- #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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