# Tagged Questions

**83**

votes

**3**answers

6k views

### Has the Lie group E8 really been detected experimentally?

A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum ...

**71**

votes

**13**answers

18k views

### why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...

**59**

votes

**26**answers

5k views

### What would you want on a Lie theory cheat poster?

For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...

**45**

votes

**3**answers

2k views

### Is “semisimple” a dense condition among Lie algebras?

The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some ...

**43**

votes

**7**answers

7k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...

**40**

votes

**15**answers

5k views

### What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty ...

**34**

votes

**5**answers

3k views

### Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...

**34**

votes

**2**answers

2k views

### What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...

**34**

votes

**1**answer

999 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 ...

**33**

votes

**11**answers

4k views

### What is significant about the half-sum of positive roots?

I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible.
Let $\mathfrak{g}$ be a semisimple Lie algebra ...

**33**

votes

**7**answers

4k views

### Why the Killing form?

I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...

**32**

votes

**4**answers

3k views

### What role does the “dual Coxeter number” play in Lie theory (and should it be called the “Kac number”)?

While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into ...

**31**

votes

**1**answer

2k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**30**

votes

**1**answer

534 views

### Is there a geometric construction of hyperbolic Kac-Moody groups?

Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras was connected to ...

**28**

votes

**8**answers

5k views

### “Modern” proof for the Baker-Campbell-Hausdorff formula

Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula?
All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and
are not at all geometric ...

**28**

votes

**1**answer

783 views

+250

### Roadmap to Geometric Representation Theory (leading to Langlands)?

I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own.
I'm becoming increasingly fascinated by stuff ...

**26**

votes

**4**answers

4k views

### Motivating the Casimir element

Weyl's theorem states that any finite-dimensional representation of a finite-dimensional semisimple Lie algebra is completely reducible. In my mind, the "natural" way to prove this result is by way ...

**26**

votes

**1**answer

504 views

### Why do the adjoint representations of three exceptional groups have the same first eight moments?

For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...

**25**

votes

**8**answers

3k views

### Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up.
In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...

**25**

votes

**2**answers

2k views

### Why do Lie algebras pop up, from a categorical point of view?

Groups pop up as automorphism groups in any category.
Rings pop up as endomorphism rings in any additive category.
Is there a similar way to attach a Lie algebra to an object in a category of a ...

**25**

votes

**4**answers

2k views

### A mysterious Heisenberg algebra identity from Sylvester, 1867

I am trying to understand two papers by James Joseph Sylvester:
P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of ...

**24**

votes

**2**answers

3k views

### Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...

**24**

votes

**1**answer

1k views

### Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?

The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...

**23**

votes

**2**answers

809 views

### Why should affine lie algebras and quantum groups have equivalent representation theories?

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra ...

**22**

votes

**6**answers

3k views

### Nice proofs of the Poincaré–Birkhoff–Witt theorem

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra with an ordered basis $x_1 < x_2 < ... < x_n$.
We define the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ to be the ...

**22**

votes

**1**answer

1k views

### What's the status of the following relationship between Ramanujan's $\tau$ function and the simple Lie algebras?

Qiaochu asked this in the comments to this question. Since this is really his question, not mine, I will make this one Community Wiki. In MR0522147, Dyson mentions the generating function $\tau(n)$ ...

**21**

votes

**3**answers

1k views

### What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra?

This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras:
37602. Like some others who started graduate study in ...

**20**

votes

**17**answers

12k views

### Learning about Lie groups

Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.

**20**

votes

**7**answers

4k views

### Why is Lie's Third Theorem difficult?

Recall the following classical theorem of Cartan (!):
Theorem (Lie III): Any finite-dimensional Lie algebra over $\mathbb R$ is the Lie algebra of some analytic Lie group.
Similarly, one can propose ...

**19**

votes

**1**answer

2k views

### Isometry group of a homogeneous space

Background
Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...

**19**

votes

**2**answers

1k views

### “isotropic” subspaces of a simple Lie algebra

Let $\bf g$ be a finite-dimensional real simple Lie algebra of compact type and let $\left<-,-\right>$ denote the positive-definite inner product induced from the negative of the Killing form. ...

**18**

votes

**3**answers

2k views

### Why the BGG category O?

Given a finite-dimensional semisimple complex Lie algebra $\mathfrak{g}$, the Bernstein-Gelfand-Gelfand category $\mathcal O$ is the full subcategory of $\mathfrak g$-modules satisfying some ...

**18**

votes

**3**answers

981 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 ...

**18**

votes

**6**answers

622 views

### Automorphism group of real orthogonal Lie groups

I would like to know what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$. My working definition of $Out$ is as follows:
Let us denote by $Aut(G)$ the ...

**18**

votes

**2**answers

623 views

### Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants

Recall that there are knots in $\mathbf{R}^3$ that are not invertible, i.e. not isotopic to themselves with the orientation reversed. However, it is not easy to tell whether or not a given knot is ...

**18**

votes

**1**answer

296 views

### On a drawing in Dixmier's Enveloping Algebras

This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are ...

**17**

votes

**6**answers

3k views

### Pythagorean 5-tuples

What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...

**17**

votes

**3**answers

1k views

### Number of triples of roots (of a simply-laced root system) which sum to zero

In a paper 1105.5073, the authors took a simply-laced root system $\Delta$ of type $G=A,D,E$, and then counted the number of unordered triples $(\alpha,\beta,\gamma)$ of roots which sum to zero: ...

**17**

votes

**2**answers

530 views

### Generators of the cohomology of a Lie algebra

Fix a characteristic zero ground field. One can easily check that if $\mathfrak g$ is a simple Lie algebra, then the trilinear map map $\omega$ given by $$\omega(x,y,z)=B([x,y],z),$$ with $B$ the ...

**16**

votes

**5**answers

547 views

### Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?

Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: ...

**16**

votes

**3**answers

842 views

### Is a retract of a free object free?

I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?

**16**

votes

**2**answers

867 views

### Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...

**16**

votes

**1**answer

393 views

### What is the homomorphism between the third exterior and third symmetric power of the adjoint representation of a simple Lie algebra?

Let $\mathfrak{g}$ be the adjoint representation of a simple Lie algebra (which is not of type $A$). Then the space of intertwiners between the third exterior power of $\mathfrak{g}$ and the third ...

**16**

votes

**1**answer

2k views

### Gap in an argument in Fulton & Harris?

I'm reading through the two chapters in Fulton and Harris on the representation theory of $\mathfrak{sl}(3,\mathbb{C})$, in preparation for lecturing on them this week. I'll use F&H's notation, ...

**16**

votes

**2**answers

1k views

### Invariants for the exceptional complex simple Lie algebra $F_4$

This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise.
Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...

**15**

votes

**5**answers

4k views

### Introduction to W-Algebras/Why W-algebras?

Hi,
does anyone know of an introduction and motivation for W-algebras?
Edit: Okay, sorry I try to add some more background. W algebras occur, for example when you study nilpotent orbits: Take a nice ...

**15**

votes

**4**answers

5k views

### Outer automorphisms of simple Lie Algebras

There is, of course, a complete classification for simple complex Lie algebras. Is there a good reference which lists the group of outer automorphisms for each?

**15**

votes

**3**answers

1k views

### How are these two ways of thinking about the cross product related?

I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free ...

**15**

votes

**3**answers

2k views

### When is a finite dimensional real or complex Lie Group not a matrix group

I have a smattering of knowledge and disconnected facts about this question, so I would like to clarify the following discussion, and I also seek references and citations supporting this knowledge. ...

**15**

votes

**4**answers

1k views

### homotopy type of connected Lie groups

Is there a simple proof (short and low-tech) of the following fact:
(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to
$K\times\mathbb{R}^n$ where $K$ is a maximal ...