Lie theory, part 12 | Daniel Tubbenhauer

Okay, welcome everyone to this continuation of Lie theory. Um next time will be my almost my favorite topic in this uh series of Lie theory. But today we want to prepare for this and I tell you about well very important concepts. Um mostly simply rather solvable. Uh um and related to center and the derived algebra. Derived algebra sounds very difficult. Actually, it's not so bad. Okay. So, let's have a look. Come on. Don't disappoint me here, you stupid screen. Excellent. Um so, why these four words matter? Now I can use it. The center derived algebra is simple and solvable. Okay. So, essentially um I had this picture of the Baker-Campbell-Hausdorff formula and I told you don't remember it and here it is again. Um the only important thing is that whenever something is commutative, this funny bracket here will always vanish and so this all of this crap here just disappears. And then the log exp, which is the relation between the group and the algebra is just easy, right? So, in some sense um in Lie theory, commutativity is a measure of easiness or non-commutativity how difficult something is. So, somehow we want to measure how non-commutative an algebra is and use that as kind of a as I said as a measurement of difficulty to say um this algebra is easy or this algebra is not so easy. Okay, so the big picture is usually well, we have a theory and we want to find the basic building blocks. I show you some uh very good pictures to remember in a second. Yeah? And as I said, commutativity is kind of the main thing we need to measure because log exp is the Well, the whole spiel of Lie theory is somehow the log exp picture we had before. So, that's it. And now we are just going step by step through whatever the center is, whatever that thing is, and so on and so on. Turns out it's actually really easy. So, center and derived algebra are kind of the opposite and I like to think about the center as this thing here. Kind of. So, the calm elements, right? The elements that don't do anything, the easy elements, whatever, the center. Things that commute with everything, right? If you commute with everything, the exp block is just easy, yeah? So, center is like the easy part and I will have a nice example later where you see it's the easy part of our little Lie algebra or uh well, Lie group, whatever you want. They're the same. Anyway, up to a log exp picture, I guess. And the derived algebra is essentially the opposite. It's everything generated by brackets, right? The brackets is are the difficult things. So, everything generated by them is like the everything that moves, if you want, right? Everything that stays quiet is the center. Uh they don't do anything. Very quiet, like this picture here. And the opposite, everything that kind of moves around and does crazy stuff. That's usually the derived algebra, right? So, it's kind of heavy pretty crucial opposites of one another uh in behavior. So, well, studying centers is usually easy and studying derived algebras is usually difficult uh and you want to look for things without center because they're kind of the simple building blocks of the picture, right? I hope this makes sense. Kind of two different things. One of them, the quiet things and creating things, yeah? — [snorts] — So, of I should also point this out. Of course, the quiet things are the easier things, yeah? But they're also the less interesting one. The usual trade-off that you have in life uh sometimes when things are easy, they're also boring uh and when they're interesting, they're difficult. That's just what it is. So, here kind of exactly this type of picture. And now kind of what I want you to have in mind uh the two opposite moods and I will focus on simple first. So, all most because that's kind of what I'm doing next time. And then eventually we come back to solvable. I'll just put it up here as well. So, I really think of this as an analogy to chemistry, right? So, the one of the main first things people did in chemistry when it was kind of um well, becoming a real science because it was before it was more like alchemy, which is a bit uh creating gold type stuff. Anyway, is kind of mapping out the elements, the basic building blocks of chemistry, like gold here, for example, right? So, and turns out that this idea is so successful that it's kind of repeated everywhere, in particular in mathematics. Mathematicians usually call this the classification of something of the basic building blocks, whatever, the elements of your theory, yeah? So, really repeated everywhere. And uh very roughly, simple just means uh no substructure. So, the elements of the theory, yeah? So, the Lie algebra is simple if it kind of is an element of the theory. And

solvable is literally it's not too bad. So, repeated brackets eventually die out. So, and somehow the algebra gets easier and easier as we go, yeah? So, one notion says something like about the basic non-commutative blocks, right? Simple. Well, essentially, the other says uh it's essentially commutative. So, very different type of behaviors between a simple and solvable. So, you know, I say it again. My picture mostly focuses on simple because that's what we're going to do next. But solvable is kind of the not really the opposite. It's kind of the opposite. It's also a measurement of how difficult something is. Yeah, but if commutative is easy, then solvable is whatever, semi-easy, if you want. Yeah? But simple is like the basic building blocks, the elements of your theory. So, you want to eventually write down a periodic table of elements, yeah? That's what people usually like to do. And mathematicians traditionally call that a classification. Anyway, so the tension here comes as I said from very different things. So, these are they sound very similar, but they're actually very different in in spirit. And why do we want that? So, here's another element picture. Um So, essentially, the simples are like the elements, and elements are reasonably easy to classify. Um generally, algebras are more like compounds. They're made out of simples. Um and just because you understand the elements doesn't imply you understand the compounds, yeah? Compounds are usually much more difficult. There are way more of them and there's a huge step between the two. But understanding the elements is kind of a first good step. And that's exactly what we're trying to do next time. We're trying to understand the simple Lie algebras, which doesn't tell you well, not too much about a general Lie algebra, right? A general Lie algebra is more like my little compound here. This is my highest zoom. Fine. Whatever. My little compound here. Um so, compounds are naturally difficult than elements. And in general, it gets even more difficult and you kind of have mixtures of different things and you want to kind of decompose and that's of course a whole another universe if you just know what the elements are. But the elements are always a good start. As I said, I focus mostly on simple here and not solvable. But essentially, those two notions are built to split our little mixture zoo here, yeah? This or huge huge uh well, maybe not This huge kind of mixture of different very differently behaving uh mixtures, com- compounds into easy families, if you want, right? The easiest families. So, solvable Lie algebra is pretty easy. The simple Lie algebra is classifiable. Something like this, yeah? And a lot of the structure theory, structure theory is important, but again, doesn't tell you all that much about the general Lie algebra, right? Um but of course we do it. That's just how mathematics works. It's like understanding that this picture is a general picture. So, there's some general pattern in mixtures. Also, mixtures are much more difficult than elements and then you can kind of build them from the elements. The analog in mathematics is structure theory. And much of the structure theory of Lie algebras, Lie theory, whatever, is well is essentially we classify or understand these simple objects and then try to understand how they assemble together, right? So, that's essentially how works as well. Let's have a look at the matrix examples to make this a bit clearer. Um a scalar matrix is that thing. Just to be clear here. So, a scalar matrix is not a diagonal matrix or a scalar matrix is a diagonal matrix, but not every diagonal matrix is a scalar matrix. Literally just what whatever. Lambda lambda, right? Always the same scalar on the diagonal. And in our little [snorts] matrix examples, if you take our favorite Lie uh algebra GLN, then the scalar matrices that commute with everything uh that's easy to check that they commute with everything, but also they're the only matrices that commute with everything. So, the center is just scalars. And of course, you can just identify them with the complex numbers in this case. So, center really easy. And the derived algebra is literally the rest and that is SLN and that's the simple Lie algebra. So, that's what remains, right? So, the derived algebra picks out the genuinely the non-Abelian part, right? The non-commutative part. So, in matrices, almost everything is non-commutative. That's kind of one of the main features of matrices. Um of course, I'm talking here about N not one. N not N equal one is a bit it's a bit boring in this case. But for N bigger than one, two by two matrices, three by three matrices, they're highly non-commutative, which is in some sense their defining feature. And of course, diagonal matrices feel feel very easy, right? They're very easy. Uh so the solvable things are

like diagonal matrices. And the simple things are like sln. That's literally what it is. The simple building blocks of the story. And yeah, we push that further next time when I just tell you that you can actually classify the simple Lie algebras. So the periodic table of simple Lie algebras. Anyway, I hope you enjoyed this video and I will talk to you next time.