I've been keeping quiet lately and trying to get some work done
for a change.  Here's a new paper that studies categorified groups
and Lie groups:

Higher-Dimensional Algebra V: 2-Groups
John C. Baez and Aaron D. Lauda

Abstract:

A 2-group is a "categorified" version of a group, in which the
underlying set G has been replaced by a category and the
multiplication map m: G x G -> G has been replaced by a functor.  
Various versions of this notion have already been explored;
our goal here is to provide a detailed introduction to two,
which we call "weak" and "coherent" 2-groups.  A weak 2-group
is a weak monoidal category in which every morphism has an
inverse and every object x has a "weak inverse": an object
y such that x tensor y and y tensor x are isomorphic to 1.
A coherent 2-group is a weak 2-group in which every object x
is equipped with a specified weak inverse x* and isomorphisms
i_x: 1 -> x tensor x*, e_x: x* tensor x -> 1 forming an
adjunction.  We define 2-categories of weak and coherent
2-groups, construct an "improvement" 2-functor which turns
weak 2-groups into coherent ones, and prove this 2-functor
is a 2-equivalence of 2-categories.  We also internalize
the concept of coherent 2-group, which gives a way to define
topological 2-groups, Lie 2-groups, affine 2-group schemes,
and the like.  We conclude with a tour of examples.  
Diagrammatic methods are emphasized throughout - especially
string diagrams.

This paper will soon appear on the mathematics arXiv, but
their computer seems unable to draw some of the pictures
correctly, so I urge you to try this PDF version instead:

http://math.ucr.edu/home/baez/hda5.pdf

The next paper in this series is due out soon and it will
study categorified Lie algebras.  This is all part of my
evil scheme to categorify all of math and then physics - see

http://math.ucr.edu/home/baez/gauge/

for how these categorified Lie groups and Lie algebras can
be used in a version of gauge theory based on the parallel
transport of string-like objects as well as point particles.

I'll be talking about this stuff in Lisbon soon, at the
Workshop on Higher-Order Geometry and Categorification:

http://www.math.ist.utl.pt/~rpicken/CHOG2003

It's happening on July 23rd-24th, and there will be a bunch
of talks on the intersection of n-category theory and mathematical
physics:

JOHN BAEZ, Univ. California at Riverside
    "Categorified Lie groups, Lie algebras, bundles and connections"
LAWRENCE BREEN, Univ. Paris 13
    "Differential geometry of gerbes and de Rham diagrams"
LOUIS CRANE, Kansas State Univ.
    "Two-Categories in differential geometry and state sums"
STEPHAN STOLZ, Univ. Notre Dame
    "What is an elliptic object?"

Contributed talks:

Higher-Dimensional Algebra VI: Lie 2-algebras
John C. Baez and Alissa S. Crans

The theory of Lie algebras can be categorified starting from a
new notion of "2-vector space", which we define as an internal
category in Vect.  There is a 2-category Vect having these
2-vector spaces as objects, "linear functors" as morphisms and
"linear natural transformations" as 2-morphisms.  We define a
"semistrict Lie 2-algebra" to be a 2-vector space L equipped
with a skew-symmetric bilinear functor satisfying the Jacobi
identity up to a linear natural transformation called the
"Jacobiator", which in turn must satisfy a certain law of its
own.  This law is closely related to the Zamolodchikov tetrahedron
equation, and indeed we prove that any semistrict Lie 2-algebra
gives a solution of this equation, just as any Lie algebra gives
a solution of the Yang-Baxter equation.  We construct a 2-category
of semistrict Lie 2-algebras and prove that it is 2-equivalent
to the 2-category of 2-term L-infinity algebras in the sense
of Stasheff.  We also study strict and skeletal Lie 2-algebras,
obtaining the former from strict Lie 2-groups and using the
latter to classify Lie 2-algebras in terms of 3rd cohomology
classes in Lie algebra cohomology.

This paper is available here:

http://www.arxiv.org/abs/math.QA/0307263

but the PDF version on my website looks a tiny bit better:

http://math.ucr.edu/home/baez/hda6.pdf

This kind of models were considered long ago (1983-84) by Nepomechie and
Orland, and somewhat less long ago by myself (1990); see
http://www.arxiv.org/abs/math-ph/0205017 for references. A similar
lattice model was recently considered in

http://www.arxiv.org/abs/hep-th/0304074
Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics
Authors: Hendryk Pfeiffer

The continuum version can be formulated as a gauge theory in loop space.
However, loop space is quite intractable, and besides one looses manifest
locality in ordinary space. As we discussed last year, one can probably
use some kind of gerbe-like structure instead, although the excitement
about gerbes from last year seems to have faded.

J A de Azcarraga, J M Izquierdo
"Lie groups, Lie algebras, cohomology and some applications in physics"
Cambridge U Press 1995

Moreover, H^1_rho(g,V) = H^2_rho(g,V) = 0 for any representation,
including the trivial one. As a special case, Whitehead's lemma states
that H^2_0(g,C) = 0.

So the only non-trivial 3-cocycle is the one you mention in your paper,
c(J^a, J^b, J^c) = f^ab_d k^dc, where f^ab_d are the structure constants
and k^dc the Killing metric. If I understand your paper correctly, this
means that your example 51 in fact exhausts the Lie 2-algebras, at least
the ones associated with semi-simple Lie algebras. Is this an interesting
result, or is it disappointing?

You also mention that any Lie 2-algebra gives a solution of Zamolodchikov
tetrahedron equation. Could you explicitly describe how this solution looks?

For those who have not heard of the tetrahedron equation, it is a generalization
of the Yang-Baxter (YB) equation. Recall that the YB equation,

  R_12 R_13 R_23 = R_23 R_13 R_12      

involves matrices R, which act on the triple tensor product V@V@V, where V is
some vector space, and R_ij acts non-trivially on the i:th and j:th factor
only. E.g., R_12 = R @ 1. Similarly, the tetrahedron equation acts on the
six-tuple tensor product V@V@V@V@V@V, where the factors are labelled by
pairs 1 <= i < j <= 4. An R-matrix now acts non-trivially on 3 factors only,
e.g. R_123 (which you might alternatively call R_12,13,23) acts on
V_12 @ V_13 @ V_23 and as the unit operator on V_14 @ V_24 @ V_34. In this
notation, the tetrahedron equation reads

  R_123 R_124 R_134 R_234 = R_234 R_134 R_124 R_123.

This can be continued to a hierarchy of p-simplex equations, with YB being
p = 2 and tetrahedron p = 3. The first equation in this hierarchy reads

  R_1 R_2 = R_2 R_1,

i.e. the matrices R_1 and R_2 (which act on the same vector space) commute.

These equations have natural interpretations as zero-curvature conditions
in p-form lattice gauge theory. Note that the p=1 equation can be written as

  R_1 R_2 (R_1)^-1 (R_2)^-1 = 1,

which means that the holonomy around a plaquette is unity. We can remember
this as "square = 1". Similarly, the YB equation can be rewritten as

 R_12 R_13 R_23 R_21 R_31 R_32 = 1,       (R_ji = (R_ij)^-1)

I have been recently trying to extend some of my effort on lattice EM
to more general gauge groups. This requires that I know something
about gauge theory :O

The first thing I am struggling with is coming up with a meaningful
lattice version of a fiber bundle. From what little I know so far, you
have some fiber bundle E, base manifold M, and a projection map
pi:E->M. However, I'm already stuck at this point because the lattice
version of this seems not so easy. For example, consider the cotangent
bundle. I would think of a section w in /\^1(K)of the cotangent bundle
T*(K) to be a 1-cochain on a simplicial complex K. However, 1-cochains
are associated with 1-simplices and not with points. So the projection
map seems like it should rather be something like

pi: T^*(K) -> K_1,

where K_1 is the 1-skeleton of K. In other words, the fibers should be
over edges and not nodes.

Does that make any sense?

Similarly, a 2-cochain is a section of the (simplicial version of the)
exterior bundle (is that what it's called?) with fibers over
2-simplices.

It seems to me like you should have different projection maps
corresponding to different degree simplices.

Has anyone tried to formalize a simplicial version of a fiber bundle?
Is there a such thing as a Lie algebra(group?)-valued simplicial
cochain?

I bring this up in this thread because your post is the first time I
have seen anyone mention placing the vector spaces on the edges. This
makes a lot more sense to me than placing the vector spaces on the
nodes. But then parallel transport should be from edge to edge across
faces, which sounds like it is related to 2-groups and other things
that I am totally clueless about :)

Best regards,
Eric

I'll have to point this out when I go back and tie up some
of the loose ends in my paper.

J. A. de Azcarraga, J. M. Izquierdo, J. C. Perez Bueno
An introduction to some novel applications of Lie algebra
cohomology and physics, available at
http://www.arXiv.org/abs/physics/9803046

but perhaps not carefully enough.

(What if g is not semisimple?   Can H^3_rho(g,V) be nonzero
for some nontrivial irrep of g in this case?)

Now, the theory of affine Lie algebras and quantum groups
is not the sort of thing that grows on trees, so it actually
confirms my hunch, slightly, to hear that there aren't tons
of other "semisimple Lie 2-algebras" out there.

[x,y] = xy - yx

at least in lots of examples.  

So, it shouldn't be surprising that any Lie algebra gives a Yang-Baxter
operator.  More precisely, a Lie algebra L over a field k gives a
Yang-Baxter operator on the vector space

L' = k + L

as follows:

B: L' tensor L' -> L' tensor L'

is given by

B((a,x) tensor (b,y)) = (b,y) tensor (a,x) + (1,0) tensor (0,[x,y])

Get it?  The "correction term" added on to the usual way
of switching things comes from the Lie bracket.  And, the
Yang-Baxter equation follows from the Jacobi identity!  There's
an even more conceptual explanation of this formula involving
quandles, but we're saving that for our next paper...

Anyway, in a Lie 2-algebra the Jacobi identity holds only
up to a natural transformation, the "Jacobiator".  So, the
Yang-Baxter equation holds only up to a natural transformation,
the "Yang-Baxterator".  Topologically this corresponds to the
*process of doing the 3rd Reidemeister move*.  But the Jacobiator
satisfies a certain equation of its own, and this corresponds
to the Zamolodchikov tetrahedron equation.

H^n(g,V) = \sum_{i+j=n} H^i(S,K) \otimes H^j(R,V)^L

By Whitehead lemmas, the terms with i=1,2 vanish, so we have

H^3(g,V) = H^3(R,V)^L +  H^3(S,K) \otimes V^L

In article <4b8cc0a6.0307310032.5f4fc...@posting.google.com>,

By this result, example 51 exhausts all the finite-dimensional
Lie 2-algebras in characteristic 0 which are not equivalent to
strict skeletal ones.  

But the *strict* skeletal ones are also quite interesting.
These are the ones where the Jacobi identity holds *on the nose*,
not just up to isomorphism.  And these are the ones where we have
a Lie algebra g, a representation rho of g on a vector space V,
and a *vanishing* element of the 3rd cohomology group of g with
coeficients in V.  

There are lots of these, of course - Whitehead's theorem
doesn't rule them out.  One way to get Lie 2-algebras of
this sort is from semidirect products of Lie algebras.  
An example is the Lie 2-algebra of the "Poincare 2-group".
This is the basis of Crane's new approach to quantum gravity.
See his recent papers for more details:

Louis Crane and David N. Yetter, Measurable categories and 2-groups,
available as http://www.arXiv.org/abs/math.QA/0305176.

Louis Crane and Marnie D. Sheppeard, 2-Categorical Poincare
representations and state sum applications, available as
http://www.arXiv.org/abs/math.QA/0306440.

In other words, particles of different mass correspond to
different projective unitary representations of the Galilei
group, with the mass m picking out a specific element of
the cohomology group H^2(G,U(1)).  

When we go to special relativity, and switch from the Galilei
group to the Poincare group, these projective representations
become honest representations.  The mass, which was the generator
of a central extension of the Galilei group, now becomes a central
element of the universal enveloping algebra of the Poincare group!
Or more precisely, its square does:

m^2 = p_0^2 - p_1^2 - p_2^2 - p_3^2.

Guillemin and Sternberg explain this transmutation in detail.  
This is also the place where I learned everthing I knew about
Whitehead's theorems on Lie algebra cohomology... until you told
me some more, that is!

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

News Story from Hong Kong:

A "gorilla" was detained by police in Central yesterday after it
snatched a bunch of bananas from an elderly woman's fruit shop,
sparking a bizarre chase that ended with the woman collapsing and
being rushed to hospital.

Grandmother Tse Lai was taken to Queen Mary Hospital after the
"gorilla" - a man in cosumte as part of a film crew making a
light-hearted TV show - pounced on a banana display at the Gage
Street shop.

The shocked shop owner responded by hitting out at the beast with
a broom before chasing it from the shop and collapsing.  Witnesses
said Mrs Tse managed to hit the gorilla twice on the head before
fainting.  Her husband, Lo Sum, 90, was stunned into silence but
a neighbor called the police.

Mrs. Tse later said: "I didn't realize that it was a gorilla at
first.  All I saw was something big and black with a lot of hair.
I thought I saw a ghost so I tried to drive it away with a broom.
But when I realised it was a gorilla I collapsed."

The gorilla, an expatriate who speaks Cantonese, refused to be
identified, but said he would send a bouquet of flowers to Mrs.
Tse.

- Sunday Morning Post, August 10, 2003.

In ordinary LGT parallel transport along a link is given by an
operator U in End(V) = V@V*. If we assume that we can identify the
dual space V* with V, we have U in V@V - the link has two endpoints.
Parallel transport along some curve is the path-ordered product
UUU..U, which still sits in V@V because the longer curve still has
two endpoints. In the continuum limit, we get a path-ordered
integral.

Now consider parallel transport across a plaquette in 2-form LGT. It
is given by a four-index quantity U in End(V@V) = V@V@V@V = V^4.
Surface parallel transport across a surface S with boundary C is now
given by a surface-ordered product, obtained by contracting indices
along inner edges of the triangulation of S. However, if C consists
of |C| links, this quantity is an element in V^|C|, the |C|-fold
tensor power of V; I have somewhere used the term barbed wire. There
is no problem to define this quantity on the lattice, but one should
keep in mind that surface-ordered products act on different spaces
depending on the number of links in the boundary. Moreover, the
continuum limit would involve continuum tensor products, which seem
quite awkward.

One reason why I might sound pessimistic is that I thought about
this 2-form lattice gauge model around 1990, but I never managed to
say anything significant about it. AFAIK, nor has anybody else
which have thought along similar lines.

Let me try to say something both correct and halfway comprehensible.  

A Lie 2-algebra is a kind of hybrid of a Lie algebra and
a category.  For starters, it is a category with a vector space
of objects and a vector space of morphisms.  The Lie bracket is
a functor, and it satisfies the Jacobi identity up to a natural
isomorphism called the "Jacobiator".  There are some axioms,
which I won't bother to explain here.  

If the source of any morphism equals its target, we say our
Lie 2-algebra is "skeletal".  

If the Jacobiator is the identity we say our Lie 2-algebra is
"strict".  

In either a skeletal or strict Lie 2-algebra, the vector space
of objects will form a Lie algebra.  This needn't be true
otherwise.

There's a concept of "equivalence" of Lie 2-algebras, and
every Lie 2-algebra is equivalent to a skeletal one.

Theorem 50 of my paper with Alissa Crans classifies
Lie 2-algebras up to equivalence.  Like I just said,
very Lie 2-algebra is equivalent to a skeletal one.
This in turn is specified (up to equivalence) by:  

1) the Lie algebra of objects, g
2) the vector space V of all endomorphisms of the zero object
3) a representation rho of g on V
4) an element of the cohomology group H^3_{rho}(g,V)

Item 4 is related to the Jacobiator.  In particular,
a skeletal Lie 2-algebra will be strict iff this element
of H^3_{rho}(g,V) is zero.  

Suppose g is a simple Lie algebra over a field k of characteristic
zero.  Then we can completely understand skeletal Lie algebras
having g as the Lie algebra of objects and having some finite-
dimensional space of morphisms!  The reason is that first
of all, we know all the finite-dimensional representations of
g.  If you don't know them, you can look them up in a book.  
Second of all, if rho is some finite-dimensional irreducible
representation of g, H^3_{rho}(g,V) = {0} unless rho is the
trivial representation, in which case H^3_{rho}(g,V) = k.  
Third, if rho is reducible it will be the direct sum of
irreducible representations, and the cohomology groups just add.

More generally, if g is semisimple, it's a direct sum of simple
Lie algebra so the problem reduces pretty easily to the previous
one.