Good Math, Bad Math

The Mapping of the E8 Lie Group (Minor Update)

Posted by Mark C. Chu-Carroll on March 19, 2007
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I’ve been getting tons of mail from people in response to the announcement of the mapping of
the E8 Lie group, asking what a Lie group is, what E8 is, and why the mapping of E8 is such a big deal?

Let me start by saying that this is way outside of my area of expertise. So I fully expect that I’ll manage to screw something up as I try to figure it out and explain it – so do follow the comments, where I’m sure people who know this better than I do will correct whatever errors I make.

Let’s start with the easy part. What’s a Lie group? Informally, it’s a group whose objects
form a manifold, and whose group operation is a continuous function. We can break that down a bit, to make it a little bit clearer.

A group is a set of objects/values with a single binary operator that has a certain set of basic properties: associativity, existence of inverse, existence of identity. It’s one of the simplest constructions of abstract algebra. What’s really fascinating about it is that that simple construction – the set plus one operation will a simple set of properties – defines the entire concept of symmetry.

Groups don’t normally require any structure on their members beyond what’s required to make the group operator work properly. You can define a group whose values are a set of points, a set of numbers, a set of coins – very nearly anything you want.

But there are certain structured sets of values that we care about, which you can
use as the objects for a group. One of those is a topological space. A topological space is
just a collection of objects which have a kind of nearness/adjacency relationship between
the objects in the collection. So a group on a topological space is interesting, because what it does is define symmetry on a set of values that preserves the nearness/adjacency relationships
of the objects in the space.

Even more interesting, we can define a particular kind of topological space: a manifold, which is a sort of “smooth” topological space: a manifold is a topological space where the structure of the nearness/adjacency relations makes every small finite region of the space appear to be Euclidean.

So a Lie group is a group whose objects form a manifold, and whose group operations preserve
the manifold structure of the nearness/adjacency relations.

Moving on – what’s E8?

i-65657f78710813fc40102c13cb120239-e8plane2a.jpg

Many lie groups are based on topological spaces that whose values are representable as some collection of matrices or groups. E8 is one of those – it’s a group based on something called a root system. The root system for E8 consists of a set of 8-dimensional vectors, which fall into two families. One family consists of all 8-dimensional vectors, with
2 unit-length elements, and 6 0-length elements; things like (1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (0, -1, 0, 0, 0, 0, 0, 1), etc. The other family consists of all of the 8-dimensional
vectors whose elements are all either +1/2 or -1/2, where the sum of all of the elements are even. So (1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2) is a member of the root system, since the sum of those elements is 0; (1/2, 1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2) is not an element of the root, system, since it’s sum is 1. The beautiful image over to the right is the image of the root system of E8.

The E8 Lie group is based on that root system – it’s a massive structure with one complex dimension (complex as in complex numbers – it’s value in each dimension is a complex number) for each of the members of the root system. So its a manifold with 248 complex dimensions, or 496 real dimensions.

There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.

The other reason is just that the complete mapping of E8 is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes to store the map of E8. If you were to write it out on paper in 6-point print (that’s really small print), you’d need a piece of paper bigger than the island of Manhattan. This thing is huge.

Update: For those who claim that mathematicians have no sense of himor, I heard via Gooseana that the title of the formal presentation where they’ll be talking about the E8 map is: “The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness”.

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Comments

  1. #1 Bob O'H
    March 19, 2007

    Huh? I thought this was going to be another post about Egnor.

    Oh, it’s a subscript 8, not a g!

    Bob

  2. #2 Mike Davis
    March 19, 2007

    Thanks for your informative post and your interesting blog. After reading twice I get the nouns and adjectives, but I’m still not clear on the verb. What does it mean to map a Lie group? From or to what is it mapped, and what does the mapping involve.

    Thanks again, –mad

  3. #3 Mark C. Chu-Carroll
    March 19, 2007

    Mike:

    Sorry, this probably should have been included in the original post.

    The idea is that you can decompose the group into finite number of discrete fundamental building blocks – and then create an exhaustive description of exactly what those blocks are, what properties they have, and what the relationships are between any pair of those building blocks. It’s the same sort of idea as the image of the root system – the root system describes how to generate the full lie group. That image shows a two dimensional projection of the root system vectors and their relationships. The full map is conceptually similar, but with even more information.

  4. #4 Douglas
    March 19, 2007

    Not authoritative here, but things important to understanding a group structure:

    Its subgroup lattice – the relationship between subsets that are themselves groups.

    Its possible representations – homomorphisms onto matrices over vector spaces – that’s where dimension usually enters into discussion of a group.

    Its character - which is a function mapping from the group to the complex numbers, in a way that converts the group operation to complex multiplication. (Think of e^x.)

    I believe that’s what’s of interest here.

  5. #5 Mike Davis
    March 19, 2007

    Thanks, Mark & Douglas.

  6. #6 Susan B.
    March 19, 2007

    And, er, what precisely is a manifold? I have a sort of conceptual notion, but defines it mathematically?

  7. #7 Mark C. Chu-Carroll
    March 19, 2007

    Susan:

    It’s actually in the article, but I’ll repeat it here.

    Intuitively, a manifold is a “smooth” topological space. It’s a space where any small region of the space looks like it’s a euclidean space, even though the entire thing isnt.

    So, think of something like a sphere. A sphere is a simple manifold. Take any small region of it, and it looks just like a little section of a euclidean plane. But look at the whole sphere, and it’s smooth, but it’s definitely not euclidean.

  8. #8 Douglas
    March 19, 2007

    And a counterexample: A circle is as much a manifold as a sphere, yes? (One dimensional instead of two, though.) Little sections look like a Euclidean line. Now consider a figure eight – the point of intersection doesn’t, can’t, look like a piece of line. Too many neighbors.

  9. #9 Mikael Johansson
    March 19, 2007

    Ummmm, having kinda missed the announcement as such – what exactly has been mapped? The entire group? The root system connected to it? What does the mapping consist of?

  10. #10 Brett
    March 19, 2007

    Saying that E_8 “seems to be part of the structure of our universe” is inaccurate. There are string theories that use two copies of E_8, but there are other string models too; and neither the E_8 cross E_8 models nor any of the others look anything like physics as we know it. There may be some way to get around this, but that’s highly conjectural.

    Second, and more peripherally, groups and their representations are not sufficient to describe all types of symmetry. Supersymmetry, for example, does not have a group structure.

  11. #11 Coin
    March 19, 2007

    Mark, I’m still trying to understand all of this, but– is the “mapping” described here something like a Cayley graph, then? Are such mappings readily available for other lie groups?

    (Lie groups are pretty confusing and resources for understanding them don’t seem to be all that abundant. I’ve been trying off and on to understand the SU(n) symmetry groups for a couple months now and am making slow or no headway so far. I keep thinking this would all be easier if there were some clear or common way for visualizing complicated lie groups.)

  12. #12 Coin
    March 19, 2007

    Supersymmetry, for example, does not have a group structure.

    Why not?

    I mean, is there some property of supersymmetry that is incompatible with the group axioms, or what?

  13. #13 Mark C. Chu-Carroll
    March 19, 2007

    Coin:

    I’m sorry, but I just don’t know enough to answer the question. Like I said at the beginning of the post, I’m really out of my depth on this one. I haven’t studied Lie groups, so what I knew about them before I started researching for this post today was that they’re groups based on manifolds. I don’t know enough about the structure to make sense out of the maps. I can grasp the basic concept enough to write a sort-of glossy post like this, but I really don’t know the deep stuff this time.

    Sorry.

  14. #14 Coin
    March 19, 2007

    That’s fine, thanks anyway and thanks for your honesty on this :)

  15. #15 John Armstrong
    March 20, 2007

    Mikhail (btw: I keep wanting to say “Misha”…), what they’ve calculated is the Kazhdan-Lusztig-Vogan polynomials, which count.. well..

    This is sort of embarassing given who my advisor is, but I’m not exactly sure. I know it’s related to character theory, if you remember that from representations of finite groups. I’ve actually sent an email to Jeff to ask him if, as long as I’m in Maryland anyhow, he has a time I can stop by his office for him to tell me what the K-L-V polynomials are and what they’re good for. Failing that, I’m sure Zuckerman is talking with Vogan this week (he said he’d be in Boston), and he’ll be able to say something when I get back to New Haven. When I hear something I can digest into an accessible-to-most-mathematicians form I’ll post it on my blog.

    Basically, the upshot is that if I didn’t do knot theory I’d be right in the middle of this stuff, and as it is I have enough contacts that I can give some good reports from them. Anyone who’s interested can find my digestions of their comments at my place.

  16. #16 John Armstrong
    March 20, 2007

    Coin, supersymmetry is based on what’s called a “super Lie algebra”

    Okay, here’s the quick and dirty version: a Lie group is a manifold and a group together, and the two structures interact. The group structure induces a certain structure on the tangent space at the identity of the group. The classical case to consider is the group of invertible n-by-n matrices over the complex numbers. The tangent space at the identity is basically the space of all n-by-n matrices. The multiplicative group structure on invertible matrices leads to the “Lie algebra” structure given by the vector space of matrices and the “Lie bracket” [A,B] = AB – BA. This is antisymmetric and satisfies the “Jacobi identity”

    [A,[B,C]] = [[A,B],C] + [B,[A,C]]

    Deep breath!

    Lie algebras are what physicists really study a lot of the time because they talk about “infinitesimal” symmetries, and we can “integrate” infinitesimal changes to get actual transformations. A Lie algebra in the abstract is a vector space with a bracket operation satisfying antisymmetry and the Jacobi identity above.

    A super Lie algebra is the same thing, but built on a “super-vector space”. That’s just a pair of vector spaces, one having “degree 0″ and the other having “degree 1″. That is: one vector space for each element of the group Z_2. We call this a “Z_2-graded vector space”. We now add a bracket operation like for a Lie algebra, but we have to pay attention to the degree. When we bracket two vectors we add their degrees modulo 2. If they have the same degree we get something in degree 0, otherwise we get degree 1.

    We also have to change the constraints. Antisymmetry is replaced by “super-anticommutativity”:

    [A,B] = (-1)deg(A)deg(B)+1[B,A]

    Do you see particle statistics yet? Bosons live in degree 0, so their operators commute and satisfy anticommutation relations with everything. Fermions live in degree 1 so their operators anticommute and satisfy commutation relations with each other. The Jacobi identity also has to be modified:

    [A,[B,C]] = [[A,B],C] + (-1)deg(A)deg(B)[B,[A,C]]

    Now we go backwards: a Lie superalgebra is the tangent space at the identity of a Lie supergroup, which is a group structure on a supermanifold, which is something that “locally looks like a super-vector space”. Of course now “locally looks like” is interpreted in terms of the sheaf of rings of “smooth functions” on the manifold, which is now a sheaf of supercommutative algebras, instead of commutative algebras for regular manifolds.

    The punchline: supersymmetry transformations are described by a Lie supergroup, not a Lie group. Lie supergroups are analyzed by Lie superalgebras, not Lie algebras. E8 is a Lie algebra, not a Lie superalgebra.

  17. #17 Mikael Johansson
    March 20, 2007

    John:
    First off – it’s neither ‘Mikhail’ nor ‘Misha’ – I’m swedish, not russian, so it’s ‘Mikael’, and use ‘Michi’ as a webnick based on my german nickname.

    Furthermore, as far as I understand it, “super-” in algebras just means “graded commutative” with possibly restricted to just a Z/2-grading. Not really a big deal, and rather weird to put that name on it – but I blame physicists.

  18. #18 Mark C. Chu-Carroll
    March 20, 2007

    John:

    Drop me a note or comment here when/if you get around to writing that post; I’ll post a link to it.

  19. #19 Blake Stacey
    March 20, 2007

    Now that I have some server space and a working MovableType installation, I just have to get itex2MML up and running, and then I can offload what I know about superalgebras: commutators, anticommutators, and how to use the grandchild of string theory to solve the hydrogen atom. . . .

  20. #20 Brett
    March 20, 2007

    It’s perfectly true that there’s nothing particularly special about the mathematical structure of superalgebras. (Supergroups are another story.) However, supersymmetry (from which the structures take the “super-” in their names) itself really is an amazing thing, and its discovery overturned a century’s thinking about how symmetries could work.

    I also want to note an important correction to John Armstrong’s post. For two objects, the super-commutator is defined to be the commutator bracket [A,B] = AB – BA unless both A and B have degree one, in which case it’s the anticommutator {A,B} = AB + BA. Bosonic (degree zero) operators satisfy commutation relations, and fermionic (degree one) operators satisfy the anticommutation relation. Mr. Armstrong appears to have been confused by the relationship of these operators to boson and fermion particles.

  21. #21 Spoonwood
    March 20, 2007

    [So a Lie group is a group whose objects form a manifold, and whose group operations preserve the manifold structure of the nearness/adjacency relations.]

    Talk about linear.

  22. #22 Torbjörn Larsson
    March 20, 2007

    its discovery overturned a century’s thinking about how symmetries could work.

    I would like to thank Mark, John, Brett and the rest for a quick peek into our world of symmetries and (super)group structures. At this point when it comes to all our symmetries or dualities I feel like I’m looking through a smoked glass, dimly – I can’t tell what they all are but I feel I can recognize them when I see them. (At least with a little help here and there.)

    I blame physicists

    Always blame physicists. Of course, in this case it could be that they were first. :-)

    (I’m guessing here, to make the joke. So please correct me if I’m wrong.)

    I’m swedish

    Perhaps it is time for The Standard Lesson in Swedish Names Which Perhaps Can Give a Hint.

    The surname is telling AFAIK, for example in my case we have the genitive case “Lars son” (“Lars’ son” in english), i.e, the possessive case of ‘son of Lars’ derived in the original patronymic.

    While in US I can see the simplified “Larson” which would be the funny-looking ‘son of Lar’. In Mikaels’ case it would possibly be the more awkward ‘son Johan’ hidden in “Johanson”. In either case, these spellings are still rather rare here.

  23. #23 John Armstrong
    March 20, 2007

    Oh I’m terribly sorry, Mikael.. I was very tired last night and was basically on math-only mode. Think of it as my BIOS. mea culpa, mea culpa, mea maxima culpa.

  24. #24 John Armstrong
    March 20, 2007

    Brett: sorry, I did get the boson (commutation) and fermion (anticommutation) sides swapped. I think you’ll find that my formulæ do give the right signs, though. If either term has degree 0 then the bracket is antisymmetric, while if both are degree 1 then the bracket is symmetric. That’s what you get in the (very special) case where the bracket is given by a commutator or anticommutator as you wrote.

  25. #25 Craig Helfgott
    March 20, 2007

    One of the important things about this result is that it completes (to a certain extent) the “atlas” of finite-dimensional Lie groups.

    There are two important concepts to explain here, and I’ll start with the easy one. Finite-dimensional Lie groups were classified in the early part of the century by Cartan & Weyl. They fall into 4 infinite families (two of which can be thought of as the “rotations” in n dimensions), and 5 “exceptional” groups, of which the biggest is E_8. Because of the regular structure of the infinite families, you can (in theory) calculate anything you want about them without too much difficulty. The exceptionals are somewhat trickier. You can get from E_8 to the other 4 exceptional groups; so (again in theory) if you can calculate things for E_8, you can calculate them for the others.

    What sort of things would we want to calculate? Well, the big thing is something called a “character”. This is closely related to something called a “representation”. The idea behind a representation is that, while you can write the set of rotations in 3D (SO(3)) as a set of 3×3 matrices, you can also write it as (for example) a set of 5×5 matrices that have the same multiplication table. This represents one possible action of this group on a 5-D object.

    So we’d like to know the set of characters of representations, and, although I don’t understand much of the details, that is part of their main result.

    Now, what’s the second concept I wanted to explain? The “atlas”. This is related to another math research program that took ~50 years to complete: The Atlas of Finite Simple Groups. Now, finite groups are not Lie groups. They are (for one thing) discrete sets of symmetries as opposed to continuous set of symmetries (like the symmetries of the cube vs those of the sphere). As it turns out, the finite groups can be built up out of atomistic building blocks call finite simple groups. These f.s.g’s fall into a number of infinite families, and about 20-odd “sporadic” groups. Sound familiar?

    The Atlas was a project to get (among other things) the characters of the finite simple groups.

    This new project extends the Atlas to finite-dimensional Lie groups.

    Now, I’m not sure how characters work for Lie superalgebras, but I imagine that’s the next logical step.

  26. #26 Michael
    March 20, 2007

    There’s a longer discussion of this topic at:
    http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html

  27. #27 Coin
    March 20, 2007

    John, I think I’m going to have to read that a few times before I really get it, but thanks for the explanation.

  28. #28 Ørjan Johansen
    March 20, 2007

    For those readers who haven’t heard about Lie algebras before and think it is terribly advanced: If you know vectors, then you have probably seen an example.

    The crossed product of 3-dimensional vectors is a Lie algebra. One of the corresponding Lie groups is the group of rotations of a sphere (denoted by SO(3)).

    This is not the canonical (“simply connected”) Lie group for this algebra, however. That is the SU(2) group which quantum physicists will know about. It is somewhat like the rotations of a sphere, except that you don’t just consider which points of the sphere map to each other by a rotation, but additionally keep track of whether you have rotated the sphere entirely an even or odd number of times.

  29. #29 P. Sternberg
    March 20, 2007

    This result is rather near and dear to my heart, as it is a question resolved by combinatorial representation theory (which was my Ph.D. topic) and made use of massive parallel computation (which is what I’m currently doing a postdoc on).

    I’d like to point out that the “pictures of representations” that they give are more specifically crystal bases. Crystal basis theory is about 15-20 years old, and is still a very active and exciting topic, mostly dealing with representations of infinite-dimensional algebras.

    To give a little more detail, crystals are colored directed graphs that explicitly encode the action of about two thirds of the algebra on the vector space, and implicitly encode the action of the other third.

    If you want to see more examples of crystals, (although not for E8), check out

    http://math.ucdavis.edu/~sternberg/crystalview/

  30. #30 Deane
    March 20, 2007

    Your description of a Lie group is the same one I learned as a student, namely that it’s a group that happens to be a manifold or a manifold that happens to be a group (where group multiplication is a smooth map). I don’t know about you, but I always found this definition to be mysterious and unsatisfying, even though I am a differential geometer.

    I think the definition looks a lot more reasonable if you say the following: Almost any Lie group is a set of n-by-n matrices that is closed under multiplication and taking inverses and that is also a smooth submanifold of the space of all n-by-n matrices. There are many important concrete examples, like the set of all invertible n-by-n matrices (also known as GL(n)), or the space of orthogonal n-by-n matrices (i.e., linear transformations that preserve the Euclidean inner product and also known as O(n)).

    The associated Lie algebra also becomes completely concrete as the set of all possible velocity vectors at the identity of curves inside the Lie group that start at the identity. For example, using this definition, it is easy to verify that the Lie algebra associated with O(n) is the set of skew-symmetric n-by-n matrices.

    In fact, if you refuse to look at anything but matrix Lie groups, a lot of things that seem very abstract and obscure suddenly become concrete and easy to prove. For most of us, these are the only Lie groups we’ll ever need and therefore we never need the abstract definition of a Lie group.

  31. #31 John Armstrong
    March 21, 2007

    Deane, there’s a problem with that: what you’re talking about are the classical algebraic groups, which just happen to be Lie groups. There are plenty of Lie groups other than those, and they’re not even the only ones Sophus Lie considered.

    Anyhow, I go completely the other way. Matrices are horrible awful things to ever use unless you absolutely have to. I had to assume a certain module was free once so I could pick a basis just to finish writing the damned paper. I haven’t felt clean since.

    As far as I’m concerned, a Lie group is a group object in the category of smooth manifolds. It’s about the most natural thing you can do and doesn’t ask that I do anything so horrendously non-canonical as pick a basis and look at actual elements.

  32. #32 JP
    March 21, 2007

    It is extremely embarrassing to me that one of my current research projects involves e8, yet I (and my local colleagues) have no idea how this announcement might be related to our work. Time to do some reading…

    Also, dimension 453,060 is not /that/ big any more. One of my last projects involved computing the rank of a (dense) dimension 3^12=531,441 matrix, and we are hard at work on the dimension 3^14=4,782,969 case (granted, our matrices live in GF_3 which makes life a little easier).

  33. #33 Doug
    March 21, 2007

    Terry Gannon has two related works on the E8 relation to the Leech Lattice, the Monster and [string] physics:

    1 – arXiv paper ‘Monstrous Moonshine: the first 25 years’ [33 pages with 124 references].
    http://www.arxiv.org/PS_cache/math/pdf/0402/0402345.pdf

    Lieven le Bruyn of ‘Never Ending Books’ website, refers to this as a survey paper
    http://www.neverendingbooks.org/?p=133

    2 – The arXiv paper [1] appears to form the framework of the Gannon book, ‘Moonshine Beyond the Monster’ [477 pages, 575 reference].

  34. #34 Torbjörn Larsson
    March 22, 2007

    Jacques Distler describes the work from a physicists perspective:

    First of all, you have to realize what is being talked about is not our friend, the compact Lie group, E 8, but a distant cousin, the split real form, which I will, henceforth, denote by E˜ 8.
    [...]
    Anyone familiar with the heterotic string will recognize the “E” series of compact Lie groups:
    E 8,E 7,E 6,E 5=Spin(10), E 4=SU(5),E 3=SU(3)×SU(2)
    .
    The corresponding split real forms
    E˜ 8,E˜ 7,E˜ 6,E˜ 5=Spin(5,5), E˜ 4=SL(5),E˜ 3=SL(3,R)×SL(2,R)
    appear in the maximal supergravity theories (the dimensional reductions of 11 dimensional supergravity down to d=11−n dimensions).
    [...]
    If you are interested in studying the spectrum of such BPS states (say, to write down a U-duality-invariant formula for the entropy of blackholes in this theory), then you are interested in the representation theory of E˜ n.

    ( http://golem.ph.utexas.edu/~distler/blog/archives/001213.html )

  35. #35 Pinner Blinn
    March 27, 2007

    Article about Vogan in the Boston Globe:

    http://www.boston.com/news/globe/health_science/articles/2007/03/26/his_mind_is_on_the_eighth_dimension/

  36. #36 John Baez
    March 30, 2007

    Mike Davis wrote:

    What does it mean to map a Lie group?

    The main thing to keep in mind is that when you heard people talk of “mapping” E8, they were speaking in a poetic sort of way, not using this word in any technical sense. It’s supposed to remind of you of “mapping the trackless wastes of the Sahara desert”.

    If you want to get technical, the ATLAS team didn’t really “map” E8: they computed the Kazhdan-Lusztig-Vogan polynomials of its split real form. For more on what that means, try this and then this.

  37. #37 Mike Davis
    April 27, 2007

    Belated thanks John for your reply and links in post #36.

  38. #38 JeffyPooh
    November 18, 2007

    As I’m sure you know by now, E8 is back in the news.

    http://exceptionallysimpletheoryofeverything.blogspot.com/