Will someone please enlighten my as to what are these algebras, plus a
bit about how they are related and why? Thanks! -Jeff Baldwin

* W-algebras (there are several)
* Virasoro algebra and its W-algebra generalization
* Lie algebras (whether or not compact)
* Volterra algebra
* Gelfand-Dickey algebra (several kinds)

news:8vu9t0$eu2$1@news.state.mn.us...

I can say something about the first three items, in reverse order.

* A Lie algebra g is a vector space (over C, say) endowed with extra
structure: the bracket, satisfying the following axioms
(a,b,c arbitrary elements in g and k, k' numbers in C):

1. Anti-symmetry: [b,a] = -[a,b],
2. Linearity: [ka+k'b, c] = k[a,c] + k'[b,c],
3. Jacobi identity: [a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0.

Of all Lie algebras, the simple ones have no ideals. Recall that
an ideal is a subspace I \subset g, such that [g,I] \subset I
(i.e. [a,i] is in I for all a in g, i in I). Simple finite-dimensional
Lie algebras were classified by Cartan and Dynkin in the first half
of the previous century. The result is the celebrated list of four
series and five exceptions:

A_n = sl(n+1)
B_n = so(2n+1)
C_n = sp(2n)
D_n = so(2n)
E_6, E_7, E_8, G_2, F_4.

Simple infinite-dimensional Lie algebras were classified by Cartan
as early as 1909. There are four series and no exceptions:

W_n = vect(n)   arbitrary vector fields (v.f.) in n dimensions
S_n = svect(n)  divergence-free v.f.
H_n             Hamiltonian v.f. (n even)
K_n             contact v.f. (n odd)

These names are standard among algebraists. However, the W in W_n
stands for Witt; it is not a W algebra!

* The Virasoro algebra Vir is a particular infinite-dimensional Lie
algebra. It is the central extension of vect(1). vect(1) has the
Fourier basis L_m = -i exp(imx) d/dx, m in Z. The brackets of Vir
are given by

[L_m, L_n] = (n-m)L_m+n - c/12 (m^3-m) \delta_m+n,

where \delta_m is the Kronecker delta. The element c is central,
meaning that it commutes with all of Vir. By Schur's lemma, it can
therefore be considered as a c-number.

What is not so widely known is that vect(N) admits a Virasoro-like
extension for all N, but it is central for N=1 only. To see this,
rewrite Vir as

[L_m, L_n] = (n-m)L_m+n - m^2 n S_m+n,
[L_m, S_n] = (n+m)S_m+n,
[S_m, S_n] = 0,
m S_m = 0.

It is easy to see that the two formulations of Vir are equivalent
(I have absorbed the linear cocycle into a redefinition of L_0).
The second formulation immediately generalizes to N dimensions.
The generators are L_j(m) = -i exp(i m_k x^k) d/dx^j and
S^i(m), where x = (x^k), k = 1, 2, ..., N denotes coordinates in N-dimensional
space and m = (m_k). The Einstein convention is used (repeated indices,
one up and one down, are implicitly summed over). The defining
relations are

[L_i(m), L_j(n)] = n_i L_j(m+n) - m_j L_i(m+n)
  + (c_1 m_j n_i + c_2 m_i n_j) m_k S^k(m+n),
[L_i(m), S^j(n)] = n_i S^j(m+n) + \delta^j_i m_k S^k(m+n),
[S^i(m), S^j(n)] = 0,
m_i S^i(m) = 0.

This is an extension of vect(N) by the abelian ideal with basis S^i(m).
Geometrically, we can think of S^i(m) as a closed dual one-form, i.e.
a closed (N-1)-form. In N=1 dimensions, a closed dual one-form is a
closed zero-form is a constant function, so the extension is
one-dimensional and central in this case.

* The W algebra W_p (not the Witt algebra vect(p)!) is a central
extension of an algebra of (p-1):st order differential operators
on the circle (I think). When p=\infty, there are several algebras:
W_\infty, W_1+\infty, and w_\infty at least. The latter is simply
the algebra of all differential operators (no central extension),
i.e. functions of x and d/dx. W_2 is Virasoro  and W_1 is an
infinite-dimensional Heisenberg algebra (Note: W_2 (as a W algebra)
= W_1 (as a Witt algebra), apart from the central extension!).
W_p for p >= 3 but p < \infty are not Lie algebras, because there
are non-linear expressions on the right-hand side. Nevertheless, the
representation theory is quite analogous to that of Virasoro, although
I don't know so much about it.

* Although you didn't ask, let me tell you a bit about Lie superalgebras,
since I have become quite interested in that subject recently. A Lie
superalgebra is a Lie algebra g with a Z_2-grading. Every element a in g
has a degree d(a) = 0 or 1 (mod 2), and two of the Lie algebra axioms
are replaced by

1. Graded anti-symmetry: [b,a] = -(-)^d(a)d(b) [a,b],
3. Graded Jacobi identity:
(-)^d(a)d(c) [a,[b,c]] + (-)^d(b)d(a) [b,[c,a]]
 + (-)^d(c)d(b) [c,[a,b]] = 0.

Finite-dimensional were classified by Kac in 1975. The infinite-dimensional
case was only recently completed by Kac, Leites and Shchepochkina
(alphabethical order; they seem to disagree about who did what first).
All algebras can be realized as vector fields acting on a (n|m)-
dimensional superspace (n bosonic and m fermionic coordinates).
There is a list of ten series

vect(n|m)       arbitrary v.f. in (n|m) dimensions
svect(n|m)      divergence-free v.f.
svect~(n|m)     a deformation of svect(n|m)
h(2n|m)         Hamiltonian v.f.
sh(m)           special Hamiltonian v.f. \subset h(0|m)
le(n)           odd Hamiltonian or Leitesian v.f. \subset vect(n|n)
sle(n)          div-free Leitesian v.f.
k(2n+1|m)       contact v.f.
m(n)            odd contact v.f. \subset vect(n+1|n)
sm(n)           div-free odd contact v.f.

The odd Hamiltonian algebra also appears in physics, where it is
known as Batalin-Vilkovisky or anti-bracket algebra. sh(m) is characterized
by a vanishing Berezin integral.

There are also five exceptions, all discovered by Shchepochkina
(this phenomenon has no bosonic counterpart):

e(4|4), e(3|6), e(5|10), e(3|8), e(1|6).

Apart from the Z_2-grading, these algebras also have a Weisfeiler
gradation, going from -d to infinity (-d = -3, -2 or -1, depending on
algebra). Call the degree k subspace g_k. Clearly, g_0 is a subalgebra
and g_k is a g_0 module for every k. What is really cool is that
for e(3|6) and e(3|8), g_0 = sl(3)+sl(2)+gl(1), which is just the
non-compact form of the standard model algebra. Moreover, Kac shows
that there is an almost perfect match between degenerate e(3|6) modules
and fundamental particles in the standard model (quarks, leptons,
gauge bosons), except that the Higgs boson is traded for a pair of
charged gluons.

Some references:

I Shchepochkina: Representations Theory 3 (1999) 373-415
hep-th/9702121 (preliminary version).

V G Kac: Adv. Math 139 (1998) 1-35.

V G Kac: math.QA/9912235.

This is a humorous comment, so it probably won't get passed by the
moderators, but there is a half-serious adage in mathematics that if you
invent a new concept, experience shows that if you give it a really
uninformative or even downright misleading name, like "W-algebra" or
"K-theory" or "noncommutative geometry", it is likely to be immediately
renamed after you by your grateful/awestruck colleagues.  Sometimes, of
course, this alleged "strategy" fails, which is why mathematics is stuck
with some really awful names.  

Some wags like to name everything after Euler on the grounds that when the
(60+n)-th volume of his collected works come out, it will be found that
Euler really did invent the thing first!  (I believe it has been estimated
that the collected works of Euler will run to some 300 volumes.)

It is well known in mathematics that most "named concepts" turn out to be
named after someone other than the person who thought of it first.  To
pick just one example likely to be familiar to some physicists: the "Morse
sequence" 0110100110010110... turned out to have been discussed by Axel
Thue years before Marston Morse wrote a famous paper "introducing" this
sequence, and very recently someone discovered that Prouhet had discussed
this sequence in the nineteenth century.

Chris Hillman

Home Page: http://www.math.washington.edu/~hillman/

It is particularly well suited to solving which sorts of tasks in
physics, mathematical physics, and/or theoretical physics?

Again, it is particularly well suited to solving which additional sorts
of tasks in physics not as well handled by finite dimensional Lie algebra?

Again, it is particularly well suited to solving which sorts of tasks in
physics?

Yes, I hesitated to ask since a simple Lie algebra is currently a bit
beyond my comprehension. But it seems that the superalgebra is of more
current interest, so I thank you for your illuminating info.

"ALL" of the above, or all Lie superalgebras?

Yes, I see this sort of statement and am frustrated that I do not
understand why this "ought to be so"... hence my interest in catching up
on the math. The math seems to be directing (within bounds) the modeling
effort, and with good reason I bet. I wish to have a better feel for
that or those reasons.

Excellent. Thank you. Do you know if any of these (or something similar)
is on the web anywhere, or where I might search to find it? I'll begin
looking in my "standard" sources, but anything you can say to speed this
process will be most welcomed.

Thank you so much for your considered response.
-Jeff Baldwin

Mostly gauge theories, but also harmonic analysis and lots
of other things probably...

It's particularly interesting for two-dimensional conformal
field theories, amongst which string theory.

Seemingly to treat particles with spins higher than two.
I'm not a specialist, but there was some proposal of doing
some W-strings with higher spins.

Lie superalgebras are of course always useful when dealing
with supersymmetric theories.

Neither do I... Most physicists aren't familiar with e(3|6)...

Maxime

In article <901dmi$45...@news.state.mn.us>,
Thomas Larsson  <Thomas.Lars...@hdd.se> wrote:

I don't know if this is the most efficient way to understand this
subject.   Asking "so what's W-algebra?" is a bit like a walking
into a nuclear reactor and saying "so what does this particular
button do?"  You'll probably either get a detailed response phrased
in technical jargon, or a very general response like "it helps you
control the reactor".  

But now I'm going to ask about some buttons myself... let's hope
I don't slip and press the one that makes everything blow up!

You must mean finite-dimensional *simple* ones, right?  

This is cool!  But you must mean simple ones, right?  And
I can't help but feel there must be some other technical conditions
you need to invoke, to get a manageable classification of
infinite-dimensional Lie superalgebras.

What I really want to know, though, is this.  What does "Leistesian"
mean?   An odd version of the Hamiltonian vector fields... defined
on some super-analogue of a symplectic manifold?

Aha!  This makes me think my guess is not completely stupid.

... which is the reason why I brought it up.

Like many people, I have had a problem with the standard model.
It is clearly important, since it agrees so well with experiments,
but the gauge group and field content seem very ad hoc. The
exceptional superalgebras is the first place where I have seen
sl(3)+sl(2)+gl(1) arise naturally in a mathematically deep context.

(The gauge algebra of the standard model is really the compact
algebra su(3)+su(2)+u(1). sl(3)+sl(2)+gl(1) is the non-compact
complexification.)

I first heard about e(3|6) in talks by Kac and Rudakov in the end
of September. It appears that Kac has talked about its possible
relation to the standard model for a year. It is also remarkable
that the world's strongest mathematician moonlights in physics.

Thomas

Of course. I wrote "simple" higher up, in the paragraph on Lie algebras.
I don't think solvable or nilpotent algebras are classified even
in the bosonic case.

The technical condition is the existence of a Weisfeiler gradation
of finite depth d. This means that the algebra is graded like this:

g = g_-d + ... + g_-1 + g_0 + g_1 + g_2 + ...

where
1. [g_k, g_l] \subset g_k+l (this is the definition of a grading)
2. dim g_k is finite
3. g_-k-1 = [g_-k, g_-1] (equality, not just inclusion)
4. maybe something more.

I think these conditions mean that g can be realized as vector
fields with functions either polynomials or formal power series.
You can of course consider other classes of functions, and
even vector fields acting on sections of bundles. Such beasts are
definitely not classified, but the local information is encoded in
the "local algebras" above.

Correct. Consider a phase space with coordinates q^i and p_i.
To each function f(q,p) we associate the Hamiltonian vector field

H_f = df/dq^i d/dp_i - df/dp_i d/dq^i.

They satisfy the algebra [H_f, H_g] = H_{f,g}, where

{f,g} = df/dq^i dg/dp_i - df/dp_i dg/dq^i

is the Poisson bracket (being lazy, I haven't checked the signs - but
they *do* work out if you do it right). In the even case, q^i and
p_i have the same parity (even *or* odd, for the same value of i);
in the odd case they have opposite parity. The odd Hamiltonian
algebra was first discovered by Leites around 1979 (hence the name),
and soon afterwards rediscovered by Batalin and Vilkovisky when they
developped the antifield formalism.

Thomas

Of course, that's a nice coincidence, but it is actually just as
ad hoc as in the standard model. Most Lie groups bigger
than SU(5) (i.e. most Lie groups...) contain
SU(3)xSU(2)xU(1) as a subgroup and their natural
representations look like those of the standard model when
you break the symmetry down to SU(3)xSU(2)xU(1).
This is nothing special of e(3|6) or e(3|8).
The mathematical context is certainly deep, but the physical
context is totally ad hoc, as well (except, of course, if you
can exhibit a consistent grand-unified theory, whose gauge
algebra must for some reason (maybe anomaly cancellation)
be e(3|6).
Then, you could say you have understood something.
And you might be celebrated by the whole physics
community.
I don't think you should think that you understand the
standard model better now that you have seen this.
The mystery of the standard model gauge group remains
totally open also after this discovery.

Which is not to say that Kac is not one of the world's
strongest mathematician, which he certainly is. I respect him
greatly.

Maxime

In article <3A2E1F90.456D4...@unine.ch>,
Maxime Bagnoud  <Maxime.Bagn...@unine.ch> wrote:

I think it's premature to say it's a "coincidence".  Yes,
most simple Lie groups contain SU(3)xSU(2)xU(1) as a subgroup.
But Larsson is claiming more than that the simple Lie superalgebra
e(3|6) contains sl(3)+sl(2)+gl(1) as a Lie subalgebra.  He seems
to be claiming that it contains it *in a god-given way*.  I don't
know what a "Weisfeiler grading" is, but hopefully it's something
intrinsic to the Lie superalgebras in question, not just an ad
hoc structure thrown on top by hand.  If so, we'd be seeing
sl(3)+sl(2)+ gl(1) arise *naturally* as a sub-Lie-algebra of
an exceptional algebraic structure.  I think this is the kind
of thing we should really be looking for, to understand the
Standard Model.

Also, he seems to be claiming that there's an almost perfect match
between certain nice e(3|6) modules and the particle content of the
Standard Model.  Something similar happens for SO(10), but not really
any other grand unified model (in my opinion).  So this too is rather
special - though "almost perfect" can sometimes be very, very far from
"perfect".  A pair of charged gluons is not a very good substitute
for a Higgs!  

Of course it's been known for quite a while that SU(3)xSU(2)xU(1)
can almost be regarded as a member of the E series of exceptional
groups, and so can SU(5) and SO(10).   So it's perhaps not so
surprising that one of the "e" Lie superalgebras is also related
to SU(3)xSU(2)xU(1).  See "week119" for more on this - I'll quote
it below.

As you probably know, the E series is built from the octonions.
This is one of the reasons I'm trying to understand the octonions
in a more conceptual way: to understand the geometrical and algebraic
concepts that might - just might!!! - underly the peculiarities of
the Standard Model.

Of course you're right.  But this discovery represents a possible
avenue of progress, and such avenues are very, very hard to find.
We need to follow up all the clues we have!  Too many particle
physicists have simply given up on trying to explain the Standard
Model and its gauge group SU(3)xSU(2)xU(1).   In my opinion this
is a big mistake.  Nature has given us a big clue.  Now it's up
to us to do something with it.  

Anyway, I'll have to read those papers Larsson cited about e(3|6)
and e(3|8).   It's my job to know this kind of stuff.

.........................................................................

Also available at http://math.ucr.edu/home/baez/week119.html

April 13, 1998

This Week's Finds in Mathematical Physics - Week 119
John Baez

I've been slacking off on This Week's Finds lately because I was
busy getting stuff done at Riverside so that I could visit the
Center for Gravitational Physics and Geometry here at Penn State
with a fairly clean slate.  Indeed, sometimes my whole life seems
like an endless series of distractions designed to prevent me from
writing This Week's Finds.  However, now I'm here and ready to have
some fun....

Recently I've been trying to learn about grand unified theories, or
"GUTs".  These were popular in the late 1970s and early 1980s, when
the Standard Model of particle interactions had fully come into its
own and people were looking around for a better theory that would
unify all the forces and particles present in that model --- in short,
everything except gravity.  

The Standard Model works well but it's fairly baroque, so it's natural
to hope for some more elegant theory underlying it.  Remember how it
goes:

-------------------------------------------------------------------------
                            GAUGE BOSONS

    ELECTROMAGNETIC FORCE          WEAK FORCE         STRONG FORCE

         photon                     W+                  8 gluons
                                    W-
                                    Z  
-------------------------------------------------------------------------

                            FERMIONS

        LEPTONS                                     QUARKS

electron        electron neutrino         down quark        up quark
muon            muon neutrino             strange quark     charm quark
tauon           tauon neutrino            bottom quark      top quark
-------------------------------------------------------------------------

                        HIGGS BOSON (not yet seen)

-------------------------------------------------------------------------

The strong, electromagnetic and weak forces are all described by
Yang-Mills fields, with the gauge group SU(3) x SU(2) x U(1).  In what
follows I'll assume you know the rudiments of gauge theory, or at
least that you can fake it.

SU(3) is the gauge group of the strong force, and its 8 generators
correspond to the gluons.  SU(2) x U(1) is the gauge group of the
electroweak force, which unifies electromagnetism and the weak force.
It's *not* true that the generators of SU(2) corresponds to the W+, W-
and Z while the generator of U(1) corresponds to the photon.  Instead,
the photon corresponds to the generator of a sneakier U(1) subgroup
sitting slantwise inside SU(2) x U(1); the basic formula to remember
here is:

Q = I_3 + Y/2

where Q is ordinary electric charge, I_3 is the 3rd component of
"weak isospin", i.e. the generator of SU(2) corresponding to the
matrix

(1/2   0)
(0  -1/2)

and Y, "hypercharge", is the generator of the U(1) factor.  The role
of the Higgs particle is to spontaneously break the SU(2) x U(1)
symmetry, and also to give all the massive particles their mass.
However, I don't want to talk about that here; I want to focus on the
fermions and how they form representations of the gauge group SU(3) x
SU(2) x U(1), because I want to talk about how grand unified theories
attempt to simplify this picture - at the expense of postulating more
Higgs bosons.

The fermions come in 3 generations, as indicated in the chart above.
I want to explain how the fermions in a given generation are grouped
into irreducible representations of SU(3) x SU(2) x U(1).  All the
generations work the same way, so I'll just talk about the first
generation.  Also, every fermion has a corresponding antiparticle, but
this just transforms according to the dual representation, so I will
ignore the antiparticles here.

Before I tell you how it works, I should remind you that all the
fermions are, in addition to being representations of SU(3) x SU(2) x
U(1), also spin-1/2 particles.  The massive fermions - the quarks and
the electron, muon and tauon - are Dirac spinors, meaning that they
can spin either way along any axis.  The massless fermions - the
neutrinos - are Weyl spinors, meaning that they always spin
counterclockwise along their axis of motion.  This makes sense
because, being massless, they move at the speed of light, so everyone
can agree on their axis of motion!  So the massive fermions have two
helicity states, which we'll refer to as "left-handed" and
"right-handed", while the neutrinos only come in a "left-handed" form.

(Here I am discussing the Standard Model in its classic form.  I'm
ignoring any modifications needed to deal with a possible nonzero
neutrino mass.  For more on Standard Model, neutrino mass and
different kinds of spinors, see "week93".)

Okay.  The Standard Model lumps the left-handed neutrino and the
left-handed electron into a single irreducible representation of
SU(3) x SU(2) x U(1):

(nu_L, e_L)                                 (1,2,-1)

This 2-dimensional representation is called (1,2,-1), meaning
that it's the tensor product of the 1-dimensional trivial rep
of SU(3), the 2-dimensional fundamental rep of SU(2), and the
1-dimensional rep of U(1) with hypercharge -1.  

Similarly, the left-handed up and down quarks fit together as:

(u_L, u_L, u_L, d_L, d_L, d_L)              (3,2,1/3)

Here I'm writing both quarks 3 times since they also come in 3 color
states.  In other words, this 6-dimensional representation is the
tensor product of the 3-dimensional fundamental rep of SU(3), the
2-dimensional fundamental rep of SU(2), and the 1-dimensional rep of
U(1) with hypercharge 1/3.  That's why we call this rep (3,2,1/3).  

(If you are familiar with the irreducible representations of U(1) you
will know that they are usually parametrized by integers.  Here we are
using integers divided by 3.  The reason is that people defined the
charge of the electron to be -1 before quarks were discovered, at
which point it turned out that the smallest unit of charge was 1/3 as
big as had been previously ...

read more »

In article <3A2E1F90.456D4...@unine.ch>,
Maxime Bagnoud  <Maxime.Bagn...@unine.ch> wrote:

By the way, a small nitpick: SU(5) doesn't contain SU(3)xSU(2)xU(1);
it just contains the quotient of this group by Z/6.  Luckily it's
really this quotient group that matters for the Standard Model.
This may or may not be a clue of some sort.  It's certainly relevant
to some of the Kaluza-Klein theories and GUTs people have studied.

I enjoyed working this out for myself a while ago:

.........................................................................

Also available at http://math.ucr.edu/home/baez/week133.html

April 23, 1999
This Week's Finds in Mathematical Physics - Week 133
John Baez

[....]

And now for something completely different, arising from a thread on
sci.physics.research started by Garrett Lisi.  What's the gauge group
of the Standard Model?  Everyone will tell you it's U(1) x SU(2) x
SU(3), but as Marc Bellon pointed out, this is perhaps not the most
accurate answer.   Let me explain why and figure out a better answer.  

Every particle in the Standard Model transforms according to some
representation of U(1) x SU(2) x SU(3), but some elements of this
group act trivially on all these representations.  Thus we can find
a smaller group which can equally well be used as the gauge group
of the Standard Model: the quotient of U(1) x SU(2) x SU(3)
by the subgroup of elements that act trivially.

Let's figure out this subgroup!  To do so we need to go through all
the particles and figure out which elements of U(1) x SU(2) x SU(3)
act trivially on all of them.  

Start with the gauge bosons.  In any gauge theory, the gauge bosons
transform in the adjoint representation, so the elements of the gauge
group that act trivially are precisely those in the *center* of the
group.  U(1) is abelian so its center is all of U(1).  Elements of SU(n)
that lie in the center must be diagonal.  The n x n diagonal unitary
matrices with determinant 1 are all of the form exp(2 pi i / n),
and these form a subgroup isomorphic to Z/n.  It follows that the
center of U(1) x SU(2) x SU(3) is U(1) x Z/2 x Z/3.  

Next let's look at the other particles.  If you forget how these work,
see "week119".  For the fermions, it suffices to look at those of the
first generation, since the other two generations transform exactly
the same way.  First of all, we have the left-handed electron and
neutrino:

(nu_L, e_L)                                

These form a 2-dimensional representation.  This representation is the
tensor product of the irreducible rep of U(1) with hypercharge -1, the
isospin-1/2 rep of SU(2), and the trivial rep of SU(3).  

A word about notation!  People usually describe irreducible reps of U(1)
by integers.  For historical reasons, hypercharge comes in integral
multiples of 1/3.  Thus to get the appropriate integer we need to
multiply the hypercharge by 3.  Also, the group SU(2) here is
associated, not to spin in the sense of angular momentum, but to
something called "weak isospin".  That's why we say "isopin-1/2 rep"
above.  Mathematically, though, this is just the usual spin-1/2
representation of SU(2).  

Next we have the left-handed up and down quarks, which come in 3
colors each:

(u_L, u_L, u_L, d_L, d_L, d_L)              

This 6-dimensional representation is the tensor product of the
irreducible rep of U(1) with hypercharge 1/3, the isospin-1/2
rep of SU(2), and the fundamental rep of SU(3).  

That's all the left-handed fermions.  Note that they all transform
transform according to the isospin-1/2 rep of SU(2) - we  call them
"isospin doublets".  The right-handed fermions all transform according
to the isospin-0 rep of SU(2) - they're "isospin singlets".  First we
have the right-handed electron:

e_R                                        

This is the tensor product of the irreducible rep of U(1) with
hypercharge -2, the isospin-0 rep of SU(2), and the trivial rep of
SU(3).  Then there are the right-handed up quarks:

(u_R, u_R, u_R)                            

which form the tensor product of the irreducible rep of U(1) with
hypercharge 4/3, the isospin-0 rep of SU(2), and the fundamental rep of
SU(3).  And then there are the right-handed down quarks:

(d_R, d_R, d_R)                            

which form the tensor product of the irreducible rep of U(1)
with hypercharge 2/3, the isospin-0 rep of SU(2), and the
3-dimensional fundamental rep of SU(3).  

Finally, besides the fermions, there is the - so far unseen - Higgs
boson:

(H_+, H_0)

This transforms according to the tensor product of the irreducible
rep of U(1) with hypercharge 1, the isospin-1/2 rep of SU(2), and
the 1-dimensional trivial rep of SU(3).  

Okay, let's see which elements of U(1) x Z/2 x Z/3 act trivially on all
these representations!  Note first that the generator of Z/2 acts as
multiplication by 1 on the isospin singlets and -1 on the isospin
doublets.  Similarly, the generator of Z/3 acts as multiplication by
1 on the leptons and exp(2 pi i / 3) on the quarks.  Thus everything
in Z/2 x Z/3 acts as multiplication by some sixth root of unity.  So
to find elements of U(1) x Z/2 x Z/3 that act trivially, we only need
to consider guys in U(1) that are sixth roots of unity.  

To see what's going on, we make a little table using the information
I've described:

         ACTION OF            ACTION OF             ACTION OF
       exp(pi i / 3)            -1               exp(2 pi i / 3)
          IN U(1)             IN SU(2)               IN SU(3)

e_L         -1                  -1                     1
nu_L        -1                  -1                     1
u_L     exp(pi i / 3)           -1               exp(2 pi i / 3)
d_L     exp(pi i / 3)           -1               exp(2 pi i / 3)

e_R          1                   1                     1
u_R     exp(4 pi i / 3)          1               exp(2 pi i / 3)
d_R     exp(4 pi i / 3)          1               exp(2 pi i / 3)

 H          -1                  -1                     1

And we look for patterns!  

See any?

The most important one for our purposes is that if we multiply all three
numbers in each row, we get 1.

This means that the element (exp(pi i / 3), -1, exp(2 pi i / 3)) in U(1)
x SU(2) x SU(3) acts trivially on all particles.  This element generates
a subgroup isomorphic to Z/6.  If you think a bit harder you'll see
there are no *other* patterns that would make any *more* elements of
U(1) x SU(2) x SU(3) act trivially.  And if you think about the relation
between charge and hypercharge, you'll see this pattern has a lot to do
with the fact that quark charges in multiples of 1/3, while leptons have
integral charge.  There's more to it than that, though....

Anyway, the "true" gauge group of the Standard Model - i.e., the
smallest possible one - is not U(1) x SU(2) x SU(3), but the quotient of
this by the particular Z/6 subgroup we've just found.  Let's call
this group G.  

There are two reasons why this might be important.  First, Marc Bellon
pointed out a nice way to think about G: it's the subgroup of U(2) x U(3)
consisting of elements (g,h) with

(det g)(det h) = 1.

If we embed U(2) x U(3) in U(5) in the obvious way, then this subgroup G
actually lies in SU(5), thanks to the above equation.  And this is what
people do in the SU(5) grand unified theory.  They don't actually stuff
all of U(1) x SU(2) x SU(3) into SU(5), just the group G!   For more
details, see "week119".  Better yet, try this book that Brett McInnes
recommended to me:

4) Lochlainn O'Raifeartaigh, Group structure of gauge theories,
Cambridge University Press, Cambridge, 1986.

Second, this magical group G has a nice action on a 7-dimensional
manifold which we can use as the fiber for a 11-dimensional Kaluza-Klein
theory that mimics the Standard Model in the low-energy limit.  The way
to get this manifold is to take S^3 x S^5 sitting inside C^2 x C^3 and
mod out by the action of U(1) as multiplication by phases.  The group
G acts on C^2 x C^3 in an obvious way, and using this it's easy to see
that it acts on (C^2 x C^3)/U(1).  

I'm not sure where to read more about this, but you might try:

5) Edward Witten, Search for a realistic Kaluza-Klein theory,
Nucl. Phys. B186 (1981), 412-428.

Edward Witten, Fermion quantum numbers in Kaluza-Klein theory,
Shelter Island II, Proceedings: Quantum Field Theory and the
Fundamental Problems of Physics, ed. T. Appelquist et al,
MIT Press, 1985, pp. 227-277.

6) Thomas Appelquist, Alan Chodos and Peter G.O. Freund, editors,
Modern Kaluza-Klein Theories, Addison-Wesley, Menlo Park, California,
1987.

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