affine Lie algebras affine. Thanks!
Oliver

**4**

votes

**1**answer

434 views

### Algorithm to find exponential map of differential operators acting on function

I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon ...

**1**

vote

**1**answer

165 views

### Resolutions of Lie algebras

We have a good notion of dgc algebra resolutions of commutative algebras.
Is there an explicit construction of a dg Lie algebra resolution of a Lie algebra?

**11**

votes

**0**answers

365 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 ...

**3**

votes

**1**answer

185 views

### Quantized conserved quantities appearing from the Lie-algebra

Hi,
consider a simple situation in quantum mechanics: Your Hilbert space is $\mathcal{H}=L^2(\mathbb{R}^3)$ and you use the obvious unitary representation $\pi\colon G=O(3)\times\mathbb{R}^3\to ...

**2**

votes

**2**answers

125 views

### Connectedness of Springer Fibers

Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If ...

**0**

votes

**2**answers

151 views

### quasi-minuscule representations

Wich representations of $F_{4}$, $E_{8}$ and $G_{2}$ are quasi-minuscule?

**2**

votes

**0**answers

130 views

### Explicit Lie May structure on cosimplicial DG Lie algebras

In the paper "Homotopy Lie algebras", Schechtman and Hinich proved that any cosimplicial
differential graded Lie algebra has the structure of a 'Lie May algebra'.
If my understanding is right here, ...

**3**

votes

**1**answer

219 views

### 'Generalised' coinvariant algebras

Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, ...

**0**

votes

**0**answers

175 views

### polynomial representation of $sl_{2}(k)$

Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
...

**2**

votes

**1**answer

321 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.

**1**

vote

**0**answers

168 views

### universal enveloping algebras and commutator subalgebras

Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$,
$U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$
and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...

**2**

votes

**1**answer

264 views

### finite dimensional irreducible representation of finite dimensional nilpotent Lie algebra

Let $k$ be a field, $L$ be a finite dimensional nilpotent Lie
algebra over $k$ and $M$ be a finite dimensional irreducible
representation of $L$. Assume that there is a linear function $\rho
: ...

**5**

votes

**1**answer

429 views

### A question about the proof of Beilinson-Bernstein localisation

I'm trying to understand the proof of the Beilinson-Bernstein localisation theorem at the moment, but there's just one point where I'm having a mental block, and was wondering if anybody could clarify ...

**3**

votes

**1**answer

156 views

### Reduction of antisymmetric complex matrices

Let $E=\mathfrak{so}(n,\mathbb{C})$ be the Lie algebra of antisymmetric complex matrices. We consider the action of the complex orthogonal group $SO(n,\mathbb{C})$ on $E$ by conjugation. Is there a ...

**1**

vote

**2**answers

270 views

### Gerstenhaber bracket out of $L_\infty$ algebras

Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that ...

**2**

votes

**1**answer

181 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 ...

**8**

votes

**3**answers

746 views

### HIgher Homotopy Groups and Representation Theory

Let $G$ be a compact Lie group, and $g$ its associated Lie algebra.
In what ways do the higher homotopy groups $\pi_{n}(G)$ with $n>1$ appear in the representation theory of $G$?
As an example, ...

**6**

votes

**1**answer

218 views

### Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem?

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define
...

**5**

votes

**1**answer

176 views

### Endomorphisms in Category O and Schubert Classes

Let $\mathfrak{g}\supset \mathfrak{b}\supset \mathfrak{h}$ be a complex semisimple Lie algebra, with choice of Borel and Cartan subalgebras, $W$ the Weyl group.
W. Soergel's 'Endomorphismensatz' ...

**3**

votes

**1**answer

360 views

### Centralizers of Nilpotent Elements in Semisimple Lie Algebras

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of ...

**0**

votes

**1**answer

128 views

### A question about G-Manifolds

I am looking for a clear reason for following fact:Is there any reference ?
Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...

**3**

votes

**1**answer

210 views

### Maximal subgroups of semisimple Lie groups

The problem of finding and classifying the maximal subgroups of simple Lie groups like $SU(3)$ is well known and solved in the literature. What about maximal subgroups of semisimple groups like $SU(3) ...

**4**

votes

**2**answers

281 views

### classification for coadjoint orbits of lower or upper triangular matrices

Is there any classification for coadjoint orbits of lower or upper triangular matrices in general case $n\times n$. Is there any reference?

**3**

votes

**2**answers

196 views

### Kostant's Theorem on Principal TDS

I have a few questions concerning Kostant's work on principal three-dimensional subalgebras (TDS). Let $\frak{g}$ be a finite-dimensional complex semisimple Lie algebra, and ...

**2**

votes

**1**answer

272 views

### Can this Lie group written as a direct product?

Let $G=G_1.G_2$ be a Lie Subgroup of $SO(k) \times SO(2) \subset SO(k,2)$, where
$G_1=SU(k/2) \subset SO(k)$ and $G_2$ is a Lie subgroup of $SO(k) \times SO(2)$ isomorphic to $SO(2)$.
Let $G_1 \cap ...

**4**

votes

**2**answers

246 views

### Almost-Lie Algebras?

Are there any reasonably natural algebras whose product (bracket) almost, but does not quite, satisfy the Jacobi relation?
A priori it doesn't matter whether the bracket is anti-symmetric.
The ...

**1**

vote

**0**answers

78 views

### multiplicity of a weight in the basic representation of $\hat{sl_2}$

it is known by Weyl character formula that multiplicity of the weight $\wedge_{0}- n.\delta$
in the basic representation $L(\wedge_{0})$ of $\hat{sl_2} $is equal to the number of partitions of $n$.It ...

**4**

votes

**2**answers

391 views

### CFTs corresponding to affine Lie algebras

I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.
On the few pages leading up to page 192 in here one can see see the ...

**4**

votes

**4**answers

672 views

### Lie algebra $\mathfrak{so}(9)$ as a subalgebra of $\mathfrak{f}_4$

What are the explicit Lie algebra mono-morphisms from $\mathfrak{so}(9)$ into the exceptional Lie algebra $\mathfrak{f}_4$ and from $\mathfrak{f}_4$ to $\mathfrak{e}_6$? Are these explicitly described ...

**2**

votes

**0**answers

121 views

### simple roots of a reflection subgroup

Consider a Hermitian symmetric pair of complex Lie algebras $(\mathfrak{g},\mathfrak{k})$ and split the set of roots into compact roots (i.e. roots of $\mathfrak{k}$) and noncomapt roots $\Delta = ...

**3**

votes

**0**answers

224 views

### Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...

**2**

votes

**1**answer

240 views

### symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation
$v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...

**1**

vote

**1**answer

179 views

### A Criterion for Reductivity of Lie Subgroups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...

**0**

votes

**1**answer

117 views

### Name for ideal generated by Lie subalgebra

Let $\mathfrak{m}$ be a Lie sub-algebra of the Lie algebra $\mathfrak{g}$. Is there a name for the smallest ideal of $\mathfrak{g}$ containing $\mathfrak{m}$? It certainly exists and coincides with ...

**2**

votes

**3**answers

220 views

### Invariant subbundles of tangent bundle of flag variety

Suppose that $G$ is a complex semisimple Lie group, $P$ a parabolic subgroup of $G$. What are all of the $P$-invariant subspaces of $\mathfrak{g}/\mathfrak{p}$? In various low dimensional examples, I ...

**1**

vote

**2**answers

166 views

### character formula for demazure modules

Is there any character formula for demazure modules in arbitary kac moody settings which does not use demazure operators?

**3**

votes

**3**answers

501 views

### Finite Order Automorphisms on Complex Simple Lie Algebras

Let $L$ be a finite dimensional complex simple Lie algebra, and
let $F(L)$ be the set of all finite order automorphisms on $L$.
Suppose that we declare $f,h \in F(L)$ to be equivalent if there exists
...

**4**

votes

**2**answers

272 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 ...

**3**

votes

**1**answer

302 views

### Do the solutions of the Maurer--Cartan equation form a simplicial set?

The Maurer--Cartan equation is the equation:
$$d\gamma+\frac 12[\gamma,\gamma]=0$$
where $\gamma$ represents a degree one element in a differential graded Lie algebra $\mathfrak g^\ast$. Let's denote ...

**2**

votes

**2**answers

238 views

### Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (ie. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...

**5**

votes

**2**answers

223 views

### Lie algebra embeddings and the center of their enveloping algrabras

Let $\mathfrak{g}_1\subset \mathfrak{g}_2$ be a Lie algebra embedding. Assume both are semisimple. For instance take the standard diagonal embedding $\mathfrak{sl}(2, \mathbb{C})\subset ...

**1**

vote

**0**answers

225 views

### Why are there two Hopf algebra structures on a Kac--Moody Algebra.

For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by
$$
\Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ...

**3**

votes

**2**answers

182 views

### Reconstructing a Lie group Banach representation from the Lie algebra rep. on analytic vectors

Dear all,
I have some difficulties with the following assertion in the book of Kirillov.
Let $G$ be a connected Lie group, and T a given (!) representation of G on a Banach space V.
Let $V^\omega$ ...

**2**

votes

**1**answer

150 views

### Kostant's theorem about U(g) being free over Z(g) and a corollary of it

Hello,
$g$ is a complex semisimple Lie algebra.
There is the result that $U(g)$ is free over $Z(g)$.
There is another result: If $E$ is a finite dimensional representation of $g$, then ...

**2**

votes

**2**answers

201 views

### Covering of Verma modules by translation of a dominant Verma module

Hello,
Could anyone give a reference or proof for the following fact (which is, probably, not very difficult):
We work in category O for a semisimple complex Lie algebra. $M_{\chi}$ denotes the ...

**1**

vote

**1**answer

193 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

**0**answers

158 views

### Bracket of lyndon words?

Here is a simple question regarding the standard Lyndon basis for the free Lie Algebra. Suppose I take two lyndon words $m$ and $n$ and their standard bracketings $B(m)$ and $B(n)$ as elements in the ...