 This is the introductory lecture of an online graduate course on Lee groups and To get one thing out of the way it's pronounced Lee groups not lie groups because it's named after the Norwegian mathematician Sophus Lee and they spell things a bit differently over there So what is a Lee group? Well a Lee group as the name suggests is a group and It's also a manifold So saying it's a manifold means it looks locally like N dimensional space Over the reels for some integer n where this number n is called the dimension of the Lee group So a typical example Might be the group gln over the reels So you recall from linear algebra that gln means n by n matrices with determinant none zero and If a matrix has none zero determinant as it has an inverse and the product of two matrices with none zero determinant has none zero Determinant so this certainly forms a group and it's also a manifold because it's an open subset of n by n matrices over the reels and this is Isomorphic to R to the n squared So gln of R looks locally like R to the n squared and you can check that the group operation is continuous and smooth and so on So what I'm going to do in this introductory lectures go through some of the Simpler examples of Lee groups, so let's start with dimension zero Well, Lee groups of dimension zero are Essentially the same as discrete groups This means either groups where you've put the discrete topology on them or groups where you just completely ignore the topology completely And we would like to classify Lee groups in order to understand them and we're already completely stuck in the zero dimensional case because the classification of Discrete groups is completely hopeless Even for finite discrete groups, we can't really classify them in any meaningful way I mean we can classify the simple ones, but how the simple ones are joined together seems to be very complicated However We can sort of reduce any Lee group to a discrete group and a connected group So if we take a Lee group G then it's got a Connected component containing the identity and This is also a Lee group and it's a normal subgroup. So we can form the quotient group G modulo G zero and This group here is now Discrete or zero dimensional so in some sense we can split any Lee group into a discrete part and a Connected part and it turns out that we can more or less classify the connected Lee groups Or up to some fudging about nil potent groups that I'll mention later So a typical example of this let's take G to be the the group of none zero reels Then it has a connected component consisting of the positive reels under multiplication And if you take the quotient of the none zero reels by positive reels You get a little group with two elements because a real can either be positive or negative So this is sort of what a how a general Lee group decomposes It's got a connected normal subgroup and the quotient is discrete So The theory of Lee groups really concentrates largely on connected Lee groups. In fact some authors even Say at the condition of being connected into definition, but that's sometimes a bit inconvenient. It's Often useful to allow non-connected Lee groups So now let's move on to the dimension one case and Here there are several obvious examples. First of all, we've got the real numbers under addition Obviously a one-dimensional the group. Secondly, we've got the non zero real numbers under multiplication and There's another example. We can have we can have the circle group s1 which is contained in the Non-zero complex number. So this is the this is the complex numbers z of absolute value equal to 1 So it's called the circle group because it looks very much like a circle and These are all very closely related for example There's a homomorphism of groups from r to the non-zero reels given by the exponential map And there's a homomorphism of groups from the reels to the circle group given by I think x to e to the 2 pi i x just put the 2 pi in there out of habit and These are homomorphisms of groups and this map here makes the reels identifies the reels with the connected component of of the non-zero reels so The connected components of these two groups are the same here the reels obviously aren't isomorphic to the Component of s1, which is just s1. However, this is an example of something called a local isomorphism so What this means is that that is that if you take the reels and Addition and you take the circle group under multiplication you can take a little neighborhood of the identity in both of them for instance there and there and The this map is is an isomorphism from this little chunk to this little chunk in that it preserves multiplication wherever you define so if you sort of Live at the identity element of the group and you're kind of short-sighted. So you can't see very far You would think these two groups are really the same And In fact any connected one-dimensional lead group is Isomorphic to either the reels under addition or to the circle group s1 another way of saying this is that any One-dimensional lead group is got by taking the one-dimensional vector space and quotient out by some discrete Subgroup So the discrete subgroup is zero in this case and it's z in this case Now let's have a look at some two-dimensional examples So one obvious way of getting two-dimensional examples is to take the product of two one-dimensional examples So we get r1 times r1, which is just a two-dimensional vector space and we get r1 times s1 and we get s1 times s1 which is The famous torus group so Looks something like a torus and this looks something like an infinitely long cylinder and this of course just looks like a plane And all of these are still abelian As we're all the one-dimensional groups and in fact you can see that these are all of the form A two-dimensional vector space quotient out by a discrete group The discrete group can be zero or z or z squared in these three cases And in fact more generally any abelian lead group any abelian connected lead group Can be written like this. It's of the form it's isomorphic to r to the m times s1 to the n Which is isomorphic to r to the m plus n modulo some discrete subgroup So we know what all the connected abelian lead groups are as usual disconnected ones Classifying even abelian Groups is hopelessly complicated in general. Although of course you can do the finite ones so These are abelian groups and you can ask are there any other Any none abelian lead groups well in dimension one that aren't but in dimension two We come across the first none abelian Connected example and this is the ax plus b group So, um, the the name suggests this is the group of all linear transformations from r to r Well Which take x to ax plus b So it's the group of affine transformations of the reels And you can represent this group as matrices because if you've got until two by two matrices the form ab 01 and multiply it by x one then you maybe get ax plus b one so So This group can be thought of as as a group of all these matrices here where of course we take a not equal to zero And it's got a connected component which are the matrices with a positive Um, and this group is not abelian as you can very easily check but it is An example of a solvable Lea group um and The meaning of solvable for lea groups is much the same as the definition of solvable for discreet groups what it means is we've got um A series of subgroups g zero contained g one contained in g two and so on up to g n which is equal to g such that each quotient G n so g i over g i minus one is abelian Of course, we implicitly assume that each g i is contained in in the is a normal subgroup of the next group So in the case of two by two matrices, what are g zero and g one and g two and so on? Well, we can draw a picture of them like this. So we have ab 01 G zero is just the identity g one is just um the group of all um Possible elements b with a being equal to one and you can think of g two as being the whole group So um In this particular case this this is changed just as length one um now, let's move on to groups of dimension three And here we come across what is possibly the the single most useful lea group of them all which is the group sl to r So this is the special linear group And it consists of all matrices abc d With ad minus bc is equal to one. So this is of course just a determinant And since the determinant is multiplicative these matrices here form a group and um two by two matrices are four dimensional and we've cut the dimension down one by putting on this equation So so this is dimension three um and There's another closely related group, which is the group p sl two of r And this is the group sl two of r Modulo the group um plus or minus one by which we mean and the matrices plus one plus one And minus one minus one So You can see this forms a little normal subgroup of order two. So we get a quotient here p stands for projective Because it acts on projective space if you care about that sort of thing um, so um as a typical example of an application of this group this group is the group of automorphisms of the upper half plane in complex analysis so if the upper half plane H Has its elements denoted by tau then the action is given by abc d Acting on tau is equal to a tau plus b over c tau plus d and you notice that if um That the the matrix minus one acts trivially So so although sl two acts on the upper half plane to get the group of automorphisms We should really quotient out by it and get the group p sl two of r um and Another three-dimensional group is the sphere s three um, and this can be written as the group of unit quaternions So you remember the quaternions consist of Hamilton's quaternions are the numbers of the form a plus b i plus c j plus d k with the multiplication given by i j equals k equals minus j i j k equals i equals minus k j and k i equals j equals minus i k and Hamilton showed that this um formed a non commutative division ring and It has a sub the subgroup s three consists of all Quaternions with a squared plus b squared plus c squared plus d squared equals one And it's not very difficult to check that this is closed under multiplication Um incidentally that you can think of if if this quaternion is called z Then this number here is equal to z times z bar where z bar is the quaternion a minus bi minus cj minus dk So it's kind of a sort of analog of complex conjugation for the complex numbers By the way, we had s one being a league group and we had s three being a league group And you may think I forgot to include s two as a league group. Um, in fact s two is not a league group it is no, um It is no group structure on at all at least not one that that is um continuous In fact s zero s one and s three are the only spheres That are league groups And you can think of these as being the numbers of absolute value one inside the reels or the complex numbers or the quaternions So this corresponds. This is very close related to the fact that there are only three finite dimensional division algebras over the reels They that they correspond to the spheres that are league groups Um Um, there's a another um Um three dimensional group Which is the heisenberg group Named after the guy who invented quantum mechanics and the heisenberg group Well, it either consists of the matrices of this form Or it consists of the matrices of this form Quotient out by the normal subgroup of all matrices of this form where n is in the integers And I never quite worked out whether the heisenberg group is this or or this I think it depends on which author you read um, so this is an example of taking a A league group and quotient out by a discrete subgroup of the center and um This group turns up in quantum mechanics quite a lot Um, so suppose you look at all the transformations of functions on the reels so you can transform a function by shifting it By a constant or you can transform a function by multiplying it by a constant So by multiplying it by a periodic function or you can Multiply it by a constant of absolute value one And if you take the group generated by these transformations, it's three-dimensional and is Essentially the same as this bit of the heisenberg group You see we have to quotient out by the discrete subgroup because if you think of these constants as e to the i b then Then whenever well, let's put two pi i b then whenever b is an integer this becomes one um so um This is a common way of constructing Lee groups and other lee groups who can quotient out by a discrete subgroup of the center um Now you notice this group here is isomorphic to r3 so it's it's um Simply connected and if we're quotient out by a discrete subgroup of the center, it turns out this quotient here Is not simply connected, but it is fundamental group Given by the integers, which is the thing you quotient out by Okay, I haven't quite discussed fundamental groups yet, but this is a just a quick introductory survey Um conversely you can sort of reverse this process If you've got a group like this one with a non trivial fundamental group You can take a sort of covering space out of it and get back the the get back a simply connected group um, so um The idea of this is that you can reduce the classification of all the groups to the classification of simply connected lee groups because it turns out that any lee group is the quotient of a simply connected lee group by by um A discrete subgroup of the center Um, the heisenberg group is also an example of a nilpotent group so Let's just recall what a nilpotent group is So if you've got a group g Let's call it g0 We can kill off the center. So let's put g1 equals g0 modulo the center Well, um, you might think g1 doesn't have a center, but it but but it actually can have a center Because I mean it wouldn't have been a center in g0, but it might now become a center in g g1 And so you can now repeat this process you put g2 equals g1 and then you kill off the center and you sort of continue like this And if g n is the identity group g0 is called nilpotent The reason for the name nilpotent will will appear later when we talk about lee algebras um, so uh, the heisenberg group is nilpotent because if you look at it the center is consist of um, essentially these matrices but by which I mean a and c are zero and you still keep the ones here But b is allowed to be anything. So so so this is This is the center and then when we kill off that we get an abelian group. So so We then end up with that with the whole group there Upper triangular groups have a very strong tendency to be a nilpotent. So if I take a group consisting of all matrices like this Then the center consists of just the matrices like this Meaning I keep ones down the diagonal and set everything else equal to zero And if you kill off this the center of what's left is given by this subgroup And if you kill off that the center of what's left is given by this subgroup and the remaining center is And the what's left is now abelians. So so it's equal to its whole center. So we get a sort of chain of Centers of quotients increasing like that so um all um upper triangular groups with ones down the diagonal are nilpotent um Conversely lee showed that any connected simply connected lee group Is a closed subgroup of a group like this. So simply connected nilpotent implies that it's a closed subgroup of something like this Well, I mean possibly a bigger matrix, of course So you may think well this almost classifies nilpotent groups. Well, it turns out that it doesn't really because um that there are immensely large numbers of different non isomorphic closed subgroups of the group of upper triangular matrices If you try and classify nilpotent groups, you can do so in small dimensions But the problem very rapidly just becomes completely out of control Um So now let's have a look at Say dimension six So here we get the l'orentz group So the l'orentz group consists of all the rotations of spacetime So um, I mean we we could do the the group of rotations o4 of r Which as you know is that is the group of all rotation all linear transformations of r preserving the um quadratic form x1 squared plus x2 squared Plus x3 squared plus x4 squared Um, when you do special relativity use a group o1 comma 3 r Which means use the quadratic form x0 squared minus x1 squared minus x2 squared minus x3 squared um, there's a Some people like putting the minus sign there sort of in front of all these in fact there seem to be two Physicists seem to be dividing in a two roughly equal subsets who are not on speaking terms with each other Who have different ideas about where you should put the minus signs in this but anyway um, so um, uh, this is um This group has several components. In fact, this group was four components um, the reason is that um It in in in this group you can reverse time or you can reverse Space in other words reflect space in a mirror and you can of course also reverse time and space And physicists get really excited about that because they want to know whether their theories are are um Um invariant under time reversal under space reversal And at one time it was thought to be obvious that all physical theories were were invariant under time reversal and space reversal But much to everyone's surprise the weak interaction in in quantum field theory turns out to be none invariant under space inversion and later it was discovered that um Things aren't even invariant under time reversal or or even under the combination of these two So the four components of the Lorentz group turn up quite a lot in physics If you take the connected component um This turns out to um um Have a double cover called a spin group So this maps onto the connected component of um 0 1 comma 3 are And is an example of the construction I was talking about that this this group here has a A non trivial fundamental group so we can take covers of it and one of these is is is the Is is the spin group and we're going to be constructing spin groups out of clifford algebras later on The spin group turns out to be locally isomorphic to The two by two matrices over the complex numbers In particular, this says this is two-dimensional representations meaning actions on two-dimensional Vector spaces, which this group doesn't have and this is the cause of things like fermions and electrons Um by the way, this isomorphism between these groups is a typical example of an accidental isomorphism And one of the confusing things about lee groups is that is that is that lee groups of small dimension have large numbers of accidental Isomorphism with each other, you know various sorts of groups which don't look as if they're the same turn out to be locally isomorphic to each other Um So, um, let's have a look in dimension eight One very famous example is the group su 3 So u 3 means 3 by 3 unitary matrices And s as usual means determinant 1 so the 3 by 3 unitary matrices of dimension 9 and This group therefore is dimension 8 and this appears an awful lot in particle physics First of all, it's the group that controls the gauge group of quantum chroma dynamics It's also a group involved in the flavor symmetries of um old style Theories the strong interaction. So sometimes known as the eight-fold way named by gal man. I believe So this is a fairly typical example of a simple lee group Which means Almost the same as simple in group theory It's got no normal subgroups except in the theory of lee groups who kind of allow Discrete subgroups of the center. So this actually has a center of order 3 But you ignore that when you when you talk about the group being when groups being simple dimension 10 One well known example is a Poincare group And this is related to the Lorentz group So the Lorentz group is the set of all rotations of spacetime the Poincare group is the set of all Translations and Rotations of spacetime. So what we get is a Semi direct product of the group of all translations of spacetime, which is a normal subgroup And on top of that you have sitting the group of all rotations of spacetime fixing a point So what we have here is this is more or less a product of simple groups In fact, it is actually a simple group and what we have here is a solvable group Well, it's actually an abelian group not just a solvable group But what this is supposed to illustrate is Almost any lee group is more or less a semi direct product of a solvable group By a product of simple groups and about the simplest non-trivial example of this That occurs in practice Maybe the Poincare group which which splits as a semi direct product of translations and a product of simple groups And this almost gives us a classification of lee groups because lee showed that any solvable group is a subgroup of opera triangular matrices For some n the closed subgroup of this group here and In the group of opera triangular matrices, you see it's got a subgroup That's a nilpotent group where I mean if you put ones on the diagonal and the quotient is abelian So solvable groups aren't too far from nilpotent groups. They're just a nilpotent group with an abelian group sitting on top of that If you do finite group theory, solvable groups are vastly more complicated than that This is This is only for connected lee groups So this sort of reduces the classification of all lee groups to nilpotent groups and to simple groups So I'll finish off by just saying a little bit about simple groups It turns out we can classify all the simple lee groups at least the ones that are Um Connected and not zero dimensional and so on so It's actually a little bit easier to do the ones that are defined over the complex numbers So complex lee groups are defined very much like real lee groups except that they have to be complex manifolds not real manifolds And there are some obvious ones. First of all, there's s l n of c For n greater than or equal to one and there are the orthogonal groups o n of c For n greater than or equal to three and there's the symplectic group O sp2 n of c for n greater than or equal to one These are the so-called classical groups At least over the complex numbers over the reals the classical groups are rather more numerous Um and in addition we find killing found Five more he found five more which are called g2 f4 e6 e7 and e8 by Someone who wasn't very imaginative about naming things So the dimensions of these groups are 14 52 78 133 and 248 And killing really ought to be a lot better known. Um, he was a kind of really modest shy guy And invented a lot of things that were later named after other people So later on in this course, we will be talking about the vile group and the coxswain number and the carton And the carton form and things like that and these were all actually first invented by killing And then people kind of forgot he invented them And he discovered these five exceptional lead groups and a lot of the credit is given to carton who Who came along a lot later than killing and just filled in a few holes in killing's work um Anyway, so killing more or less showed that the complex simple lead groups were given by this list here Um So what we'd be doing later on in the course is showing how to construct all these Lea groups. Um, I've just finished by drawing pictures of them Um, there's a famous way of drawing pictures of all these lea groups. So we draw a picture of sl2 by drawing a point So each point means sl2 And then all of these other groups found by killing are obtained by sticking together copies of the group sl2 Of the complex numbers. This is one of the reasons I said sl2 was so important all other groups are kind of Built out of copies of it in some sense So the groups sln Have a so-called dink and diagram That looks like that And what this means is each dot is a copy of sl2 And if the dots are not joined it means the copies of sl2 Commute with each other and if the dots are joined by line It means the copies of sl2 sit inside each other in the same way that copies of two Two copies of sl2 sit inside sl3 So This is a sort of picture of the group sln The orthogonal groups o and look like this Except for some of them which look a bit different and look like this I'll sort of explain this later And the symplectic groups look like this And f g2 Looks like this f4 looks like this And e6 e7 e8 Look like this This and this So one of the major goals of this course will be to explain what all these dink and diagram mean diagrams mean and show How to construct the simple lead groups by starting with these dink and diagrams Okay, so One of the problems with lead groups is they're rather complicated as topological manifolds I mean, they have a very complicated apology And you can simplify them greatly by instead looking at the tangent space And this is what we will be discussing next lecture where we find the tangent space of a lead group at the origin Is something called a lead algebra