Lie theory, part 11 | Daniel Tubbenhauer

Okay, welcome everyone to this continuation of the theory of lies. We're still lying a lot. I'm lying every day. I'm lying all the time. It's very sad. Hopefully [snorts] I'm telling the truth right now because we are well, I don't know hopefully. But anyway, we'll see. We are talking about the Campbell Baker Campbell Hausdorff formula. Which is something you should have seen at once at least once in your life and then you will never use it again. I don't remember a single time when I used that formula ever. I don't even remember how the formula looks like. I only know what it does. And that's essentially what we are going to discuss. So will I even pull up the formula? Maybe. I don't remember anymore. We'll see. We'll see. But it's an interesting thing. Okay. So here's again my favorite picture of Oh yeah, why commutativity matters. Hopefully at this point I've convinced you that commutativity matters. If you open the door and you walk through or you walk through the door and then you open the door, usually that is actually different. I commute matters. The algebra we add in the league group we multiply logarithm exponential map. And the Baker Campbell Hausdorff formula literally compares those. What is lost on the way? How you need to do that? How the bracket enters the picture? Okay. So what we do is near the identity we take e to the x and e to the y in the group and ask but take a logarithm of it. So Baker Campbell Hausdorff formula says that the logarithm of that thing if they commute it's just e to the x plus y so it's just x plus y. But what happens if they don't commute? What is that? There should be some formula in x and y. And you could quit the video here. There is the formula for it. It's called the Baker Campbell Hausdorff formula. That's essentially it. I never well, as I said, I never I've never really used this formula. It's not I mean it's obviously important because it compares how the logarithm exponential map actually behave on league groups or non-commuting objects. But it's not nice enough to actually well, we'll see. We keep on going. The easy case when everything commutes. So the bracket is zero. That's literally what it might means and nothing happens, right? e to the x plus y is just e to the x times e plus y. So the logarithm of that thing is just x plus y. As you would expect it from the inverse of the exponential function. Inverse here you mirror around the identity axis and you have this nice inverse. When everything commutes they're really nice friends of one another. Or kind of anti-friends if you want. They undo one another like anti-particles, I guess. Anti-friends. Let's say anti-friends. Yeah, so the formula is the formula that I have showed you is very boring and easy in that case. It doesn't deserve a name. Maybe it would deserve a name but it should be not of some mathematicians like Baker Campbell and Hausdorff which were like last century's mathematicians. But maybe more like from the 16th century or something. My historical knowledge here ends very quickly. But anyway, so this doesn't deserve a name, right? It's just just what the logarithm does anyway. And then comes the formula and it's it's not that great. It has infinitely many terms in general. It's it's not that great. But maybe we just can remember the first part. So you have what we expect and then a correction term which involves the bracket. Excellent. So if the bracket vanishes then everything vanishes and we are happy. But non-commutativity means the bracket doesn't vanish, right? Remember the bracket usually is just the commutator, right? And if the commutator doesn't vanish it's really just bracket xy is xy minus yx. So literally if it commutes it's zero, right? And if it doesn't commute it's not zero. So if the bracket doesn't vanish you have something non-trivial. Oh, I actually showed you that's not really the formula. But this is roughly how the formula looks like. Oh, let's discuss it at least for one second just for its glorious magnificent disgusting presence, I guess. So you take first order brackets, you take second third order brackets, you take fourth order brackets, you take fifth order brackets. Fine. So let's not look at it anymore and let's just keep on going. Let's just focus on the first term. So essentially, right? Highest higher terms involve nested brackets. Of course all the brackets fifth order brackets blah blah blah. So eventually the league algebra itself, the whole league algebra with all of its structure controls the whole expansion. Which is again a good reason why league algebra is not just a linear space but it's also the bracket operation. You can see that. Without the bracket there would be literally nothing you can do. — [snorts] — So literally the bracket somehow remembers the non-commutativity as a story. That's essentially the formula. And meaning of a memorizing really do we

really need to memorize the Baker Campbell Hausdorff? Nah, we don't need to. If you want to you're free to. Some people like to memorize the first 5 million digits of pi. If you want to do that, go for it. But I you don't have to. Not the point, right? The important thing is usually not the full series. Usually you just take some terms of it anyway. And what it means conceptually is a little bit more important. Right? It translates a product into a sum and there will be some loss somewhere. correction terms somewhere. That's just what it is. This is literally a near the identity statement, right? Locally globally the logarithm is just an absolutely disgusting map. The logarithm is already an absolutely disgusting map in real numbers. Here's a complex logarithm which is an absolutely disgusting map which has multiple branches and you have this Riemann surface type story. Oh my goodness, we don't want to do that. So locally, yeah, great. And globally, yeah, no. Essentially, but that's the relationship between league groups and league algebras anyway. Really only good locally. It's fine. And this is this is local. And globally, yeah, we don't want to think about it. Let's not do it. Let me not do it. Why do we care? Well, essentially it is the way to go between one and the other, right? Because if it's a league group it's a logarithm of the league group the league algebra. Oh god. Oh my goodness. The league algebra is a logarithm of the league group. And the league group is the exponential of the league algebra. We kind of want to know how exponential and logarithm really behave with one another. So it's a really important tool kind of for the structure theory of league groups, of course. Yeah. So this is essentially explaining why league algebra maps are the correct things of local league group maps and whatever, right? Locally everything is kind of really nice. That's what why we want to do it. Yeah. And yeah, so next time I will talk about differential structures and how actually homomorphisms can be differentiated to get league algebra maps. But the formula itself which you don't need to remember as I said. Which I haven't even written down. There's a closed formula to write down all the funny coefficients for example and all the funny terms that appear. Yeah, let's not do it. Anyway, the formula itself just explains that our league group and our league algebra are almost the same object. But non-commutativity makes it just life a little bit more difficult as it usually does. That's why life is so If life would be commutative, oh my goodness, that would be good or bad. I actually can't tell. But it would be probably easier. But life is not commutative and league groups are also not commutative. Anyway, I hope you enjoyed this video and I also hope to see you next time. And I realize now next time apparently I'm trying to do that and not that. Oh, we'll see what will happen next time. Okay, I still hope you enjoyed this video and I hope to see you next time.