 This lecture is part of an online course about Lie groups and will be about some of the weird things that happen in characteristic P that's greater than zero So most of the time we've been talking about Lie groups over the real numbers So we can have things like the general linear group or the orthogonal group and lots of other groups like that and For much of this theory, there's no particular reason why you should work with the real numbers rather than some other field For example, if you've got any field whatsoever We can still define the general linear group over the field the orthogonal group over the field and so on So these are examples of things called algebraic groups so so Lie groups you define them to be a real manifold with a group structure algebraic groups you you can define them as Algebraic varieties with a group structure in other words, they're defined by some polynomial equations and A lot of the theory of Lie groups also applies to algebraic groups over any field for example for any group We can get a Lie algebra Which is of course going to be a Lie algebra over the field K And a homomorphism of groups gives rise to a homomorphism of Lie algebras and so on however, there are also some quite big differences between groups over the reals and groups over fields and and this lecture is going to be about some of the differences So so let's just start with an example Here I'm going to give examples of complex Lie groups so a complex Lie group is just like a real Lie group except you work with a complex manifold and Won't really matter because the ones we're going to look at are Very simple. So here are three examples first of all we can take the complex numbers under addition That's a one-dimensional complex Lie group Secondly, we can take the non-zero complex numbers under multiplication A third example would be an elliptic curve. This is a little bit trickier So an elliptic curve is something that looks like y squared equals x cubed plus b x plus c except We should add points at infinity which I not going to go into detail about and Elliptic curves together with a point of infinity all have a natural group structure on them and All of these have the same Lie algebra The Lie algebra is just the complex numbers with with the bracket being zero of course because it's one dimensional Now just as for real Lie groups for complex Lie groups if two things have the same Lie algebra They're locally isomorphic and we can write down local isomorphisms fairly easily So to go from the additive complex numbers to the multiplicative complex numbers, of course, we use the exponential function So z maps to e to the 2 pi i z and this identifies the non-zero complex numbers as the complex numbers modulo the subgroup of integers So these two are locally isomorphic as expected The local isomorphism between the complex numbers and elliptic curve is a little bit trickier and involves the viastrice p function So we take z to a point of the project of plain given by We need the derivative of the viastrice function and the Viastrice function and then I guess one So this would be in projective coordinates. I'm not going to cover the viastrice functional projective coordinates this lecture Because this is just a brief example anyway, this map from the Complex numbers to the elliptic curve identifies the elliptic curve as the complex numbers modulo a lattice So a lattice is isomorphic to z squared and a typical example might be all numbers m plus n i for m n in z So we see what's happening is we've got three locally isomorphic groups And if we take the simply connected one the others are just quotients of it by a discrete subgroup you can also find a map from This group to this group if you want So this looks pretty much like what happens for the reals Now let's see what happens over Over a field of positive characteristics, so we're going to take a field K of characteristic p greater than zero and Just as over the complex numbers we can find three algebraic groups, so we can find K plus which is the additive group of the field K or we can take the multiplicative group of none zero elements of K or we can take an elliptic curve over K Which will again be given by the points y squared equals x cubed plus bx plus c plus a pointed infinity Again, these all have the same Lee algebra And the Lee algebra is just the field K under addition. However as groups They're just totally different They're not locally isomorphic at all. In fact, there are Any homomorphism between any two of these groups just has to be the trivial homomorphism and first of all These two here affine Algebraic groups, it means they're affine varieties and this one is projective and Affine and projective varieties are sort of almost opposite of each other And These two are also completely different for instance. This one is an example of a unipotent group What this means is if you take a matrix representation of it all elements of the group Have all eigenvalues equal to one for example We can represent this group as the following subgroup of the general linear group and you see all its elements have all eigenvalues one On the other hand, this one is an example of something called a reductive Algebraic group and this means well in this particular case. It means all the elements are diagonalizable and for non-Abelian groups that definition of reductive is a bit more complicated and and Reductive groups and unipotent groups are sort of implacable enemies of each other You can think of reductive groups as being good and unipotent groups as being evil When you when you study groups if anything is going wrong or causing complications It's because there is a unipotent group there somewhere causing trouble. So so these These groups of nothing to do with each other in characteristic p notice. There's no exponential function Because if you try and write down the exponential function You notice it's got denominators and these denominators don't always make sense in characteristic p So we we can't write down the functions between these groups that we had in the characteristic zero case because they're first They're not algebraic and second. They've got denominators and so on In particular, we see that Homomorphisms of lee algebras in general don't really say anything at all about homomorphisms of groups that there are simply no non-trivial homomorphisms between any of these groups even though the lee algebras are the same So so the the lee algebra and characteristic p greater than zero just tells you less about the group than in characteristic zero So now here's another example where characteristic p behaves oddly. Let's work over the reals We notice that high order differential operators Can be written in terms of First order Differential operators in other words These are more or less vector fields. So for example, if we take the lee algebra to be the left invariant differential operators of order One on a group G then we know that it's universal enveloping algebra is the left invariant differential operators of or of any order on the group G in characteristic P greater than zero this this just totally fails the high-order differential operators in General don't have all that much to do with the first order differential operators. In fact, this works even in the simplest examples Let's just take G to be the group of real numbers under addition So this is we're doing the characteristic zero case first then the differential operators Are easy to figure out the left invariant differential operators are just spanned by the operators D by dx to the n no problem at all. It's just a they just form a polynomial algebra Now let's try and work over Z so Here we can define an analog of differential operators acting on the ring Z of x and We can form differential operators D by dx to the n as before but we can there's a sort of extra wrinkle we can do because we can form the differential operator D by dx to the n and Divide it by n factorial and this gives you a perfectly good differential operator because it takes x to the k to K choose n X to the k minus n and you notice this is always an integer so so so we get more Differential operators and you might expect because of this extra possibility of dividing by n factorial You might wonder what is what actually is a differential operator in in these more general cases Well e is a differential operator of order less than n you can define that as saying that the commutator of e with multiplication by any function has order Less than we would n minus 1 so you can sort of define differential operators in general, but by this condition here Anyway, now let's look over what happens if you look over a field k of characteristic p And we want to know what are the differential operators on the polynomials in k? well here we can get the differential operators which are a sort of Divided power of D and these just take x to the k to k choose n x to the k minus n so it's just reduction of this differential operator mod p in some sense and Now you notice that these differential operators are not generated by the first order operator D one which is just D by DX in in fact D by DX to the power of p Is just equal to zero in characteristic p as you can easily check So this first order differential operator just generates a small finite dimensional subspace of the space of all higher order differential operators In particular in characteristic zero if you know the first order operators You know all of them in some sense, but in characteristic greater than zero Knowing the first order operators is only a small piece of what you need And so In the previous lecture We had this result that over the reels if we've got the Universal enveloping algebra, which over the reels is all high-order differential operators So in this algebra L is the subspace of Primitive elements So we recall the primitive elements are one such that delta of v is equal to v tensor one plus one tensor v and Delta is the map from u of L to u of L tensor u of L which for L in L takes L to L tensor one plus one tensor L and this is a homomorphism of algebra So What happens in characteristic greater than zero? well, this just fails and We gave an example of this last lecture. I'll just quickly recall it. So again, let's just take a one-dimensional additive group then the leap that then the Universal enveloping algebra of its leap algebra is just the polynomial ring in In D so we get 1d d squared d cubed and so on d to the p and so on and You can't really interpret these as differential operators because the piece power of The differential operator d by dx would actually be zero, but it's not zero in the universal enveloping algebra This is just the the the polynomial algebra in D On the other hand if we compare this with the differential operators The differential operators are spanned by 1d d squared d cubed and so on to d to the p and so on and These ones are just none zero multiples of each other, but these ones are definitely not zero none zero multiples of each other and the these two Things have nothing to do with each other. This one's a differential operation. This one just isn't or it's zero or whatever So in In this one What are the primitive elements? Well the primitive elements are easy to figure out They're spanned by d but d to the p is also primitive and d to the p squared is primitive and D to the p cubed is primitive and so on in general if you take a primitive element its p-th power is also primitive On the other hand in this one We can work out the primitive elements. We define you can define the Co-product delta by saying delta of d to the p K is the sum of d to the i tensor d to the k minus i and this time the primitive elements are Just spanned by d so So what this suggests is that in Positive characteristic the correct analog of the universal enveloping algebra is not the universal enveloping algebra Algebra of differential operators over the of the group and So when when you're working with groups of characteristic p you you don't really work with universal enveloping algebras What people often do in fact is they work with something called a formal group and the formal group turns out to be very closer related to the To the ring of differential operators over the field It's in some sense a sort of Something to do with the dual of this ring, but I won't go into that I should mention in characteristic p there are even stranger examples So we can get nought dimensional groups of order P so these are in some sense finite groups, but they are Connected Which doesn't seem to make sense if you're working over the reels. I mean a finite finite House door set can't be connected But in characteristic p the very weird things like this can happen an example of this is the group mu p Which is just the p th roots of One in some sense so its coordinate ring is K of x Over x to the p minus one and if you want to know more about this you can go to a Course on algebraic groups in characteristic p. I'm just mentioning this as a warning about just how strange things can get So The last example I'm going to present is that many lee algebras many lee algebras In characteristic p greater than zero Do not correspond to groups At least not in any easy way. So over the reels any real lee algebra. There's always a lee group associated to it So in positive characteristic this fails and let's start with an example in characteristic zero called the vit algebra So the vit algebra is more or less Polynomial vector fields on the line. So we're going to define l n minus one equals x to the n d by d x So we're going to take the lee algebra to have a basis of these and if you work out the common That the commutator l m l n Well, this is just x to the m d by d x x to the n d by d x minus x to the n d by d x x to the m d by d x Which if you work it out is equal to n minus m x to the m plus n minus 1 d by d x which is equal to n minus m l m plus n so So we have this nice infinite dimension lee algebra now in characteristic P greater than zero that we can define a variation of it We just define it to be spanned by operators l m with m only to find modulo P So this is now an algebra of dimension P and We use the same rule as before l m l n Equals n minus m l m plus n and This is a finite dimension lee algebra and it does not correspond to any Simple algebraic group And in fact there are quite a lot of finite dimension lee algebras in characteristic P, which just don't correspond to Algebraic groups and characteristic P in any obvious way and Sure Chevrolet classified the Algebraic groups and characteristic P the simple algebraic groups and over an algebraic you closed fields in characteristic P and Really amazed everyone by showing it's the same more or less the same as the classification of simple lee groups and Characteristic zero and this was a big surprise to everybody because they knew that the lee algebras in characteristic P We're completely different to the lee algebras and characteristic zero However, the lee groups and characteristic P at least the simple ones Are quite close to the ones in characteristic zero. So again, this is an example where the Lee algebras don't really seem to have all that much to do with the lee groups Okay, so next lecture we're going back to the case of algebras over the reels