**3**

votes

**1**answer

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

**1**answer

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

**3**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**5**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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 ...