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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17
votes
3answers
779 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 ...
1
vote
0answers
90 views

Exact sequence of L-infinity-algebras

We call a sequence of $L_\infty$-algebras (weak) maps $$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$ is exact if it is exact on the the underlying chain complexes level. Thought I don't know ...
6
votes
1answer
216 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 ...
3
votes
1answer
199 views

Lie algebras with a one-dimensional maximal subalgebra

Let L be a Lie algebra with a one-dimensional maximal subalgebra. Is the following true? Over a perfect field of characteristic 0 or p > 3, every such finite-dimensional Lie algebra is either ...
4
votes
1answer
161 views

Decomposing tensor products of modules for the orthogonal/symplectic groups in characteristic zero

I would like to know if there is a perfect analogue of the classical Littlewood-Richardson rule for decomposing tensor products of simple modules for the orthogonal/symplectic groups in characteristic ...
2
votes
0answers
99 views

Maurer-Cartan elements of the extension of an $L_{\infty}$-algebra

Let $g$ be a nilpotent $L_{\infty}$-algebra. For every commutative differential graded algebra $A$, one can form the extension $g\otimes A$ and endow it with a nilpotent $L_{\infty}$-algebra ...
1
vote
0answers
92 views

A Isomorphism between the extension group and cohomology group of Lie algebras [closed]

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
7
votes
1answer
235 views

Restriction of highest-weight representations to Heisenberg subalgebras

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $\tilde{\mathfrak{g}}=\mathfrak{g}((t))\oplus \mathbf{C}K\oplus \mathbf{C}d$ its Kac-Moody extension ($K$ is the level and $d$ ...
12
votes
2answers
368 views

Introducing division strategically in operads to accommodate formulas like the Baker-Campbell-Hausdorff formula

I am not as familiar with operad terminology as I'd like to be, so I might be missing some well-known term in the area. If so, I'd appreciate any pointers to the correct terms. Consider the following ...
2
votes
0answers
53 views

Is it possible to compute the Iwahori Decomposition using the Chavalley Commutator Formulas?

Ideally, I would like a constructive, algorithmic proof of this fact. I have convinced myself that it is true, but my "proof" is not pretty. I would like to know if a more attractive or intuitive ...
9
votes
0answers
194 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
142 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 ...
1
vote
1answer
125 views

The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular

This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We ...
0
votes
1answer
180 views

non-locally simple $\mathcal{g}$-modules

I'm interested in an example of a simple $\mathcal{g}$-module $M$ over some locally simple Lie algebra say $\mathcal{g}\simeq gl(\infty)$ such that $M$ is not isomorphic to a direct limit of simple ...
4
votes
2answers
278 views

lie algebras, Kac Moody, and quantum mechanics book

Hi all, I've just finished a graduated course on Kac-Moody algebras, and I'm really looking for some reading in regard to their applications to Quantum Mechanics. Can you help?
-1
votes
1answer
244 views

identifying dual of lie algebra of general linear groups

Is there any reference for the following fact? I am looking for a nice and simple proof. Assume that $G=GL(n,C)$, the group of invertible $n\times n$ matrices with complex entries. Why can the dual ...
3
votes
1answer
119 views

Zero-divisors in a graded Lie algebra

Let $\mathfrak{g}$ be positively graded Lie algebra over $\mathbb{Q}$, concentrated in even degrees. Question: If $\mathfrak{g}$ is not free, must there exist linearly independent elements ...
6
votes
5answers
405 views

Applications of Chevalley Restriction Theorem

Let $G$ be a simple linear algebraic group (over $\mathbb{C}$, say) and $\mathfrak{g}$ be its Lie algebra, $\mathfrak{t}\subset \mathfrak{g}$ the Lie algebra of a maximal torus in $G$ and $W$ the ...
3
votes
3answers
402 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
1answer
242 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
1answer
445 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
1answer
392 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
1answer
162 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
0answers
353 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
1answer
182 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
2answers
122 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
2answers
140 views

quasi-minuscule representations

Wich representations of $F_{4}$, $E_{8}$ and $G_{2}$ are quasi-minuscule?
2
votes
0answers
128 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
1answer
215 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
0answers
173 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
1answer
306 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
0answers
164 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
1answer
258 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
1answer
413 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
1answer
151 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
2answers
268 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
1answer
168 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
3answers
721 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
1answer
216 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
1answer
171 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
1answer
330 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
1answer
127 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
1answer
195 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
2answers
274 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
2answers
190 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
1answer
268 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
2answers
244 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
0answers
75 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
2answers
389 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
4answers
671 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 ...