- #1
John Baez
Also available as http://math.ucr.edu/home/baez/week238.html
August 16, 2006
This Week's Finds in Mathematical Physics (Week 238)
John Baez
NASA is trying to built up suspense with this "media advisory":
1) NASA, NASA Announces Dark Matter Discovery,
http://www.nasa.gov/home/hqnews/2006/aug/HQ_M06128_dark_matter.html
which says simply:
Astronomers who used NASA's Chandra X-ray Observatory will host
a media teleconference at 1 p.m. EDT Monday, Aug. 21, to announce
how dark and normal matter have been forced apart in an extraordinarily
energetic collision.
Hmm! What's this about?
Someone nicknamed "riptalon" at Slashdot made a good guess. The media
advisory it lists the "briefing participants" as Maxim Markevitch, Doug
Clowe and Sean Carroll. Markevitch and Clowe work with the Chandra
X-ray telescope to study galaxy collisions and dark matter. Last
November, Markevitch gave a talk on this work, which you can see here:
2) Maxim Markevitch, Scott Randall, Douglas Clowe, and Anthony H. Gonzalez,
Insights on physics of gas and dark matter from cluster mergers, available at
http://cxc.harvard.edu/symposium_2005/proceedings/theme_energy.html#abs23
So, barring any drastic new revelations, we can guess what's up.
Markevitch and company have been studying the "Bullet Cluster", a
a bunch of galaxies that has a small bullet-shaped subcluster zipping
away from the center at 4,500 kilometers per second.
It seems that one of the rapidly moving galaxies in this subcluster
has hit a bystander galaxy - I'm not sure, but a high-speed collision
of galaxies occurred. When this kind of thing happens, the *gas* in
the galaxies is what actually collides; stars are too thinly spread to
hit very often. And when the gas collides, it gets hot. In this case,
it heated up to about 160 million degrees and started emitting X-rays
like mad! In fact, this may be hottest known galactic cluster.
That's fun. But that's not reason to call a press conference. The
cool part is not the crashing of gas against gas. The cool part is
that the dark matter in the galaxies was unstopped - it kept right
on going!
How do people know this? Simple. Folks can see the *gravity* of
the dark matter bending the light from galaxies further away!
So: X-rays show the gas here, but gravity shows most of the mass is
somewhere else. That's good evidence that dark matter is for real.
For more try these:
3) Maxim Markevitch, Chandra observation of the most interesting
cluster in the Universe, available at astro-ph/0511345.
4) M. Markevitch, A. H. Gonzalez, L. David, A. Vikhlinin, S. Murray,
W. Forman, C. Jones and W. Tucker, A Textbook Example of a Bow Shock
in the Merging Galaxy Cluster 1E0657-56, Astrophys.J. 567 (2002), L27.
Also available as astro-ph/0110468.
5) Eric Hayashi and Simon D. M. White, How rare is the bullet cluster?,
Mon. Not. Roy. Astron. Soc. Lett. 370 (2006), L38-L41, available as
astro-ph/0604443.
So, dark matter is seeming more and more real. In fact, last year
folks found evidence for "ghost galaxies" made mainly of dark matter
and cold hydrogen, with very few stars:
6) PPARC, New evidence for a dark matter galaxy,
http://www.interactions.org/cms/?pid=1023641
It thus becomes ever more interesting to find out what dark matter
actually *is*. The lightest neutralino? Axions? Theoretical
physicists are good at inventing plausible candidates, but finding
them is another thing.
Since I'd like to send this off in time to beat NASA, I won't say a
lot more today... just a bit.
Dan Christensen and Igor Khavkine have discovered some fascinating
things by plotting the amplitude of the tetrahedral spin network -
the basic building block of spacetime in 3d quantum gravity - as
a function of the cosmological constant.
They get pictures like this:
7) Dan Christensen and Igor Khavkin, Plots of q-deformed tets,
http://jdc.math.uwo.ca/spinnet/
Here the color indicates the real part of the spin network amplitude,
and it's plotted as a function of q, which is related to the
cosmological constant by a funky formula I won't bother to write down
here.
You can get some nice books on category theory for free these days:
8) Jiri Adamek, Horst Herrlich and George E. Strecker,
Abstract and Concrete Categories: the Joy of Cats, available at
katmat.math.uni-bremen.de/acc/acc.pdf
9) Robert Goldblatt, Topoi: the Categorial Analysis of Logic,
available at
http://cdl.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010
10) Michael Barr and Charles Wells, Toposes, Triples and Theories,
available at http://www.case.edu/artsci/math/wells/pub/ttt.html
The first two are quite elementary - don't be scared of the title
of Goldblatt's book; the only complaints I've ever heard about it
boil down to the claim that it's too easy!
You can also download this classic text on synthetic differential
geometry, which is an approach to differential geometry based on
infinitesimals, formalized using topos theory:
11) Anders Kock, Synthetic Differential Geometry, available at
http://home.imf.au.dk/kock/
He asks that we not circulate it in printed form - electrons are
okay, but not paper.
Next I want to say a *tiny* bit about Koszul duality for Lie
algebras, which plays a big role in the work of Castellani on
the M-theory Lie 3-algebra, which I discussed in "week237".
Let's start with the Maurer-Cartan form. This is a gadget that shows
up in the study of Lie groups. It works like this. Suppose you have
a Lie group G with Lie algebra Lie(G). Suppose you have a tangent
vector at any point of the group G. Then you can translate it to the
identity element of G and get a tangent vector at the identity of G.
But, this is nothing but an element of Lie(G)!
So, we have a god-given linear map from tangent vectors on G to the
Lie algebra Lie(G). This is called a "Lie(G)-valued 1-form" on G,
since an ordinary 1-form eats tangent vectors and spits out numbers,
while this spits out elements of Lie(G). This particular god-given
Lie(G)-valued 1-form on G is called the "Maurer-Cartan form", and
denoted omega.
Now, we can define exterior derivatives of Lie(G)-valued differential
forms just as we can for ordinary differential forms. So, it's
interesting to calculate d omega and see what it's like.
The answer is very simple. It's called the Maurer-Cartan equation:
d omega = - omega ^ omega
On the right here I'm using the wedge product of Lie(G)-valued
differential forms. This is defined just like the wedge product of
ordinary differential forms, except instead of multiplication of
numbers we use the bracket in our Lie algebra.
I won't prove the Maurer-Cartan equation; the proof is so easy you
can even find it on the Wikipedia:
12) Wikipedia, Maurer-Cartan form,
http://en.wikipedia.org/wiki/Maurer-Cartan_form
An interesting thing about this equation is that it shows
everything about the Lie algebra Lie(G) is packed into the
Maurer-Cartan form. The reason is that everything about the
bracket operation is packed into the definition of omega ^ omega.
If you have trouble seeing this, note that we can feed omega ^ omega
a pair of tangent vectors at any point of G, and it will spit out
an element of Lie(G). How will it do this? The two copies of omega
will eat the two tangent vectors and spit out elements of Lie(G).
Then we take the bracket of those, and that's the final answer.
Since we can get the bracket of *any* two elements of Lie(G) using
this trick, omega ^ omega knows everything about the bracket in
Lie(G). You could even say it's the bracket viewed as a geometrical
entity - a kind of "field" on the group G!
Now, since
d omega = - omega ^ omega
and the usual rules for exterior derivatives imply that
d(d omega) = 0
we must have
d(omega ^ omega) = 0
If we work this concretely what this says, we must get some identity
involving the bracket in our Lie algebra, since omega ^ omega is just
the bracket in disguise. What identity could this be?
THE JACOBI IDENTITY!
It has to be, since the Jacobi identity says there's a way to take
3 Lie algebra elements, bracket them in a clever way, and get zero:
[u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0
while d(omega ^ omega) is a Lie(G)-valued 3-form that happens to vanish,
built using the bracket.
It also has to be since the equation d^2 = 0 is just another way
of saying the Jacobi identity. For example, if you write out the
explicit grungy formula for d of a differential form applied to a
list of vector fields, and then use this to compute d^2 of that
differential form, you'll see that to get zero you need the Jacobi
identity for the Lie bracket of vector fields. Here we're just
using a special case of that.
The relationship between the Jacobi identity and d^2 = 0 is actually
very beautiful and deep. The Jacobi identity says the bracket is
a derivation of itself, which is an infinitesimal way of saying that
the flow generated by a vector field, acting on vector fields, preserves
their Lie bracket! And this, in turn, follows from the fact that the
Lie bracket is *preserved by diffeomorphisms* - in other words, it's
a "canonically defined" operation on vector fields.
Similarly, d^2 = 0 is related to the fact that d is a natural operation
on differential forms - in other words, that it commutes with
diffeomorphisms. I'll leave this cryptic; I don't feel like trying
to work out the details now.
Instead, let me say how to translate this fact:
d(d omega) = 0 IS SECRETLY THE JACOBI IDENTITY
into pure algebra. We'll get something called "Kozsul duality".
I always found Koszul duality mysterious, until I realized it's
just a generalzation of the above fact.
How can we state the above fact purely algebraically, only
using the Lie algebra Lie(G), not the group G? To get ourselves
in the mood, let's call our Lie algebra simply L.
By the way we constructed it, the Maurer-Cartan form is "left-invariant",
meaning it doesn't change when you translate it using maps like this:
L_g: G -> G
x |-> gx
that is, left multiplication by any element g of G. So,
how can we describe the left-invariant differential forms on G
in a purely algebraic way? Let's do this for *ordinary* differential
forms; to get Lie(G)-valued ones we can just tensor with L = Lie(G).
Well, here's how we do it. The left-invariant vector fields on G
are just
L
so the left-invariant 1-forms are
L*
So, the algebra of all left-invariant diferential forms on G
is just the exterior algebra on L*. And, defining the exterior
derivative of such a form is precisely the same as giving the
bracket in the Lie algebra L! And, the equation d^2 = 0 is
just the Jacobi identity in disguise.
To be a bit more formal about this, let's think of L as a graded
vector space where everything is of degree zero. Then L* is the
same sort of thing, but we should *add one to the degree* to think
of guys in here as 1-forms. Let's use S for the operation of "suspending"
a graded vector space - that is, adding one to the degree. Then
the exterior algebra on L* is the "free graded-commutative algebra on SL*".
So far, just new jargon. But this let's us state the observation
of the penultimate paragraph in a very sophisticated-sounding way.
Take a vector space L and think of it as a graded vector space
where everything is of degree zero. Then:
Making the free graded-commutative algebra on SL* into a *differential*
graded-commutative algebra is the same as making L into a Lie algebra.
This is a basic example of "Koszul duality". Why do we call it
"duality"? Because it's still true if we switch the words
"commutative" and "Lie" in the above sentence!
Making the free graded Lie algebra on SL* into a *differential*
graded Lie algebra is the same as making L into a commutative algebra.
That's sort of mind-blowing. Now the equation d^2 = 0 secretly
encodes the *commutative law*.
So, we say the concepts "Lie algebra" and "commutative algebra" are
Koszul dual. Interestingly, the concept "associative algebra" is its
own dual:
Making the free graded associative algebra on SL* into a *differential*
graded associative algebra is the same as making L into an associative
algebra.
This is the beginning of a big story, and I'll try to say more later.
If you get impatient, try the book on operads mentioned in "week191",
or else these:
13) Victor Ginzburg and Mikhail Kapranov, Koszul duality for quadratic
operads, Duke Math. J. 76 (1994), 203-272. Also Erratum, Duke Math.
J. 80 (1995), 293.
14) Benoit Fresse, Koszul duality of operads and homology of partition
posets, Homotopy theory and its applications (Evanston, 2002),
Contemp. Math. 346 (2004), 115-215. Also available at
http://math.univ-lille1.fr/~fresse/PartitionHomology.html
The point is that Lie, commutative and associative algebras are all
defined by "quadatic operads", and one can define for any such operad
O a "dual" operad O* such that:
Making the free graded O-algebra on SL* into a *differential*
graded O-algebra is the same as making L into an O*-algebra.
And, we have O** = O, hence the term "duality".
This has always seemed incredibly cool and mysterious to me.
There are other meanings of the term "Koszul duality", and if
really understood them I might better understand what's going on
here. But, I'm feeling happy now because I see this special case:
Making the free graded-commutative algebra on SL* into a *differential*
graded-commutative algebra is the same as making L into a Lie algebra.
is really just saying that the exterior derivative of left-invariant
differential forms on a Lie group encodes the bracket in the Lie algebra.
That's something I have a feeling for. And, it's related to the
Maurer-Cartan equation... though notice, I never completely spelled out
how.
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
August 16, 2006
This Week's Finds in Mathematical Physics (Week 238)
John Baez
NASA is trying to built up suspense with this "media advisory":
1) NASA, NASA Announces Dark Matter Discovery,
http://www.nasa.gov/home/hqnews/2006/aug/HQ_M06128_dark_matter.html
which says simply:
Astronomers who used NASA's Chandra X-ray Observatory will host
a media teleconference at 1 p.m. EDT Monday, Aug. 21, to announce
how dark and normal matter have been forced apart in an extraordinarily
energetic collision.
Hmm! What's this about?
Someone nicknamed "riptalon" at Slashdot made a good guess. The media
advisory it lists the "briefing participants" as Maxim Markevitch, Doug
Clowe and Sean Carroll. Markevitch and Clowe work with the Chandra
X-ray telescope to study galaxy collisions and dark matter. Last
November, Markevitch gave a talk on this work, which you can see here:
2) Maxim Markevitch, Scott Randall, Douglas Clowe, and Anthony H. Gonzalez,
Insights on physics of gas and dark matter from cluster mergers, available at
http://cxc.harvard.edu/symposium_2005/proceedings/theme_energy.html#abs23
So, barring any drastic new revelations, we can guess what's up.
Markevitch and company have been studying the "Bullet Cluster", a
a bunch of galaxies that has a small bullet-shaped subcluster zipping
away from the center at 4,500 kilometers per second.
It seems that one of the rapidly moving galaxies in this subcluster
has hit a bystander galaxy - I'm not sure, but a high-speed collision
of galaxies occurred. When this kind of thing happens, the *gas* in
the galaxies is what actually collides; stars are too thinly spread to
hit very often. And when the gas collides, it gets hot. In this case,
it heated up to about 160 million degrees and started emitting X-rays
like mad! In fact, this may be hottest known galactic cluster.
That's fun. But that's not reason to call a press conference. The
cool part is not the crashing of gas against gas. The cool part is
that the dark matter in the galaxies was unstopped - it kept right
on going!
How do people know this? Simple. Folks can see the *gravity* of
the dark matter bending the light from galaxies further away!
So: X-rays show the gas here, but gravity shows most of the mass is
somewhere else. That's good evidence that dark matter is for real.
For more try these:
3) Maxim Markevitch, Chandra observation of the most interesting
cluster in the Universe, available at astro-ph/0511345.
4) M. Markevitch, A. H. Gonzalez, L. David, A. Vikhlinin, S. Murray,
W. Forman, C. Jones and W. Tucker, A Textbook Example of a Bow Shock
in the Merging Galaxy Cluster 1E0657-56, Astrophys.J. 567 (2002), L27.
Also available as astro-ph/0110468.
5) Eric Hayashi and Simon D. M. White, How rare is the bullet cluster?,
Mon. Not. Roy. Astron. Soc. Lett. 370 (2006), L38-L41, available as
astro-ph/0604443.
So, dark matter is seeming more and more real. In fact, last year
folks found evidence for "ghost galaxies" made mainly of dark matter
and cold hydrogen, with very few stars:
6) PPARC, New evidence for a dark matter galaxy,
http://www.interactions.org/cms/?pid=1023641
It thus becomes ever more interesting to find out what dark matter
actually *is*. The lightest neutralino? Axions? Theoretical
physicists are good at inventing plausible candidates, but finding
them is another thing.
Since I'd like to send this off in time to beat NASA, I won't say a
lot more today... just a bit.
Dan Christensen and Igor Khavkine have discovered some fascinating
things by plotting the amplitude of the tetrahedral spin network -
the basic building block of spacetime in 3d quantum gravity - as
a function of the cosmological constant.
They get pictures like this:
7) Dan Christensen and Igor Khavkin, Plots of q-deformed tets,
http://jdc.math.uwo.ca/spinnet/
Here the color indicates the real part of the spin network amplitude,
and it's plotted as a function of q, which is related to the
cosmological constant by a funky formula I won't bother to write down
here.
You can get some nice books on category theory for free these days:
8) Jiri Adamek, Horst Herrlich and George E. Strecker,
Abstract and Concrete Categories: the Joy of Cats, available at
katmat.math.uni-bremen.de/acc/acc.pdf
9) Robert Goldblatt, Topoi: the Categorial Analysis of Logic,
available at
http://cdl.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010
10) Michael Barr and Charles Wells, Toposes, Triples and Theories,
available at http://www.case.edu/artsci/math/wells/pub/ttt.html
The first two are quite elementary - don't be scared of the title
of Goldblatt's book; the only complaints I've ever heard about it
boil down to the claim that it's too easy!
You can also download this classic text on synthetic differential
geometry, which is an approach to differential geometry based on
infinitesimals, formalized using topos theory:
11) Anders Kock, Synthetic Differential Geometry, available at
http://home.imf.au.dk/kock/
He asks that we not circulate it in printed form - electrons are
okay, but not paper.
Next I want to say a *tiny* bit about Koszul duality for Lie
algebras, which plays a big role in the work of Castellani on
the M-theory Lie 3-algebra, which I discussed in "week237".
Let's start with the Maurer-Cartan form. This is a gadget that shows
up in the study of Lie groups. It works like this. Suppose you have
a Lie group G with Lie algebra Lie(G). Suppose you have a tangent
vector at any point of the group G. Then you can translate it to the
identity element of G and get a tangent vector at the identity of G.
But, this is nothing but an element of Lie(G)!
So, we have a god-given linear map from tangent vectors on G to the
Lie algebra Lie(G). This is called a "Lie(G)-valued 1-form" on G,
since an ordinary 1-form eats tangent vectors and spits out numbers,
while this spits out elements of Lie(G). This particular god-given
Lie(G)-valued 1-form on G is called the "Maurer-Cartan form", and
denoted omega.
Now, we can define exterior derivatives of Lie(G)-valued differential
forms just as we can for ordinary differential forms. So, it's
interesting to calculate d omega and see what it's like.
The answer is very simple. It's called the Maurer-Cartan equation:
d omega = - omega ^ omega
On the right here I'm using the wedge product of Lie(G)-valued
differential forms. This is defined just like the wedge product of
ordinary differential forms, except instead of multiplication of
numbers we use the bracket in our Lie algebra.
I won't prove the Maurer-Cartan equation; the proof is so easy you
can even find it on the Wikipedia:
12) Wikipedia, Maurer-Cartan form,
http://en.wikipedia.org/wiki/Maurer-Cartan_form
An interesting thing about this equation is that it shows
everything about the Lie algebra Lie(G) is packed into the
Maurer-Cartan form. The reason is that everything about the
bracket operation is packed into the definition of omega ^ omega.
If you have trouble seeing this, note that we can feed omega ^ omega
a pair of tangent vectors at any point of G, and it will spit out
an element of Lie(G). How will it do this? The two copies of omega
will eat the two tangent vectors and spit out elements of Lie(G).
Then we take the bracket of those, and that's the final answer.
Since we can get the bracket of *any* two elements of Lie(G) using
this trick, omega ^ omega knows everything about the bracket in
Lie(G). You could even say it's the bracket viewed as a geometrical
entity - a kind of "field" on the group G!
Now, since
d omega = - omega ^ omega
and the usual rules for exterior derivatives imply that
d(d omega) = 0
we must have
d(omega ^ omega) = 0
If we work this concretely what this says, we must get some identity
involving the bracket in our Lie algebra, since omega ^ omega is just
the bracket in disguise. What identity could this be?
THE JACOBI IDENTITY!
It has to be, since the Jacobi identity says there's a way to take
3 Lie algebra elements, bracket them in a clever way, and get zero:
[u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0
while d(omega ^ omega) is a Lie(G)-valued 3-form that happens to vanish,
built using the bracket.
It also has to be since the equation d^2 = 0 is just another way
of saying the Jacobi identity. For example, if you write out the
explicit grungy formula for d of a differential form applied to a
list of vector fields, and then use this to compute d^2 of that
differential form, you'll see that to get zero you need the Jacobi
identity for the Lie bracket of vector fields. Here we're just
using a special case of that.
The relationship between the Jacobi identity and d^2 = 0 is actually
very beautiful and deep. The Jacobi identity says the bracket is
a derivation of itself, which is an infinitesimal way of saying that
the flow generated by a vector field, acting on vector fields, preserves
their Lie bracket! And this, in turn, follows from the fact that the
Lie bracket is *preserved by diffeomorphisms* - in other words, it's
a "canonically defined" operation on vector fields.
Similarly, d^2 = 0 is related to the fact that d is a natural operation
on differential forms - in other words, that it commutes with
diffeomorphisms. I'll leave this cryptic; I don't feel like trying
to work out the details now.
Instead, let me say how to translate this fact:
d(d omega) = 0 IS SECRETLY THE JACOBI IDENTITY
into pure algebra. We'll get something called "Kozsul duality".
I always found Koszul duality mysterious, until I realized it's
just a generalzation of the above fact.
How can we state the above fact purely algebraically, only
using the Lie algebra Lie(G), not the group G? To get ourselves
in the mood, let's call our Lie algebra simply L.
By the way we constructed it, the Maurer-Cartan form is "left-invariant",
meaning it doesn't change when you translate it using maps like this:
L_g: G -> G
x |-> gx
that is, left multiplication by any element g of G. So,
how can we describe the left-invariant differential forms on G
in a purely algebraic way? Let's do this for *ordinary* differential
forms; to get Lie(G)-valued ones we can just tensor with L = Lie(G).
Well, here's how we do it. The left-invariant vector fields on G
are just
L
so the left-invariant 1-forms are
L*
So, the algebra of all left-invariant diferential forms on G
is just the exterior algebra on L*. And, defining the exterior
derivative of such a form is precisely the same as giving the
bracket in the Lie algebra L! And, the equation d^2 = 0 is
just the Jacobi identity in disguise.
To be a bit more formal about this, let's think of L as a graded
vector space where everything is of degree zero. Then L* is the
same sort of thing, but we should *add one to the degree* to think
of guys in here as 1-forms. Let's use S for the operation of "suspending"
a graded vector space - that is, adding one to the degree. Then
the exterior algebra on L* is the "free graded-commutative algebra on SL*".
So far, just new jargon. But this let's us state the observation
of the penultimate paragraph in a very sophisticated-sounding way.
Take a vector space L and think of it as a graded vector space
where everything is of degree zero. Then:
Making the free graded-commutative algebra on SL* into a *differential*
graded-commutative algebra is the same as making L into a Lie algebra.
This is a basic example of "Koszul duality". Why do we call it
"duality"? Because it's still true if we switch the words
"commutative" and "Lie" in the above sentence!
Making the free graded Lie algebra on SL* into a *differential*
graded Lie algebra is the same as making L into a commutative algebra.
That's sort of mind-blowing. Now the equation d^2 = 0 secretly
encodes the *commutative law*.
So, we say the concepts "Lie algebra" and "commutative algebra" are
Koszul dual. Interestingly, the concept "associative algebra" is its
own dual:
Making the free graded associative algebra on SL* into a *differential*
graded associative algebra is the same as making L into an associative
algebra.
This is the beginning of a big story, and I'll try to say more later.
If you get impatient, try the book on operads mentioned in "week191",
or else these:
13) Victor Ginzburg and Mikhail Kapranov, Koszul duality for quadratic
operads, Duke Math. J. 76 (1994), 203-272. Also Erratum, Duke Math.
J. 80 (1995), 293.
14) Benoit Fresse, Koszul duality of operads and homology of partition
posets, Homotopy theory and its applications (Evanston, 2002),
Contemp. Math. 346 (2004), 115-215. Also available at
http://math.univ-lille1.fr/~fresse/PartitionHomology.html
The point is that Lie, commutative and associative algebras are all
defined by "quadatic operads", and one can define for any such operad
O a "dual" operad O* such that:
Making the free graded O-algebra on SL* into a *differential*
graded O-algebra is the same as making L into an O*-algebra.
And, we have O** = O, hence the term "duality".
This has always seemed incredibly cool and mysterious to me.
There are other meanings of the term "Koszul duality", and if
really understood them I might better understand what's going on
here. But, I'm feeling happy now because I see this special case:
Making the free graded-commutative algebra on SL* into a *differential*
graded-commutative algebra is the same as making L into a Lie algebra.
is really just saying that the exterior derivative of left-invariant
differential forms on a Lie group encodes the bracket in the Lie algebra.
That's something I have a feeling for. And, it's related to the
Maurer-Cartan equation... though notice, I never completely spelled out
how.
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
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