 This video is the third part of the Wienberg lecture on Wienberg's algorithm and Katz-Mudin algebras. It might not make a whole lot of sense unless you've seen the first two parts of the lecture, so I'll put links to them in the description of the video below. Now if I think of it, it might not make a lot of sense even if you have seen the first two parts of the lecture. What we're going to discuss in this lecture is whether you can assign nicely algebras to Wienberg's group. So to recall what we did in the first two parts of this lecture, Wienberg looked at the lattices i,n,1. So this is the odd unimodular Laurentian that is consisting of ornums m1 up to mn, mn plus 1, with all the mi in z, and m1 squared plus plus mn squared minus mn plus 1 squared is the norm of this vector, the square of its length. Notice there's a minus sign here. And Wienberg calculated the Dinken diagrams of the reflection groups, and you remember he found these rather intriguing diagrams that two of them looked like this, for example. Now, there's one obvious way to get Dinken diagrams, which is that if you take a Lie algebra, the Lie algebra, you might want it to be finite dimensional and semi-simple, then this gives you a Dinken diagram. For example, if we take the Lie algebra of all n by n matrices over the reals, then this gives us a Dinken diagram, which looks like this, with n minus 1 points. So this is the a n minus 1 Dinken diagram. And we can ask, do Wienberg's Dinken diagrams also correspond to various Lie algebras? So what happens is we can look at spherical reflection groups, and these correspond to the usual finite dimensional Lie algebras. We also looked at Euclidean reflection groups. And these give the so-called affine Lie algebras, more or less. So a typical example of this would be the group of n by n matrices with coefficients that are Laurent polynomials in the reals. Actually, that's not quite true. There's a little extra fuss you have to do about the center of the Lie algebra, but I won't worry about this. And finally, you can ask, what about hyperbolic reflection groups? So for example, we can look at Wienberg's examples of the lattices i, n1 and the even lattice i, i, n1. And we can ask, what's the algebras to these give rise to? And I'll be discussing this lecture, what we want to put here. So Katz and Moody gave one answer to this. What they found was a way of associating any to any Dinken diagram, some sort of Lie algebra. This is called a Katz Moody algebra. And so for the finite dimension Euclidean Dinken diagrams, this gives the finite dimension and Euclidean Lie algebras. So what we need to do is to see, how do you get from a Dinken diagram to Lie algebra? So what does a Dinken diagram look like? Well, a Dinken diagram has a bunch of points and the points have various numbers of lines between them. The way you get from a Dinken diagram to Lie algebra is as follows. First of all, for any point, you get a copy of the Lie algebra SL2. So that's just little two by two matrices A, B, C, D with A plus D is zero under the usual Lie bracket. And the lines between the points tell you how these two little copies of SL2 interact. For example, if you have two points like this, this means that the two corresponding copies of SL2 commute with each other. And if you have two copies of SL2 and a single line between them, then what happens? This means the two copies of SL2 don't quite commute. They sort of interact in the way that, well, there are two obvious copies of SL2. There's one copy here and there's another copy there. And you see they sort of overlap in some sense. And a line between two copies of SL2 means they overlap in this sense. And if you assign more lines, then more complicated things happen. Well, that's a sort of rather informal way of describing the algebra of a Dinken diagram. More precisely, you can write down the Harish-Chandra-Sare relations. So what we do is we take numbers E, I, F, I and H, I for each point. So these are going to be generators of the Lie algebra. And now we want some relations telling you how these generators fit together. So the relations look like this. First of all, H, I, H, J commute with each other. This means the H, I... If you take all the H, I's, they form an abelian Lie algebra called the Cartan algebra, Cartan sub-algebra, Lie algebra. The Cartan sub-algebra sort of behaves a bit like the diagonal matrices of the N by N matrices. Next, we introduce relations saying what the EIs and the HIs interact, how they interact with each other. So we have H, I, E, J equals A, I, J, E, J and H, I, F, J is minus A, I, J, F, J. These numbers A, I, J are mysterious constants making up something called the Cartan matrix. And the Cartan matrix is determined by the number of lines between points in the Lie algebra. So what's going on here is this says the numbers E, E, J and H, J, sorry, E, J and F, J are eigenvectors of the Cartan sub-algebra H. So let's call capital H for the abelian Lie algebra generated by this. And this just means they're eigenvectors. In the case of the general linear group, the EIs and the FJs are things that live just above or just below the diagonal. So you can think of the EIs would have a non-zero entry in one of these positions and the FIs would have a non-zero entry in one of these positions and be the zero somewhere else. So the EIs and the HIs sort of give you the diagonal entries and the EIs and the FIs give you the entries that are just above or just below the diagonal. Next we have some more relations saying E, I, F, J equals naught for I, not equal to J and H, I, if I is equal to J. And what this relation says is that the generators E, I, F, I and H, I form a little sub-algebra that looks like SL2 of the real. So it's the three-dimensional, the algebra that everybody knows about. And in matrix form these are just the little 2 by 2 blocks on the diagonal. So a typical one of these, the algebras might look like a 2 by 2 block there. So these are the relations that are easy to understand. We also have some rather more complicated relations, one of which says that EI, EI, SON, EI, EJ is equal to zero and the same for the FIs. So what do these say? Well, first of all you have to say how many EIs you put there but it turns out it doesn't really matter how many EIs you put there. I mean, people have some complicated formula depending on these numbers A, I, J but in fact it turns out that provide you've got a sufficiently large number of EIs that all these relations are equivalent. And what these say is that the operations taking a Libra with EI or with FI are locally nilpotent. What this means is they're not quite nilpotent but they're nilpotent if you sort of apply them to any finite dimensional subspace. In the finite dimensional case, this says they are actually nilpotent. This is very nice because it means that when you act with SL2R on the cat's moody algebra, the cat's moody algebra splits up as a sum of finite dimensional representations of SL2R which is good because we know all about the finite dimensional representations of SL2R. So these relations sort of say the cat's moody algebra behaves very nicely as a representation of all these SL2Rs. So cat's moody pointed out you would take any dink and diagram whatsoever just write down these relations and this gives you the algebra. And as I said for the finite dimensional reflection groups, this gives you the finite dimensional of the algebras and for affine reflection groups, this gives you the affinely algebras. So this gives us one answer to our question. If we've got any hyperbolic reflection group, we get a dink and diagram and from the dink and diagram we get a cat's moody algebra. So, well, that seems to answer the question but it's an entirely satisfactory answer because it turns out the cat's moody algebras you get are kind of a bit wild and difficult to understand and what we really like is a nice lean algebra here. And the question is what do you mean by a nicely algebra? Well, everybody agrees the finite dimensional the algebras and the affinely algebras are nice. So let's see what's nice about them. Well, what's nice about them is we can write down an explicit character formula which was discovered by Hermann Weier and it's called the Weier character formula. The finite dimensional one as well. Cat's noted that it could be extended to all cat's moody algebras to the Weier cat's formula. It's not completely straightforward extending it because if you look at all the proofs of the Weier character formula and finite dimensions none of them actually work if the le algebra is infinite dimensional but cat's discovered that if you take one of these proofs and kind of stretch it a little bit you can get it to work for infinite dimensions. The main problem cat's had to solve is this proof used the Casimir operator which doesn't quite make sense for infinite dimensional le algebras but cat's discovered a clever way to regularize it so it did make sense and was then able to push through a proof. Now I'm going to be most interested in the case of the character formula for the one dimensional trivial representation. It sounds kind of stupid writing down the character formula for the one dimensional representation because we know the character of the one dimensional representation. But the Weier cat's character formula says that the character of a representation is equal to a certain sum over a certain product and this is rather nice because if we know the character is one then we're going to get a nice identity saying the sum, a certain sum is equal to a certain product. So we get an identity saying the sum is a product which is the Weier denominator formula and the Weier denominator formula looks like this and what we do is we take a sum over something called the Weier group and this is equal to a product over something where these are something called a positive roots and row that appears all over the place here and here is something called the Weier vector. This is just a sign plus or minus what we're depending on the determinant of the element of the Weier group. So I just give a couple of examples of the Weier denominator formula for finite dimensional and affine algebras. So for finite dimensions the Weier denominator formula is just the Vandermonde identity at least for the le algebras a and minus one. So you remember the Vandermonde identity says if you've got a matrix 1, 1, 1 x0 x1 x2 y0 x0 squared x1 squared x2 squared and you want to know it's determinant. This is the product over i less than j of x i minus x j up to a sign which I always get wrong and you see what we have here is a product over i less than j and you can see that i less than j corresponds to three upper triangle entries here which are the three positive roots of a little 2 by 2 matrix and the sum is over if you expand out this determinant you get six terms which correspond to the six entries of the symmetric group S3 on three points and this is just the Weier group of the le algebra of 3 by 3 matrices. So here this is really a sum over the Weier group and this is a product over positive roots. So the usual Vandermonde identity is a typical example of the Weier denominator formula in finite dimensions. The simplest example of the Weier denominator formula for affine algebras is the famous Jacobi triple product identity and to get the Jacobi triple product identity what you do is you draw this picture here and what this is is it's really the root system of it's the root system of SL2 are with coefficients this and the wrong polynomials so here what we have is the root system of SL2 and it sort of gets shifted to the right or the left whenever you sort of multiply by Z and what I'm going to do is I'm going to take the positive roots to be these points here and to each positive root I'm going to assign a factor as follows so this is going to be here I'm going to put 1-q squared, 1-q to the 4 and so on here I'm going to put 1-qz here I'm going to put 1-qz to the minus 1 here I'm going to put 1-q cubed z to the minus 1 here 1-q cubed z and so on so what we get is a big product that looks like a product over n greater than 0 of 1-q to the 2n 1-q to the 2n-1z 1-q to the 2n-1z to the minus 1 and if you expand out this product what you would mostly expect to get is some sort of horrible complicated mess it's fairly obvious that all the terms of this product when you expand it out are going to lie inside a certain parabola and normally you'd expect to have something very complicated on the inside but it turns out that all terms inside the parabola vanish what's happening is there's sort of massive amounts of cancellation going on so what we're left with is the terms that are exactly on the parabola which are easy to work out it's just sum over n of minus 1 to the n q to the n squared z to the n so it's very easy to remember this identity if you remember this picture of the root lattice of obviously algebra here then the product is just the positive roots and the sum is just the sum over the boundary of the obvious region containing the terms of this product and Mcdonald found such identities or all the other affine root systems they called them Mcdonald identities many of them were sort of found by Dyson who sort of overlooked the connection with the algebras which he was very frustrated by Mcdonald came up with these identities slightly before affine-cats moody algebras were well known and cat sort of said that when he looked at the Mcdonald identities and looked at affine-cats algebras it was just sort of immediately obvious that the Mcdonald identities were the denominator formula for the affine-cats moody algebras so that was a very nice explanation the Mcdonald identities were originally a bit puzzling because the sum was, the product was not only over the roots of the affine reflection group which were these terms and these terms but there were some extra terms here which were a bit puzzling but these turned out to be the norm zero roots of the corresponding affine algebra so the reason why we get nice identities is we know, first of all we know the simple roots now knowing the simple roots tells us what the vial group is just the reflection group of the simple group and this tells us what the sum is in the vial denominator formula we need to know the root multiplicities I didn't quite make these obvious but if you go back to the vial denominator formula we should really take several factors if the root alpha has multiplicity greater than one so there should really be a factor here where you put in the multiplicity of alpha so we also know the root multiplicities in both the finite dimensional and affine cases and this tells us what the product is in the vial denominator formula and now the problem is that if we write down a Dinkindragon we certainly know what the simple roots are but the root multiplicities seem to be very complicated in general now for finite dimensional algebras they're easy they're just one but affine algebras they're much more complicated they're usually either one or the rank or something something a bit related they're easy to describe so what do the root multiplicities of these more general cats moody algebras look like well people did a lot of computer calculations and there are some examples in cats' books so here's the root multiplicities of a little rank two if you look at them the multiplicities it's rather hard to say anything much about them let me just magnify a bit so you can see so if I zoom in a bit so you can see the multiplicities I mean they're not too difficult to calculate by computer but it's hard to say anything very interesting about them and you can write a product formula for them to some extent a more interesting case is it was done by Frenkel and Feingold who looked at this rank 3-le algebra this rank 3-le algebra this turns up as the reflection group of the modular group so its vial group is essentially just gl2 over the integers and if you look at the multiplicities something quite interesting has happened first of all you see that various numbers are repeating quite often so we've got a number 30 appearing 3 times so 3 different orbits of roots and then we get numbers 42 appearing 3 times and 56 and so on and if you look at the multiplicities well there's a sequence 2 3 5 7 11 and every number theorist will immediately recognize these numbers as the values of the partition function they're also primes for the first few but that's just a coincidence they're really values of the partition function so what's the partition function well the partition function is just the number of ways of writing the number as a sum of positive integers so the partition numbers look like this and for example the fourth entry is 5 and that's because we can write 4 is equal to 4 or 3 plus 1 or 2 plus 2 or 2 plus 1 plus 1 or 1 plus 1 plus 1 plus 1 and these numbers are usually noted by pn and Euler found this very nice formula for p of n sum of pn times q to the n is just equal to product over n greater than 0 of 1 over 1 minus q to the n so it's 1 over 1 minus q 1 over 1 minus q squared and so on so that looks very exciting we've got a lot of numerical evidence that we're getting the partition function appearing here except we don't if you go on further you know you go up to here for example again you can see one of the coefficients is 627 but the other is only 626 so we're getting the partition function for quite a long way except it suddenly starts breaking down and as you as you go further it breaks down more and more so we've got something rather puzzling going on here it almost looks as if we've got a nice formula for multiplicity but then it goes wrong so I will explain later on why this the algebra almost has root multiplicities that are given by the partition function well the most striking hyperbolic reflection group was Conway's reflection group so you remember Conway calculated the reflection group of the 26th dimensional even Lorenzian lattice and found that it's dink and diagram is just the leach lattice for details see the second part of this lecture so the obvious thing to do is to look at the what are the multiplicities of the corresponding Katz-Mudi algebra so we're taking the whole leach lattice as a dink and diagram but so it's got an infinite number of vertices and it's sort of very complicated and looking at Katz-Mudi algebra and Igor Frenkor made this absolutely amazing discovery he showed that multiplicities of a root alpha is at most p24 of 1 minus alpha squared over 2 so what's p24 well the sum of p24n q to the n is the product of n greater than 0 of 1 over 1 minus q to the n to the 24 so it's like the partition function except you've got this 24th power here so these series numbers get big quite quickly so it's 1 plus 24q plus 324q squared and so on and furthermore he showed that there are lots of vectors that have exactly this multiplicity so we get equality if alpha beta equals 1 where beta squared is equal to 0 and beta does not correspond to the leach lattice so you remember there are 23 orbits of vectors of norm 0 correspond to the 23 neomy lattices and if a root is in a product 1 with at least one of those then the multiplicity is exactly 24 if beta does correspond to the leach lattice the multiplicity actually becomes less than this number here and the way Igor Frenkor proved this is he used string theory more precisely he used the no ghost theorem from string theory which predicts the dimension of some sort of space of strings to be given by this number here in 26 dimensions so the you know string theory has this problem that predicts space time is 26 dimensions which is a real headache for physicists but is really great for mathematicians because it's actually the right dimension for this lattice here as Igor Frenkor discovered so we can also ask how good is Igor Frenkor's upper bound for other vectors well let's take a look at some vectors alpha in the 26th dimensional lattice and see what the multiplicity is so if alpha squared is equal to 2 and then there's one orbit and the multiplicity is just one so it's exact if alpha squared equals 0 then there are 24 times infinity orbits that's because there are 24 orbits of primitive norm 0 vectors and you can multiply any of them by a positive integer and for 23 of these the multiplicity is equal to 24 and for 1 the multiplicity is equal to 0 that's the leaf lattice causing trouble alpha squared is equal to minus 2 and things start getting really complicated there are 121 orbits and you can work out the multiplicity for each of these and it turns out 119 have multiplicity 324 exactly equal to Frenkor's upper bound and two of them have multiplicity slightly less and for alpha squared is minus 4 it's similar so there are 665 orbits and most of them have multiplicity 3200 which is the next coefficient and there's a small hand for which have multiplicity slightly less and you see this is suspiciously similar to what was happening with the example calculated by Feingold and Frenkor that many of the roots have multiplicity is given by some nice coefficient but eventually the roots start having multiplicity a little bit less than that so something is almost working but starts to break down after a while and so how can we explain this well this gives a strong hint that is a slightly larger algebra with multiplicity exactly p24 of 1 minus alpha squared over 2 and what can this algebra be well it can't be a cat's moody algebra because that doesn't quite work but it turns out you can generalise the notion of the cat's moody algebra so this is a generalisation of cat's moody algebras and the generalisation is quite simple so the cat's moody algebra depends on these numbers aij and it always assumes that aii is equal to whenever ii is whenever 2 of these entries is the same then the diagonal entry is 2 and this corresponds to the fact that all simple roots have norm greater than 0 the norm of the simple root is closer to aij the exact way depends on how you normalise things well what we should do is we can also allow aii to be less than or equal to 0 and most of the theory of cat's moody algebras goes through with some slight changes so these correspond to so-called imaginary simple roots where the norm is less than or equal to 0 the term imaginary is a bit unfortunate because it conflicts with another use for the term imaginary in the theory of Li algebras here the term imaginary means that the norm which is roughly the square root of aii is an imaginary number rather than a real number and we need to modify the relation slightly so if aii is less than or equal to 0 we discard the relations that say ei, ei and so on ej is equal to 0 and of course we discard the ones and so on and we just keep a tiny piece of them we keep the relation ei fj equal to 0 if aij is 0 so other than that the relations are very similar to the relations for cat's moody algebras and again the whole theory of cat's moody algebras extends the slightly more general case in particular we get a character formula and a denominator formula which is a little bit more complicated because the imaginary simple roots give some extra correction terms so another way of thinking of the difference is that these simple roots of aii was 2 all correspond to copies of SL2 of r as we saw earlier the simple roots of aii or 0 correspond to copies of the Heisenberg algebra which is another three-dimensional of the algebra it's a sort of degeneration of SL2 in some sense the ones with aii are less than 0 also correspond to copies of SL2 of r except that I mentioned earlier that the cat's moody algebras is a sum of finite dimensional representations of this SL2 r well if aii is less than 0 this is no longer true and the cat's moody algebra is in general some infinite dimensional discrete series representations of SL2 so there are some differences but most of the theory goes through and so we can ask can we find the algebra one of these more general forms of cat's moody algebra we can so the algebra can be described as follows so this is going to be the root lattice and for each of these vectors we need to describe the multiplicity and the multiplicity of alpha is going to be p of 1 minus p so p24 of 1 minus alpha squared over 2 for alpha not 0 and 26 of alpha is equal to 0 so this is a really hugely algebra if we sort of draw the light cone then here it's got a lot of vectors of norm 2 each of which has multiplicity 1 and then it's got a bunch of vectors of norm 0 on the light cone each with multiplicity 24 and this light cone is living in 26 dimensional rents in space and then it's got some more vectors lying on the surface of vectors with alpha squared equals minus 2 and these vectors all the vectors lying on here have multiplicity 24 and then there are more vectors lying on the next surface above it or with multiplicity 3200 and then as you go further up the multiplicity you get bigger and bigger and bigger so this light algebra has a very narrow waist in some sense but it gets huge up here and it gets huge down there and we can ask what is its denominator formula and you can write down its denominator formula explicitly it looks like this first of all we take a sum of the vial group and we take the sign and then we take omega of sum over tau n of e to the n row and this is equal to e to the row times product over alpha greater than 0 1 minus e to the alpha the malt of alpha and this multiplicity is just given by this expression here and what's going on is this looks just like the denominator formula or finite dimensional affine the algebra except we've got these extra correction terms these come from the imaginary simple roots here tau of n is Romanogen's tau function you remember it's defined like this so sum of tau n q to the n is just q times product over 1 minus q to the n 24 which is you know it's 2 minus 24 q squared and so on so so this counts as a nicely algebra because we have a simple expression for both the simple roots that are just given by points of each lattice and the root multiplicities they're just given by this values of p24 now we can ask can we find similarly algebras for the other reflection groups and this turns out to be rather difficult there's one case in which you can do this suppose we take an automorphism sigma of the leach lattice and actually strictly speaking we don't take an automorphism of the leach lattice we take an automorphism of a certain double cover of a leach lattice but I'm not going to worry about that too much and we can look at the fixed point lattice lambda sigma where we just take all points of the leach lattice fixed by sigma then this lattice behaves in a way that's very similar to the leach lattice in particular we can find reflection groups whose dink and diagram is not quite this lattice but something similar so what we can do is we can take things like lambda sigma plus a little two-dimensional Laurentian lattice plus a little two-dimensional lattice with some other number there and this is similar to the 26-dimensional Laurentian lattice in particular you can define its reflection group and the dink and diagram turns out to be describing terms of several cosets of this fixed sub lattice and Peter Nieman looked at this for certain sigma of prime order and worked out the corresponding denominator formulas you get and when sigma has ordered 23 you find something really quite interesting the fixed sub lattice lambda sigma is dimension equal to 2 it has determinant 23 and I don't know what that's doing there and the corresponding Laurentian lattice contains the lattice of 2 minus 2 minus 2 2 minus 1 minus 1 which was studied by Brenkel and Feingold so just recall that Brenkel and Feingold were studying the lattice corresponding to this cot and matrix or this lattice and that was the one whose root most species were given by values of the partition function so here this lattice is now going to be four-dimensional but it's containing three-dimensional lattice and what Peter Nieman found was that the Lie algebra corresponding to this two-dimensional fixed sub lattice of lambda has root multiplicities related to the coefficients of 1 over eta of tau what's eta of tau well eta of tau is equal to q to the 1 over 24 times product of 1 minus q to the n to the 1 where n is equal to 0 where q is equal to e to the 2 pi i tau now you notice the coefficients of this are going to be very similar to the coefficients of the partition function so this is going to give you q to the minus 1 over 24 times 1 plus q plus 2 q squared plus 3 q cubed and so on where these are the values of the partition function and then we need to multiply it by eta of 23 tau which is going to give us 1 plus q to the 23 plus 2 q to the 46 and so on yes we should have a q to the minus 23 over 24 and what now you see but what's going on here is the coefficients so the first 23 coefficients are given by values of the partition function but after that they start getting a little bit bigger than the partition function and the root multiplicities aren't always given exactly by the coefficients of this function it's a little bit more complicated I won't go into exact details so many of the coefficients are given by the coefficients of this function so this sort of explains the root multiplicities found by Bencore and Feingold it turns out that these numbers here aren't really values of the partition function they're really coefficients of this function here and the Lie algebra we should be looking at is not quite this one but a certain rank for Lie algebra that contains this as a very large subspace I should say by the way that Franco doesn't agree with me that this explains his results he wants to see a Lie algebra whose coefficients are given exactly by the partition function not by this slightly modified form Neil Scheithau is systematically looked at all the automorphisms of the Lie lattice with a non-trivial fixed point sub lattice and describe the Lie algebras you get from all of these so we get a whole family of Lie algebras whose root multiplicities and simple roots are known explicitly and parameterised by certain elements of Conway's group of symmetries of the Lie lattice so this gives you a family of reflection groups related to similar to Conway's reflection group all of which are related to very nicely algebras well next you can ask what about Wienberg's other reflection groups so Wienberg also found reflection groups of say the E10 Dink and Diagram for the 10 dimensional even Lorenz and Lattice and you can ask do these correspond to nicely algebras and the answer is yes and I will be discussing this in the final part of this lecture in the next video and the key point here is that to construct these Lie algebras we need to use automorphic forms so I'll be explaining the relation of Wienberg's reflection groups to certain automorphic forms