Questions tagged [lie-algebras]
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 groups and differentiable manifolds.
2,180
questions
142
votes
14
answers
47k
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 ...
105
votes
3
answers
9k
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 ...
79
votes
26
answers
6k
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, ...
74
votes
7
answers
19k
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 discussion....
72
votes
6
answers
7k
views
Surprisingly short or elegant proofs using Lie theory
Today, I was listening to someone give an exhausting proof of the fundamental theorem of algebra when I recalled that there was a short proof using Lie theory:
A finite extension $K$ of $\mathbb{C}$...
67
votes
11
answers
12k
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 (...
59
votes
8
answers
12k
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 ...
58
votes
3
answers
4k
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 ...
54
votes
9
answers
9k
views
Nice proofs of the Poincaré–Birkhoff–Witt theorem
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$, with an ordered basis $x_1 < x_2 < ... < x_n$.
We define the universal enveloping algebra $U(\mathfrak{g})$ of $\...
53
votes
5
answers
9k
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 ...
52
votes
5
answers
7k
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 ...
50
votes
5
answers
9k
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 ...
45
votes
16
answers
8k
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 \frac{(-x)^n}{(n+1)!}...
44
votes
2
answers
3k
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 ...
41
votes
9
answers
6k
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 ...
40
votes
1
answer
4k
views
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 ...
39
votes
2
answers
2k
views
Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?
In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031),
Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the ...
38
votes
1
answer
4k
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. ...
37
votes
18
answers
23k
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.
37
votes
3
answers
3k
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 $\mathfrak{...
34
votes
2
answers
3k
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 ...
34
votes
1
answer
1k
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 ...
33
votes
2
answers
1k
views
A question about subspace in ${\bigwedge}^2({\mathbb R}^n)$
Let $E$ be a linear subspace of ${\bigwedge}^2({\mathbb R}^n)$. What is the minimal dimension of $E$ that guarantees $E$ contains a nonzero element of the form $X\wedge Y$, with $X, Y\in{\mathbb R}^n$?...
33
votes
1
answer
4k
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$ ...
33
votes
3
answers
6k
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. ...
31
votes
8
answers
9k
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 (...
31
votes
4
answers
3k
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 ...
30
votes
7
answers
7k
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 ...
30
votes
3
answers
6k
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 ...
30
votes
4
answers
11k
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?
30
votes
0
answers
991
views
Follow-up to Steinberg's problem (12) in his 1966 ICM talk?
Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
29
votes
4
answers
3k
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 ...
28
votes
3
answers
5k
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 ...
27
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)$ ...
27
votes
1
answer
2k
views
Is there an accessible exposition of Gelfand-Tsetlin theory?
I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...
27
votes
1
answer
851
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 ...
26
votes
3
answers
4k
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 manner....
26
votes
2
answers
4k
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 ...
26
votes
1
answer
2k
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, ...
25
votes
3
answers
2k
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 $(p(n))_{n\in \mathbb{N}}$...
25
votes
4
answers
6k
views
Formal geometry
[Edit (June 20, 2010): I posted an answer to this which summarizes one that I received verbally a few weeks after posting this question. I hope it is useful to someone.]
I am presently seeking ...
25
votes
1
answer
691
views
The de Rham complex of the octonionic projective spaces
The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
24
votes
6
answers
5k
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 ...
24
votes
2
answers
1k
views
Lie groups generated by finitely many Lie algebra elements
Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A standard fact is that $G$ is generated by $\exp(\mathfrak{g})$, i.e. every $g \in G$ can be written as $g=\exp(x_1)\cdots\exp(x_n)$ ...
23
votes
6
answers
7k
views
Introduction to W-Algebras/Why W-algebras?
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 ...
22
votes
6
answers
2k
views
About the definition of E8, and Rosenfeld's "Geometry of Lie groups"
I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...
22
votes
5
answers
7k
views
The Jacobi Identity for the Poisson Bracket
It is well known that if $M, \Omega$ is a symplectic manifold then the Poisson bracket gives $C^\infty(M)$ the structure of a Lie algebra. The only way I have seen this proven is via a calculation in ...
22
votes
3
answers
2k
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: $\...
22
votes
5
answers
2k
views
Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
22
votes
1
answer
2k
views
Modern reference for maximal connected subgroups of compact Lie groups
What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line?
I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits ...