 So this lecture is part of an online mathematics course on group theory and will be about the quaternion group and the quaternions. So just recall from last lecture we had a quaternion group with eight elements plus or minus one plus or minus i plus or minus j plus or minus k and these satisfy the relations i j equals k j k equals i k i equals j and i squared equals j squared equals k squared equals minus one and you can represent the elements of this group as matrices so we can put i equals i minus i j equals one minus one and k equals i i and of course one is just represented by the matrix the identity matrix and minus one by that and we can actually form a ring of quaternions so what we do is we take all matrices the form a plus bi plus cj plus dk with a b c and d real so this is isomorphic to r to the four the four dimensional vector space with coordinates a b c and d and we can define a ring multiplication on it using these rules notice this multiplication isn't commutative because i j is minus j i j k is minus k j and k i is minus i k so you sometimes get minus signs whenever you change the order of things so this ring was discovered by Hamiltonian and is called the ring of Hamiltonian quaternions the term quaternion means a group of four things Hamilton may have got it from the from the bible where it says when Herod arrested St Peter he delivered him to four quaternions of soldiers and a quaternion of soldiers is just a is just a group of four soldiers if you want to see this you have to use an old bible because modern bible is just translated as a squad of soldiers or something boring like that but anyway quaternions are a sort of analog of the complex numbers and a lot of things you can do with complex numbers you can do with quaternions so you remember complex numbers you just have things a plus bi with a and b real and we just have i squared equals minus one and there aren't any j's or k's and what you can do with a complex number is you can take its complex conjugate so the complex conjugate of a plus bi is a minus bi and we know that if z is a complex number then z times its complex conjugate let's put z equals a plus bi is a squared plus b squared which is always greater than or equal to zero and this can be called the norm of a complex number z and it has the properties that norm of z1 z2 is the norm of z1 times the norm of z2 so the norm of z is just the square of its absolute value and you can do exactly almost exactly the same thing for the quaternions so if a quaternion is a plus bi plus cj plus dk then we set its complex conjugate quaternion conjugate to be a minus bi minus cj minus dk and again we find z times z bar is equal to a squared plus b squared plus c squared plus d squared which is greater than or equal to zero and you may think when you multiply z by z bar you're going to get all these funny cross terms involving i times j but these all cancel out because here you've gotten i times j and here you've gotten j times i and ij plus j i is zero so they cancel out so and again you can check if you define this to be the norm of z then norm of z1 z2 is the norm of z1 times the norm of z2 and you've got to be a bit careful about proving this because quaternions are not commutative so if we find the norm of z1 z2 is equal to z1 z2 times z1 z2 bar and now conjugation has the property that z1 z2 bar is equal to z2 bar z1 bar and you should notice here that there's a change in order so this is fairly easy just to check explicitly anyway using that we see this is z1 z2 z2 bar z1 bar and now we've got a z2 z2 bar and z2 z2 bar is it's a real multiple of one so commutes with everything so this is now z1 z1 bar z2 z2 bar which is norm of z1 norm of z2 well what can we do with this well if you've got the complex numbers then the complex numbers of absolute value one which is the same as saying the norm is equal to a one form a circle s1 and this is a group so using the complex numbers you can see the circle s1 is a group under because we can just identify it as complex numbers of absolute value one and use complex multiplication we do the same thing for the quaternions we take the quaternions with norm of z equals one or the absolute value of z equals one because of course the absolute value of a quaternion is just defined to be the square root of its norm and these are the solutions of a squared plus b squared plus c squared plus d squared equals one so it's a sphere s3 so s3 is also a group and this is kind of well you may think maybe all spheres are groups but in fact it's very rare for a sphere to be a group the only spheres that are groups are s0 s1 and s3 so s0 is the things of norm z equals one in r so it's just the group of two elements plus or minus one not dimensional sphere is just two points this is the things of norm one in c and this is the things of norm one in the Hamiltonian quaternions and of course this is a non-commutative group whereas these two groups are commutative so how much one of the reasons Hamilton was very interested in quaternions is that they give a very easy way to define rotations in three-dimensional space and Hamilton actually spent a lot of time trying to rewrite all of physics in terms of quaternions using this people rediscover quaternions and Clifford algebra is about every 10 or 20 years or so and get very excited about them latest incarnation is called geometric algebra but anyway so how can we have how can we use quaternions to describe rotations in three-dimensional space what we do is we make three-dimensional space equal to the set of quaternions of the form bi plus cj plus dk so these are the sort of imaginary quaternions where we don't where we take real part a of the quaternion to be zero and suppose we take a vector v in r3 and suppose we take a quaternion g in s3 thought of as a subset of the quaternions then you can see that g v g to the minus one is also in r3 this is quaternion multiplication and moreover it preserves length because the norm of g v g to the minus one is equal to the norm of v because it's the norm of g times the norm of g to the minus one is just one and the norm of v is just the square of its length so this actually preserves lengths so it also preserves parity so this is actually a rotation of three-dimensional space if you do this transformation you get a rotation this is actually turns out to be very useful in computer games of all things because 3d computer games involve large numbers of rotations as you're changing your point of view and so on and the obvious way to represent rotations is as three by three matrices well a three by three matrix has nine entries but you can instead represent rotations as a quaternion and this only uses four entries and furthermore multiplication of quaternions is more efficient than multiplication of three by three matrices so if you represent rotations by quaternions it sort of slightly speeds up your computer game um well you may think this means that s3 is the group of rotations of three-dimensional space well that's not quite true what we get is we get a homomorphism from s3 onto the group of rotations of three-dimensional space this means the special orthogonal group of dimension three over the reals which are just three by three orthogonal matrices that they just as you remember from linear algebra these represent rotations in r3 well it turns out this group has a non-trivial kernel and the kernel is the group of order two consisting of plus or minus one you can see if g is plus or minus one then this is just equal to v and it's not difficult to check that conversely if g v g to the minus one is equal to v for all v then the only possibility is g is plus or minus one and what we have here is called a double cover of s03 of r um double covers of groups are actually quite common in mathematics so a double cover is a group mapping onto this group here whose kernel is order two and well a true way of getting a double cover would just be to take the product of s03 with a the group of order two but this is a different double cover and a much more interesting one um it's um actually a blue dot tree at this point to demonstrate the soup plate trick in order to illustrate this double cover so let me try and do that um so the soup plate trick involves taking a soup plate like this and you imagine it full of soup that this isn't actually full of soup but you pretend it is so it has to be held upright and for some weird reason you wish to rotate the soup plate by 360 degrees anti-clockwise so you do it like this and if you rotate it by 360 degrees anti-clockwise not very surprising that your arm gets a 360 degree twist in it so i'm going to rotate it by another 360 degrees anti-clockwise and you expect your arm is now going to get a double twist in it but instead the twist kind of disappears so what does this have to do with the um double cover of s03 well uh oh here we go um well this uh the group s03 of rotations represents possible positions of the soup plates i mean you you you you you can rotate the soup plate by any element of s03 so s03 is something to do with the positions of the soup plate and this extra factor of plus or minus one is is a group of order two and corresponds to the fact that you can have a 360 degree twist in your arm while holding the soup plate but if you do two 360 degrees then then this then the square of this element is one so that twist in your arm sort of disappears um so another application of this is this group here is actually called by physicists the spin group and it turns up quite a lot in quantum mechanics so in quantum mechanics um if you look at the space of possible wave functions of a particle you might expect that it's actually on by rotations of three-dimensional space this is doing none relativistic quantum mechanics but it turns out that quite often this doesn't quite work um that works fine for the particles called bosons which have integral spin but there are particles like electrons which are called fermions which have half integral spin and you discover that these particles are actually actually on by this group here so this double cover sort of underlies the fact that um electrons exist and we wouldn't exist if this double cover didn't exist well there's another rather useful thing you can do with this double cover what we can do is use another sheet and we take this double cover one goes to plus or minus one goes to s three goes to s o three r goes to one and now inside s o three we can take some sort of finite group of rotations well you could there are some rather uninteresting groups of rotations like cyclic groups but the three most interesting finite groups of rotations are as everybody knows the group of rotations of these platonic solids so we can take the tetrahedral octahedral or icosahedral group so we could have um in particular for platonic solids we get these groups of order 12 24 and 60 well what we can do is we can take the inverse images of these groups in s three so here we now get groups of order 24 48 and 120 these groups here are called the binary tetrahedral octahedral and icosahedral groups and they're all double covers of these groups here um for example we have a map one goes to plus or minus one goes to this group of order 120 to this group of order 160 to one where this is the icosahedral group and this is the double cover of the icosahedral group called the binary icosahedral group and the obvious question is is this group just a product of the icosahedral group with a group of order two and the answer is it isn't the point is s three has only two as it has only one element of order two which is the element minus one and this is easy to see because if z squared if z has ordered two then z squared equals one z is in s three so z is z times z bar is also equal to one and these two equations imply z equals z bar that means z must be real let's be a multiple of a so z is just equal to a for a real and since z squared equals one we find a equals plus or minus one so s three is only one element of order two well obviously that means this group here has only one element of order two which is this element here and if it was the product of this group by this group here there are lots of elements of order two because the icosahedral group has loads of elements of order two so we found some central extension of the icosahedral group which is not a product of the icosahedral group with plus or minus one and we'll be talking more about this group later when we get to groups of order 120 I just mentioned briefly that quaternions can not only be used to describe rotations of three-dimensional space they can also be used for four-dimensional space so I'll just briefly say what we do so we identify four-dimensional space with the set of all quaternions a plus bi plus cj plus dk and now suppose we've got two elements g in s3 and h in s3 now if we take a vector in r4 we can map v to gv to h or possibly h to the minus one and for any g and h this turns out to be an element of s04 of r it seems to check it preserves the norm because these both norm one so it's a rotation so this gives us a map from the group s3 times s3 to s04 of r and it's not difficult to check it's onto and as you can probably guess this map isn't injective it's got a kernel of order two consisting of the element that's minus one in both of these because if you multiply v by minus one on the left and the right then we get the identity map so s04 of the reals turns out to be isomorphic to s3 times s3 quotient out by a group of order two so the next lecture we will be talking about a bit more about the dihedral group and and we'll prove and we'll use Burnside's lemma in order to count orbits of a group on a set