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.

Filter by
Sorted by
Tagged with
141 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 ...
Olivier Bégassat's user avatar
104 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 ...
Richard Borcherds's user avatar
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....
Theo Johnson-Freyd's user avatar
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 (...
Justin Campbell's user avatar
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 ...
Ryan Reich's user avatar
  • 7,173
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 ...
Theo Johnson-Freyd's user avatar
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 $\...
user332's user avatar
  • 3,878
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 ...
Timothy Chow's user avatar
  • 78.1k
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 ...
zroslav's user avatar
  • 1,412
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 ...
Jim Humphreys's user avatar
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)!}...
Theo Johnson-Freyd's user avatar
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 ...
Theo Johnson-Freyd's user avatar
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 ...
Anton Geraschenko's user avatar
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 ...
Saal Hardali's user avatar
  • 7,549
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 ...
Jim Humphreys's user avatar
38 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.
Daniel Erman's user avatar
  • 2,895
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. ...
Nathaniel Bottman's user avatar
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{...
Yonatan Harpaz's user avatar
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 ...
John Baez's user avatar
  • 21.3k
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 $X, Y\in{\mathbb R}^n$?...
Yuval's user avatar
  • 637
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 $M$ ...
José Figueroa-O'Farrill's user avatar
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. ...
Selene Routley's user avatar
32 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 ...
Matthias Künzer's user avatar
31 votes
8 answers
8k 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 (...
Mark.Neuhaus's user avatar
  • 2,004
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 ...
Jim Humphreys's user avatar
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 ...
Theo Johnson-Freyd's user avatar
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 ...
bradhd's user avatar
  • 497
30 votes
4 answers
10k 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?
blt's user avatar
  • 1,203
30 votes
0 answers
989 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" (...
Jim Humphreys's user avatar
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 ...
darij grinberg's user avatar
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 ...
user2013's user avatar
  • 1,653
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. ...
Ben Webster's user avatar
  • 43.9k
27 votes
1 answer
848 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 ...
Will Sawin's user avatar
  • 135k
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....
Qiaochu Yuan's user avatar
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 ...
algori's user avatar
  • 23.2k
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, ...
Noah Snyder's user avatar
  • 27.8k
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 $(p(n))_{n\in \mathbb{N}}$...
Dietrich Burde's user avatar
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 ...
David Jordan's user avatar
  • 6,053
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 ...
Nadia SUSY's user avatar
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 ...
mikhail skopenkov's user avatar
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 $g=\exp(x_1)\cdots\exp(x_n)$ ...
Lorenz Haber's user avatar
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 ...
Jan Weidner's user avatar
  • 12.8k
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 ...
Pierre's user avatar
  • 2,145
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 ...
Paul Siegel's user avatar
  • 28.8k
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: $\...
Yuji Tachikawa's user avatar
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 ...
Benjamin's user avatar
  • 2,069
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 ...
Allen Knutson's user avatar

1
2 3 4 5
44