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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3
votes
1answer
332 views

SU(6) -> SU(3) branching rule

I read in at least one paper and in the wiki below http://en.wikipedia.org/wiki/Quark_model that the 56 symmetric irrep of SU(6) breaks down into 10^{3/2} + 8^{1/2} irreps of SU(3)xSU(2). Here the ...
1
vote
1answer
147 views

Degree bounds when restricting an irrep of a compact Lie group to a torus

I am not sure of the right terminology, but here goes. Let $G$ be a compact, connected, simply connected, non-abelian Lie group. For any choice of one-dimensional torus $S\subset G$, and any ...
12
votes
3answers
933 views

Motivation of Virasoro algebra

I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with ...
4
votes
2answers
364 views

Dense subgroups of Lie Groups

SETUP: Let $G$ be a connected Lie group, and $H\subset G$ be a FINITELY GENERATED dense subgroup. I am interested in knowing what kind of information one can infer on the complexity of $H$. I am ...
2
votes
2answers
240 views

For a Weyl group, what is the connection between its exponents and lengths of its elements?

The following seems to be true: if $|W_q| := \sum {q^{l(w)}}$, where the sum is taken over the elements $w$, then $|W_q| = \prod {(1 + q +...+ q^{e_i})}$, where the product is taken over the exponents ...
10
votes
1answer
330 views

What are the simple Lie superalgebras of type E?

Background Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...
0
votes
1answer
116 views

why no Lie algebra degenerate to a rigid algebra? Why the closure of a rigid algebra forms the irreducible component of variety of Lie algebras?

Hi I just started working on degeneration and contractions, I would like to know: why no Lie algebra degenerate to a rigid algebra?(rigid algebra:an algebra whose orbit is zariski open) Why the ...
8
votes
2answers
422 views

A certain theorem about finite-dimensional Lie algebras over an algebraically closed field with zero characteristic.

Using Engel's Theorem and Lie's Theorem, one can easily establish the following result: Let $ \frak{g} $ be a finite-dimensional Lie algebra over an algebraically closed field $ \mathbb{F} $ of ...
1
vote
2answers
168 views

length function of a coxeter group with respect to two different simple systems are equal or not?

Is there any relation between length function of a coxeter group with respect to two different simple systems as two simple systems are weyl conjugates of one another?
0
votes
0answers
65 views

approximation in Lie algebras

Let $x_{1}$, $x_{2}$, $x_{3}$ three disctinct closed points of a curve $X$ over an algebraically closed field k. Let G a connected reductive group and $\mathfrak{g}$ his Lie algebra. I fix a Borel ...
7
votes
1answer
264 views

One more question about PBW

Let $k$ be a commutative ring with unit and $L$ be a Lie $k$-algebra. Let $U(L)$ be the universal enveloping $k$-algebra of $L$ (one can define it as a quotient of the tensor algebra, as it is ...
1
vote
0answers
229 views

Complex Finite Dimensional Representation of GL(N,C)

What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$? We already know all the complex finite dimensional linear representation of SU(N).
34
votes
0answers
733 views

Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?

In their seminal 1979 paper here, Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corresponding Iwahori-Hecke algebra. In particular they showed how to pass from a standard ...
7
votes
3answers
430 views

Why/when classification of simple objects is “simple” ? E.g. (unknown) classification of simple Lie algebras in char =2,3…

Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem. I wonder what is known/expected for char p=2,3 ? More vague ...
1
vote
2answers
189 views

Symmetric and Exterior products of sl(n,C)-module

Let M be the $sl(n,C)$-representation of the inclusion $sl(n,C)\hookrightarrow gl(n,C)$. Let q be a symbol. $f(q)=1-M q + \wedge^2Mq^2-...+(-1)^n\wedge^nMq^n$ $g(q)=\sum_{i=0}^\infty Sym^iM \; q^i$ ...
5
votes
2answers
431 views

Representation ring of SU(n)?

What's the structure of representation ring of SU(n)? What are the representations of generators?
4
votes
1answer
310 views

Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

I've recently read the following line in an interesting paper: It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...
7
votes
0answers
257 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
2answers
217 views

Commutator formula in infinite dimensions

The commutator formula states that for A,B elements of a Lie algebra, $\lim_{n\to \infty}\left\{ ...
2
votes
2answers
430 views

finding highest weight of dual of a representation of a semisimple lie algebra

If V is an irreducible representation of a semi simple lie algebra having highest weight λ then what will be the highest weight of the corresponding irreducible representation V∗ (Dual of V)?
6
votes
1answer
112 views

Embedding of F(4) in OSp(8|4)?

Is the superconformal algebra in five dimensions, $F(4)$, a subalgebra of the (maximal) six-dimensional superconformal algebra $OSp(8|4)$?
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 ...
4
votes
3answers
299 views

Good even grading and principal Levi type

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e$ be a nilpotent element in it. In the theory of finite W-algebras one often encounters the following two conditions: 1) $e$ is ...
8
votes
1answer
270 views

On q-Demazure operators

Setup Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements ...
0
votes
0answers
130 views

complex reductive Lie groups which are not defined over the real numbers

Hello Someone knows something about the complex reductive Lie groups (the complexification of a compact Lie group) which are not defined over the real numbers. I would like to know, if is it ...
3
votes
0answers
109 views

Number of submodules in $\wedge^2 V$ and $S^2V$ isomorphic to $\mathfrak{g}$

Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $\mathfrak{g}\subset\mathfrak{so}(V)$ be an orthogonal irreducible representation. It can be shown that the number of ...
10
votes
1answer
455 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 ...
6
votes
2answers
238 views

About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$

Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) ...
4
votes
2answers
281 views

Weights of restricted modules of some Cartan type Lie algebras

Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$. Denote by $L_{[p]} $ the minimal $p$-envelope of ...
2
votes
2answers
325 views

A remark in Jantzen's 'Lectures on Quantum Groups'

In Jantzen's AMS text 'Lectures on Quantum Groups' he makes the following remark (p.187, preface to Chapter 9): "For general (complex semisimple f.d. Lie algebra) $\frak{g}$ we can consider for each ...
2
votes
1answer
170 views

The real group orbits on the flag variety always contains the holomorphic directions?

Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of ...
7
votes
1answer
315 views

How to get Haar measure on a compact Lie group, given the complexification?

This is the first in what may be a series of questions on the theme "a Banach algebraist/Bear Of Little Brain needs help with algebraic geometry". $\newcommand{\Cplx}{{\mathbb ...
3
votes
2answers
416 views

Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural symplectic structure?

Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$). My questions is: it is always true that we have ...
4
votes
2answers
443 views

Failure of Jacobson Morozov in positive characteristics

The Jacobson-Morozov theorem that any nilpotent $e$ in the lie algebra of a simple algebraic group $G$ can be embedded in an $sl_2$-triple, has a restriction (in terms of the coxeter number) on the ...
4
votes
2answers
167 views

Module in category O not generated by a finite set of HWVs.

For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category $\mathcal O$" under the impression that any module in $\mathcal O$ has a finite ...
5
votes
2answers
471 views

Kostant's theorem on invariant polynomials in positive characteristic

Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of ...
9
votes
2answers
440 views

Does any identity holding in all finite-dimensional Lie algebras hold in all Lie algebras?

Equivalently, is the free Lie algebra on finitely many generators over a fixed field $k$ (say of characteristic not equal to $2$) residually finite-dimensional in the sense that any nonzero element ...
6
votes
0answers
154 views

The meaning of a “subcomplex” of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex $(\wedge^{\cdot} \mathfrak{g}^* ...
4
votes
0answers
130 views

The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
10
votes
5answers
645 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 ...
3
votes
2answers
428 views

German term for “restricted Lie algebra” ?

Can anyone tell me the German term for "restricted Lie algebra" ? Many thanks in advance ! Kind regards, Stephan Kroneck.
2
votes
1answer
184 views

Decomposition of Lorentz-like groups

When studying the Lorentz group $O(1,3)$, one can decompose it into four parts... physicist usually called these Proper-orthochronuos $\mathscr{L}^{\uparrow}_+$, Proper-asynchronous ...
5
votes
0answers
137 views

On an interesting subalgebra of the functions on the cotangent bundle of the flag variety

Setup Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the ...
5
votes
3answers
643 views

Nilpotent Lie Algebras

Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. Given $\xi\in\frak{g}$, what is known about the intersection of $im(ad_{\xi})$ (the image of ...
2
votes
2answers
543 views

Maurer-Cartan structure equation derivation

Dear all. I'm a theoretical physicist trying to understand the structure equations and their geometrical significance, this for their gravitational applications. I know the relation between the Lie ...
3
votes
1answer
628 views

A possible mistake in Kac's “Infinite Dimensional Lie Algebras”

I have a paperback 3rd edition and on page 65 you can find Proposition 5.8. My question is about part (c): If $A$ is of indefinite type, then $$ \overline{X} = \{ h \in \{ \frak h_{\mathbb{R}} ...
2
votes
1answer
155 views

Stabilizers and Quotients of a Nilpotent Lie Algebra

Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. What conditions on $\xi\in\frak{g}$ ensure the existence of a (canonical or non-canonical) surjective morphism ...
1
vote
1answer
163 views

Generic Stabilizers in a Nilpotent Lie Algebra

Let $\frak{g}$ be a finite-dimensional nilpotent complex Lie algebra, and consider the adjoint Lie algebra representation of $\frak{g}$. What is known about the Lie-algebraic structure of the ...
1
vote
1answer
203 views

Module given by generators and relations

Let $\frak G$ be a Lie algebra and let $M$ be a $\frak G$-module generated by a vector $v$ satisfying some set of defining relations denoted by $R$. I mean, $M = U(\frak G)/\langle R \rangle$, where ...
5
votes
0answers
110 views

Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$ ?

Let $n\in\mathbb N$. Let $k$ be a commutative ring. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $x_n$ over $k$. (The multiplication in ...