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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139 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 ...
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0answers
89 views

complexification of the real lie algebra sp(m,n)

I am unable to verify the fact that the complexification of the real lie algebra sp(m,n) is sp(2(m+n),C), where sp(m,n) is the set of endomorphism preserving the hermitian bilinear form over the ...
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2answers
265 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
155 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
689 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
207 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
168 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
282 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
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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
171 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
249 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
179 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
260 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
238 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 ...
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0answers
72 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
359 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
661 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
0answers
109 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
0answers
203 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
1answer
236 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
1answer
167 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
1answer
114 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
3answers
213 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 ...
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2answers
133 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
3answers
473 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
2answers
262 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
1answer
281 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
2answers
219 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
2answers
212 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 ...
0
votes
0answers
199 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
2answers
166 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
1answer
145 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
2answers
194 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
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
0answers
144 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 ...
1
vote
2answers
246 views

What are the symmetric and anti-symmetric representations of $6\times6$ of $SU(6)$ in $SU(3)\times SU(2)$?

A 6-dimensional ( fundamental) representation of $SU(6)$ becomes (3,2) representation in $SU(3)\times SU(2)$. We can decompose $6\times 6$ of $SU(6)$ into 21-dimensional symmetric and 15-dimensional ...
0
votes
0answers
136 views

Nontrivial copies of SO(r) in SO(n)

If $G=SO(n)=SO(\mathbb R^n)$ and $r\leq n$, it is easy to find a closed subgroup $H\leq G$ that is isomorphic to $SO(r)$, just let $S\subseteq\mathbb R^n$ be an $r$-dimensional subspace and let ...
3
votes
2answers
291 views

degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$

How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.
2
votes
0answers
129 views

Harmonic analysis and non-symmetric Macdonald polynomials?

I have recently been reading a lot about Macdonald polynomials, the symmetric and the non-symmetric ones. One thing that strikes me is that the symmetric Macdonald polynomials admit a positive theory, ...
2
votes
3answers
354 views

Dimension of Unipotent Radicals

A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...
2
votes
2answers
414 views

smallest simplest $E_8$ -module

What is the smallest simplest(non-trivial) $E_8$ -module ?
3
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0answers
230 views

The normalizer a maximal compact subgroup of a semi-simple Lie group

Let $G$ be a semi-simple real Lie group such that $|\pi_0(G)|<\infty$ and let $K$ be a maximal compact subgroup of $G$. Q1: How does one prove that $N_G(K)=K$? So I know a nice (and low-tech) ...
10
votes
1answer
782 views

decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...
4
votes
2answers
543 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 ...
0
votes
1answer
124 views

L a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L).

Let L be a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L). (Der is short for derivation). In order to show that ad(L)=Der(L), the proof I followed proves that that the ...
12
votes
2answers
544 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 ...
4
votes
2answers
248 views

Are maximal connected semisimple subgroups automatically closed?

(Yet another question in a series demonstrating my rather embarrassing ignorance of standard Lie theory... I hope this is not too basic for MO!) To be a little more precise: let $G$ be a real ...
2
votes
2answers
322 views

Around the socle filtration of a Verma module

Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$. Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of ...
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 ...
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1answer
420 views

parabolic subalgebras and Cartan decomposition

Let $\mathfrak{g}$ be a complex simple Lie algebra and $\mathfrak{k}$ its complex subalgebra such that $(\mathfrak{g},\mathfrak{k})$ is a Hermitian symmetric pair; $\mathfrak{g}= ...