Week 5
I think I'll start out this week's list of finds with an elementary
introduction to Lie algebras, so that people who aren't "experts" can
get the drift of what these are about. Then I'll gradually pick up
speed...
1) Vyjayanthi Chari and Alexander Premet, Indecomposable restricted
representations of quantum sl_2, University of California at Riverside
preprint.
Vyjayanthi is our resident expert on quantum groups, and Sasha, who's
visiting, is an expert on Lie algebras in characteristic p. They have
been talking endlessly across the hall from me and now I see that it has
born fruit. This is a pretty technical paper and I am afraid I'll never
really understand it, but I can see why it's important, so I'll try to
explain that!
Let me start with the prehistory, which is the sort of thing everyone
should learn. Recall what a Lie algebra is... a vector space
with a "bracket" operation such that the bracket [x,y] of any two vectors
x and y is again a vector, and such that the following hold:
a) skew-symmetry: [x,y] = -[y,x].
b) bilinearity: [x,ay] = a[x,y],
[x,y+z] = [x,y] + [x,z]. (a is any number)
c) Jacobi identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.
These conditions, especially the third, may look sort of weird if you
are not used to them, but the examples make it all clear. The easiest
example of a Lie algebra is gl(n,C), which just means all nxn complex
matrices with the bracket defined as the "commutator":
[x,y] = xy - yx.
Then straightforward calculations show a)-c) hold... so these conditions
are really encoding the essence of the commutator.
Now recall that the trace of a matrix, the sum of its diagonal entries,
satisfies tr(xy) = tr(yx). So the trace of any commutator is zero, and
if we let sl(n,C) denote the matrices with zero trace, we see that it's a
sub-Lie algebra of gl(n,C) - that is, if x and y are in sl(n) so is [x,y],
so we can think of sl(n,C) as a Lie algebra in its own right. Going from
sl(n,C) to gl(n,C) is essentially a trick for booting out the identity
matrix, which commutes with everything else (hence has vanishing
commutators). The identity matrix is the only one with this property,
so it's sort of weird, and it simplifies things to get rid of it here.
The simplest of the sl(n,C)'s is the Lie algebra sl(2,C), affectionately
known simply as sl(2), which is a 3-dimensional Lie algebra with a
basis given by matrices people call E, F, and H for mysterious reasons:
E = 0 1 F = 0 0 H = 1 0
0 0 1 0 0 -1
You will never be an expert on Lie algebras until you know by heart that
[H,E] = 2E, [H,F] = -2F, [E,F] = H.
Typically that's the sort of remark I make before screwing up by a
factor of two or something, so you'd better check! This is a
cute little multiplication table... but very important, since sl(2) is
the primordial Lie algebra from which the whole theory of "simple" Lie
algebras unfolds.
Physicists are probably more familiar with a different basis of sl(2),
the Pauli matrices:
sigma_1 = 0 1 sigma_2 = 0 -i sigma_3 = 1 0
1 0 i 0 0 -1
For purposes of Lie algebra theory it's actually better to divide each of
these matrices by i and call the resulting matrices I, J, and K,
respectively. We then have
IJ = -JI = K, JK = -KJ = I, KI = -IK = J, I^2 = J^2 = K^2 = -1
which is just the multiplication table of the quaternions! From the
point of view of Lie algebras, though, all that matters is
[I,J] = 2K, [J,K] = 2I, [K,I] = 2J.
Given the relation of these things and cross products, it should be no
surprise that the Pauli matrices have a lot to with angular momentum
around the x, y, and z axes in quantum mechanics.
If we take all *real* linear combinations of E,F,H we get a Lie algebra
over the *real* numbers called sl(2,R), and if we take all real linear
combinations of I,J,K we get a Lie algebra over the reals called
su(2). These two Lie algebras are two different "real forms" of sl(2).
Now, people know just about everything about sl(2) that they might want
to. Well, there's always something more, but I'm certainly personally
satisfied! I recall when as an impressionable student I saw a book by
Serge Lang titled simply "SL(2,R)," big and fat and scary inside. I
knew what SL(2,R) was, but not how one could think of a whole book's
worth of things to write about it! A whole book on 2x2 matrices??
Part of how one gets so much to say about a puny little Lie algebra like
sl(2) is by talking about its representations. What's a representation?
Well, first you have to temporarily shelve the idea that sl(2) consists
of 2x2 matrices, and think of it more abstractly simply as a
3-dimensional vector space with basis E,F,H, equipped with a Lie algebra
structure given by the multiplication table [H,E] = 2E, [H,F] = -2F,
[E,F] = H. If this is how I'd originally defined it, it would then be
a little *theorem* that this Lie algebra has a "representation" as 2x2
matrices. And it would turn out to have other representations too.
For example, there's a representation as 3x3 matrices given by sending
E to
0 1 0
0 0 2
0 0 0,
F to
0 0 0
2 0 0
0 1 0,
and H to
2 0 0
0 0 0
0 0 -2
In other words, these matrices satisfy the same commutation relations as
E,F, and H do.
More generally, and more precisely, we say an n-dimensional
representation of a Lie algebra L (over the complex numbers) is a linear
function R from L to nxn matrices such that
R([x,y]) = [R(x),R(y)]
for all x,y in L. Note that on the left the brackets are the brackets
in L, while on the right they denote the commutator of nxn matrices.
One good way to understand the essence of a Lie algebra is to figure out
what representations it has. And in quantum physics, Lie algebra
representations are where it's at: the symmetries of the world are
typically Lie groups, each Lie group has a corresponding Lie algebra,
the states of a quantum system are unit vectors in a Hilbert space, and
if the system has a certain Lie group of symmetries there will be a
representation of the Lie algebra on the Hilbert space. As any particle
physicist can tell you, you can learn a lot just by knowing which
representation of your symmetry group a given particle has.
So the name of the game is classifying Lie algebra representations...
and many tomes have been written on this by now. To keep things from
becoming too much of a mess it's crucial to make two observations.
First, there's an easy way to get new representations by taking the
"direct sum" of old ones: the sum of an n-dimensional representation and
an m-dimensional one is an (n+m)-dimensional one, for example. Another
way, not so easy, to get new representations from an old one is to look for
"subrepresentations" of the given representation. In particular, a
direct sum of two representations has them as subrepresentations.
(I won't define "direct sum" and "subrepresentation" here... hopefully
those who don't know will be tempted to look it up.)
So rather than classifying *all* representations, it's good to start by
classifying "irreducible" representations - those that have no
suprepresentations (other than themselves and the trivial 0-dimensional
representation). This is sort of like finding prime numbers... they are
"building blocks" in representation theory. But things are a little bit
messier, alas. We say a representation is "completely reducible" if it
is a direct sum of irreducible representations. Unfortunately, not all
representations need be completely reducible!
Let's consider the representations of sl(2,C). (The more sophisticated
reader should note that I am implicitly only considering finite-
dimensional complex representations!) Here life is as nice as could
be: all representations are completely reducible, and there is just
one irreducible n-dimensional representation for each n, with the
2-dimensional and 3-dimensional representations as above. (By the
way, I really mean that there is only one irreducible n-dimensional
representation up to a certain equivalence relation!) Physicists -
who more often work with the real form su(2) - call these the spin-0,
spin-1/2, spin-1, etc. representations. The "spin" of a particle is,
in mathematical terms, just the thing that tells you which
representation of su(2) it corresponds to!
Now let me jump up several levels of sophistication. In the last few
years people have realized that Lie groups are just a special case of
something called "quantum groups"... nobody talks about "quantum Lie
algebras" but that's essentially a historical accident: quantum groups
are NOT groups, they're a generalization of them, and they DON'T have Lie
algebras, but they have a generalization of them - so-called
quantized enveloping algebras.
Quantum groups can be formed from simple Lie algebras, and they depend
on a parameter q, a nonzero complex parameter. This parameter - q is
for quantum, naturally - can be thought of as
hbar
e
- the exponential of Planck's constant. When we set hbar = 0 we get
q = 1, and we get back to the "classical case" of plain old-fashioned
Lie algebras and groups. Every representation of a quantum group
gives an invariant of links (actually even tangles), and these link
invariants are functions of q. If we take the nth derivative of one of
these invariants with respect to hbar and evaluate it at hbar = 0 we get
a "Vassiliev invariant of degree n" (see the article "week3" for the
definition). Better than that, when q is a root of unity each
quantum group gives us a 3-dimensional "topological quantum field
theory," or TQFT known as Chern-Simons theory. In particular, we get an
invariant of compact oriented 3-manifolds. So there is a hefty bunch of
mathematical payoffs from quantum groups. And there are good reasons to
think of them as the right generalization of groups for dealing with
symmetries in the physics of 2 and 3 dimensions. If string theory *or*
the loop variables approach to quantum gravity have any truth to them,
quantum groups play a sneaky role in honest 4-dimensional physics too.
In particular, there is a quantum version of sl(2) called sl_q(2).
When q = 1 we essentially have the good old sl(2). Chari and Premet
have just worked out a lot of the representation theory
of sl_q(2). First of all, it's been known for some time that as long as
q is not a root of unity - that is, as long as we don't have q^n = 1
for some integer n - the story is almost like that for ordinary sl(2).
Namely, there is one irreducible representation of each dimension, and
all representations are completely reducible. This fails at roots of
unity - which turns out to be the reason why one can cook up TQFTs in
this case. It turns out that if q is an nth
root of unity one can still define representations of dimension
0,1,2,3, etc., more or less just like the classical case, but only those
of dimension < n are irreducible. There are, in fact, exactly n
irreducible representations, and the fact that there are only finitely
many is what makes all sorts of neat things happen. The k-dimensional
representations with k >= n are not completely reducible. And, besides
the representations that are analogous to the classical case, there are
a bunch more. They have not been completely classified - they are,
according to Chari, a mess! But she and Premet have classified a large
batch of the "indecomposable" ones, that is, the ones that aren't direct
sums of other ones. I guess I'll leave it at that.
2) David Kazhdan and Iakov Soibelman, Representations of the quantized
function algebras, 2-categories and Zamolodchikov tetrahedra
equations, Harvard University preprint.
In this terse paper, Kazhdan and Soibelman construct a braided monoidal
2-category using quantum groups at roots of unity. As I've said a few times,
people expect braided monoidal 2-categories to play a role in generally
covariant 4d physics analogous to what braided monoidal categories do in
3d physics. In particular, one might hope to get invariants of
4-dimensional manifolds, or of surfaces embedded in 4-manifolds, this
way. (See last week's post for a little bit about the details.)
I don't feel I understand this construction well enough yet to want
to say much about it, but it is clearly related to the construction
of a braided monoidal 2-category from the category of quantum group
representations given by Crane and Frenkel (see "week2").
3) Adrian Ocneanu, A note on simplicial dimension shifting, available
as arXiv:hep-th/9302028.
Ouch! This paper claims to show that the very charming 4d TQFT
constructed by Crane and Yetter in "A Categorical construction of 4d
topological quantum field theories" (arXiv:hep-th/9301062) is
trivial! In particular, he says the resulting invariant of compact
oriented 4-manifolds is identically equal to 1. If so, it's back to
the drawing board. Crane and Yetter took the 3d TQFT coming from
sl_q(2) at roots of unity and then used a clever trick to get
3-manifolds from a simplicial decomposition of a 4-manifold to get a
4d TQFT. Ocneanu claims this trick, which he calls "simplicial
dimension shifting," only gives trivial 4-manifold invariants.
I am not yet in a position to pass judgement on this, since both
Crane/Yetter and Ocneanu are rather sketchy in key places. If indeed
Ocneanu is right, I think people are going to have to get serious about
facing up to the need for 2-categorical thinking in 4-dimensional
generally covariant physics. I had asked Crane, a big proponent of
2-categories, why they played no role in his 4d TQFT, and he said that
indeed he felt like the kid who took apart a watch, put it back
together, and found it still worked even though there was a piece left
over. So maybe the watch didn't really work without that extra piece
after all. In late March I will go to the Conference on Quantum Topology
thrown by Crane and Yetter (at Kansas State U. at Manhattan), and I'm
sure everyone will try to thrash this stuff out.
4) Abhay Ashtekar and Jerzy Lewandowski, Representation theory of
analytic holonomy C*-algebras, available as arXiv:gr-qc/9311010.
This paper is a follow-up of the paper
5) Abhay Ashtekar and Chris Isham, Representations of the holonomy
algebras of gravity and non-Abelian gauge theories, Journal of
Classical and Quantum Gravity 9 (1992), pp. 1069-1100. Also available
as arXiv:hep-th/9202053.
and sort of complements another,
6) John Baez, Link invariants, holonomy algebras and functional
integration, available as arXiv:hep-th/9301063.
The idea here is to provide a firm mathematical foundation for the
loop variables representation of gauge theories, particularly quantum
gravity. Ashtekar and Lewandowski consider an algebra of
gauge-invariant observables on the space of su(2) connections on any
real-analytic manifold, namely that generated by piecewise analytic
Wilson loops. This is the sort of thing meant by a "holonomy
algebra". They manage to construct an explicit diffeomorphism-invariant
state on this algebra. They also relate this algebra to a similar
algebra for sl(2) connections - the latter being what really comes up
in quantum gravity. And they do a number of other interesting things,
all quite rigorously. My paper dealt instead with an algebra
generated by "regularized" or "smeared" Wilson loops, and showed that
there was a 1-1 map from diffeomorphism-invariant states on this
algebra to invariants of framed links - thus showing that the loop
variables picture, in which states are given by link invariants,
doesn't really lose any of the physics present in traditional
approaches to gauge theories. I am busy at work trying to combine
Ashtekar and Lewandowski's ideas with my own and push this program
further - my own personal goal being to make the Chern-Simons path
integral rigorous - it being one of those mysterious "measures on the
space of all connections mod gauge transformations" that physicists
like, which unfortunately aren't really measures, but some kind of
generalization thereof. What it *should* be is a state (or continuous
linear functional) on some kind of holonomy algebra.
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