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14

Oct

2024

Oct

2024

We are the kind of people who are always interested in the strongest example of something, the paragon. When we eat Swiss cheese (Emmental), we want our senses to tell us that; we shouldn’t have any doubts that maybe we are eating cheddar.

This book takes the same approach to Jordan systems in mathematics. They come in lots of different forms, such as Jordan algebras and cubic norm structures, but broadly speaking at the heart of all of them are Albert algebras. If Albert algebras didn’t exist, the study of Jordan systems would be considerably less interesting, and it would often be enough to work instead with more familiar objects like associative algebras. Our new book, *Albert algebras over commutative rings: The Last Frontier of Jordan Systems*, is the first to focus on Albert algebras as an object of study.

The view that Albert algebras are the paragon of Jordan systems is very similar to what one finds for the more famous octonions, which are the strongest example of alternative algebras or composition algebras. If octonions didn’t exist, the study of alternative algebras and composition algebras would reduce to studying associative algebras. The similarity of these two situations is not just a coincidence. Albert and octonion algebras are interconnected, and the proofs of many theorems about Albert algebras rely on results about octonions. Consequently, we develop also the theory of octonion and composition algebras in our book.

The treatment of both of these subjects, Albert algebras and octonion algebras, is also done in a broader setting than in other books. Most results are proved over an arbitrary commutative ring instead of over a field of characteristic different from 2 or 3.

**One weird trick**

When you try to extend theorems that work over fields to arbitrary rings, one common phenomenon is that the theorem works for semilocal rings but not for arbitrary rings. (Linear algebra enthusiasts are especially likely to have seen this.) We experienced this in proving the theorems in this book. But when we looked at the proofs, they felt a little awkward and we wondered: Is “semilocal” the right hypothesis?

We looked around the literature and found the concept of an “LG ring” that was introduced in the early 1980s. It is a slightly more general class of rings than semilocal rings, in that it includes also the ring of all algebraic integers, which is not semilocal. The beautiful thing about LG rings is that the definition is actually easier to use to prove things — the weaker hypothesis is somehow more useful. Using the “one weird trick” of replacing “semilocal ring” with “LG ring” in our statements gave more powerful results and simplified the proofs!

**Why Jordan systems?**

One classic reason to study Jordan systems is because you are interested in something else and discover that a Jordan system naturally arises as a tool to study the thing you are really interested in. Kevin McCrimmon once described this phenomenon in Lie theory by saying “If you open up a Lie algebra and look inside, 9 times out of 10 there is a Jordan algebra … which makes it tick.” Other places where Jordan structures naturally arise are bounded symmetric domains, Moufang polygons, and Severi varieties.

Maybe the most famous phenomenon directly related to Albert algebras are the 27 lines. In the 1800s, mathematicians proved that any smooth cubic surface over the complex numbers has exactly 27 lines on it and that the incidence relations between those 27 lines are the same, regardless of which surface is chosen. It turns out that these 27 lines can be identified with 27 basis vectors of the unique Albert algebra over the complex numbers and that the multiplication in the Albert algebra is connected with geometric properties of the lines, so the geometry of the arrangement of 27 lines encodes the Albert algebra and vice versa. Now, you can depict the lines and their intersections using a graph, where each vertex (black dot) represents one of the 27 lines and you join two vertices with an edge (blue line) if the lines intersect. We have drawn exactly this graph in the attached image. It naturally lives in six dimensions, so in order to show it to you we have had to draw instead a projection of the graph into two dimensions. That projection piles three of the vertices into the same, central location, so we have indicated that in the picture by drawing concentric circles around that vertex.

One particular connection that we focus on in our book is with the study of exceptional algebraic groups, particularly those of type G_{2}, F_{4}, and E_{6}. This connection can be used in two directions, in the sense that theorems about Albert algebras can be used to prove results about the algebraic groups and vice versa. In our final chapter, we do exactly that. One notable application is that we determine all Albert algebras over the integers up to isomorphism, which Jordan theorists believed to be unknown before our work. It turns out that there is one new isomorphism class of Albert algebra over the integers beyond the ones you might naturally guess. We construct it explicitly as an isotope of one of the others, relying on computations by Elkies and Gross.

**Who is this book for?**

This book is aimed at people who are interested in octonion, composition, Albert, or Jordan algebras. The first chapter focuses on the settings of classical interest, meaning those algebras over the complex numbers, the real numbers, the rational numbers, or the integers, so it is accessible by a broader audience.

The rest of the book is about algebras over a commutative ring, which requires a little more background from algebra, roughly the level of a graduate course. We refer to general references like Bourbaki’s *Algebra* for background results on algebra and we prove (almost) everything else we need.

The book includes more than 300 exercises. These are more than just exercises in the sense of a way to practice one’s knowledge, in the sense that they also provide important additional material extending the text. A complete set of solutions to the exercises is available to download from the arXiv.

**What’s in it?**

Following the historical introduction in Chapter I focusing on the cases of classical interest, Chapters III and IV develop the theory of composition algebras over a commutative ring. We believe this is the first treatment of this subject in book form. We show that composition algebras over LG rings are obtained from the Cayley-Dickson construction, a result we have not seen in print before.

Chapters V, VI, and VII are the heart of the book. They develop the general theory of Jordan algebras (which were once called quadratic Jordan algebras), of cubic Jordan algebras, and especially Albert algebras. One significant result is an analogue for Albert algebras of the result about the Cayley-Dickson construction of composition algebras: every Albert algebra over “almost all” LG rings is obtained from the Tits construction.

This part of the book includes substantial applications to the theory of Albert algebras over fields. For one, we give a complete proof of the theorem that reduced Albert algebras over a field (of any characteristic) are classified by their quadratic trace, equivalently by their associated 3- and 5-fold Pfister quadratic forms. For another, we show that every division cubic Jordan algebra over a field has dimension 1, 3, 9, or 27 or is a purely inseparable extension over a field of characteristic 3, and we give examples of all of these.

The remaining two chapters concern connections between composition and Jordan algebras on the one hand and Lie algebras or affine group schemes on the other hand. This part of the book includes an explicit and concrete development of flat cohomology and faithfully flat descent.

**Come join us!**

We have been studying these algebras for decades and this book contains a lot of what we have learned, much of it never before presented in book form. We are delighted to share it with you!

Title: *Albert Algebras over Commutative Rings: The Last Frontier of Jordan Systems*

Authors: Skip Garibaldi, Holger P. Petersson and Michel L. Racine

ISBN: 9781009426855

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