A Taste of Jordan Algebras Kevin McCrimmon Department of Mathematics University of Virginia Charlottesville, Virginia Dedicated to the memory of Jake and Florie { in mathematico parentis To Jake (the Doktor-Vater) for his mathematical in(cid:13)uence on my research, and to Florie (the Doktor-Mutter) for helping me (and all Jake’s students) to get to know him as a warm human being. Future histories of mathematics should take into account the role of Doktor- Mutters in the fostering of mathematics. 1. INTRODUCTION iii 1. Introduction On several occasions I and colleagues have found ourselves teach- ing a 1-semester course for students at the second year of graduate study in mathematics who want to gain a general perspective on Jor- dan algebras, their structure and their role in mathematics, or want to gain direct experience with nonassociative algebra. These students typically have a solid grounding in (cid:12)rst-year graduate algebra and the Artin-Wedderburn theory of associative algebras, and a few have been introduced to Lie algebras (perhaps even Cayley algebras, in an o(cid:11)- hand way), but otherwise they have not seen any nonassociative al- gebras. Most of them will not go on to do research in nonassociative algebra, so the course is not primarily meant to be a training or breed- ing ground for research, though the instructor often hopes one or two will be motivated to pursue the subject further. This text is meant to serve as an accompaniment to such a course. It is designed (cid:12)rst and foremost to be read. It is a direct mathematical conversation between the author and a reader whose mind (as far as nonassociative algebra goes) is a tabula rasa. In keeping with the tone ofaprivateconversation, Igivemoreheuristicmaterialthaniscommon in books at this level (pep talks, philosophical pronouncements on the proper way to think about certain concepts, random historical anec- dotes, o(cid:11)hand mention of some mathematicians who have contributed to our understanding of Jordan algebras, etc.), and employ a few Eng- lish words which do not standardly appear in mathematical works. It is important for the reader to develop a visceral intuitive feeling for the subject, to view the mathematics as a living and active thing: to see isomorphisms as cloning maps, isotopes as subtle rearrangements of an algebra’s DNA, radicals as pathogens to be isolated and removed by radicalsurgery, annihilatorsas biologicalagents forkillingo(cid:11)elements, Peircers as mathematicalenzymes (\Jordan-ase") which break an alge- bra down into its Peirce spaces. Like Charlie Brown’s kite-eating trees, Jordan theory has Zel’manov’s tetrad-eating ideals (though we shall stay clear of these carnivores in our book). The book is intended for students to read on their own without assistance by a teacher. In particular, I have tried to make the proofs completeand understandable, givingmuchmoreheuristicandexplana- tory comment than is usual in graduate texts. To help the reader through the proofs in Parts III, IV (and the proof-sketches in Part II, Chapter 8), I have tried to give each important result or formula a mnemonic label, so that when I refer to an earlier result, instead of saying \by Formula 21.3(i), which of course you will remember, ..." I iv can say \by Nuclear Slipping 21.3(i)", hoping to trigger long-repressed memories of a formula involving nuclear elements of alternative alge- bras. While I wind up doing most of the talking, there is some room in Parts III and IV for the reader to participate (and stay mathemati- cally (cid:12)t) by doing exercises. The Exercises give slight extensions, or alternate proofs, of results in the text, and are placed immediately after the results; they give practice in proving variations on the previ- ous mathematical theme . At the end of each chapter I gather a few problems and questions. The Problems usually take the form \Prove that something-or-other"; they involve deeper investigations or length- ier digressions than exercises, and develop more extensive proof skills on a new theme. The Questions are more open-ended, taking the form \What can you say about something-or-other" without giving a hint whichway the answer goes; they develop proofskillsinuncharted terri- tories,incomposingamathematicalthemefromscratch(mostvaluable for budding researchers). Hints are given at the back of the book for some of the exercises, problems, and questions (though these should be consulted only after a good-faith e(cid:11)ort to prove them). PartIisinthenature ofanextended colloquiumtalk,abriefsurvey of the life and times of Jordan algebras, to provide appreciation of the role Jordan algebras play on the broader stage of mathematics. I indi- cate several applications to other areas of mathematics: Lie algebras, di(cid:11)erential geometry, and projective geometry. Since the students at this level cannot be assumed to be familiar with all these areas, the description has to be a bit loose; readers can glean from this partjust enough respect and appreciation to sanction and legitimate their in- vestment in reading further. PartIIisdesignedtoprovideanoverviewofJordanstructuretheory in its historical context. It gives a general historical survey from the originsinquantummechanicsin1934toE(cid:12)mZel’manov’sbreathtaking description of arbitrary simple algebras in 1983 (which later played a role in his Fields Medal work on the Burnside Problem). I give precise de(cid:12)nitions and examples, but omit proofs. In keeping with its nature, I have not included any exercises. Parts III and IV are designed to provide direct experience with nonassociativity,andeitherone(inconjunctionwithPartI)couldserve as a basis for a one-semester course. Throughout, I stick to linear Jordan algebras over rings of scalars containing 1/2, but give major emphasis to the quadratic point of view. PartIIIgives adevelopmentofJacobson’s classicalstructure theory for Jordan algebras with capacity, in complete detail and with full 1. INTRODUCTION v proofs. It is suitable for a one-semester course aiming to introduce students to the methods and techniques of nonassociative algebra. The details of Peirce decompositions, Peirce relations, and coordinatization theorems are the key tools leading to the Classical Structure Theorem. Part IV gives a fulltreatment of Zel’manov’s Exceptional Theorem, that the only simple i-exceptional Jordan algebras are the Albert alge- bras,closingthehistoricalsearchforanexceptionalsettingforquantum mechanics. This part is much more concerned with understanding and translating to the Jordan setting some classical ideas of associative theory, including primitivity; it is suitable for a one-semester course aiming to introduce students to the modern methods of Jordan alge- bras. The ultra(cid:12)lter argument, that if primitive systems come in only a (cid:12)nite number of (cid:13)avors then a prime system must come in one of those pure (cid:13)avors, is covered in full detail; ultra(cid:12)lters provide a useful tool that many students at this level are unacquainted with. I have dedicated the book to Nathan and Florie Jacobson, both of whom passed away during this book’s long gestation period. They had an enormous in(cid:13)uence on my mathematical development. I am greatly indebted to my colleague Kurt Meyberg, who carefully read through Part III and made many suggestions which vastly improved the exposition. I am also deeply indebeted to my colleague Wilhelm Kaup, who patiently corrected many of my misconceptions about the role of Jordan theory in diferential geometry, improving the exposition in Part I and removing (cid:13)agrant errors. My colleague John Faulkner helped improve my discussion of applications to projective geometries. I would also like to thank generations of graduate students at Virginia who read and commented upon the text, especially my students Jim Bowling, Bernard Fulgham, Dan King, and Matt Neal. vi Index of Notations General Typographical Conventions Rings of scalars (unital, commutative, associative rings) are indi- (cid:15) cated by capital Greek letters (cid:8);(cid:10): Scalars are denoted by lower case Greek letters: (cid:11);(cid:12);(cid:13);::: Almost all our algebraic systems will be alge- bras or modules over a (cid:12)xed ring of scalars (cid:8), which willalmost always 1 contain an element . 2 Mere sets are indicated by italic capital letters X;Y;Z at the end (cid:15) of the alphabet, index sets also by I;J;S. Modules and linear spaces are denoted by italic capital letters: (cid:15) A;B;C;J;V;W;:::. The zero subspace will be denoted by boldface 0 to distinguish it from the element (operator, vector, or scalar) 0. This signals a subtle and not-too-important distinction between the set 0 = 0 consisting of a single element zero, and the element itself. f g The range f(A) of some function on a set A will always be a set, while the value f(a) will be an element. Algebraic systems are denoted by letters in small caps: general (cid:15) linear algebras by A;B;C, ideals by I;J;K. Associative algebras are indicated by D when they appear as coordinates for Jordan algebras. 0 Jordan algebras are indicated by J;Ji;J; etc. Maps or functions between sets or spaces are denoted by italic (cid:15) lower case letters f;g;h;:::, morphisms between algebraic systems of- ten by lower case Greek letters ’;(cid:27);(cid:28);(cid:26), sometimes upper case italic letters T;S. Functors and functorial constructions are denoted by script cap- (cid:15) ital letters ; ; ; ;:::. F G H T Speci(cid:12)c Notations The identity map on a set X is denoted by 1X. The projection (cid:15) of a set on a quotient set is denoted by (cid:25) : X X= s; the coset or ! equivalence class of x X is denoted by (cid:25)(x) = x = [x]. 2 CartesianproductsofsetsaredenotedX Y. Moduledirectsums (cid:15) (cid:2) are indicated by V W. Algebra direct sums, where multiplication as well as addition is(cid:8)performed componentwise, are written A (cid:1) B to distinguish them from mere module direct sums. Subsets are denoted X Y, with strict inclusion denoted by (cid:15) (cid:18) X Y or X < Y; subspaces of linear spaces are denoted A B, (cid:26) (cid:20) while kernels (submodules of modules, ideals of algebras) are indicated by B/A. Left, right, and -ideals of algebras are denoted by I /‘ A; (cid:3) (cid:3) I/r A; B / A. 1. INTRODUCTION vii Involutions on algebras are indicated by a star (though invo- (cid:15) (cid:3) lutions on coordinate algebras of matrix algebras are often denoted (cid:3) x x(cid:22)). (A; ) denotes the hermitian elements x = x of an algebra ! H (cid:3) A under an involution . (cid:3) Products in algebras are denoted by x y or just xy (especially for (cid:15) (cid:1) associative products); the special symbol x y is used for the bilinear (cid:15) product in Jordanalgebras. The leftandrightmultiplicationoperators byanelementxinalinearalgebraaredenoted Lx;Rx or‘x;rx or(cid:21)x;(cid:26)x. The quadratic and trilinear products in Jordan algebras are denoted by Uxy and x;y;z , with operators Ux;Vx;y (in Jordan triples Px;Lx;y, f g in Jordan pairs Qx;Dx;y). Unit elements of algebras are denoted by 1. We will speak of (cid:15) a unit element and unital algebras rather than identity element and algebras with identity; we will reserve the term identity for identical relations or laws (such as the Jordan identity or associative law). A will denote the formal unital hull (cid:8)-algebra (cid:8)1 A obtained by formal (cid:8) adjunction of a unit element. b n n matrices and hermitian matrices are denoted n and n. (cid:15) (cid:2) M H Matrices are denoted by X = (xij); their traces and determinants are denoted by tr(X); det(X) respectively, and the transpose is indicated tr by X . If the matrix entries come from a ring with involution, the adjoint (the conjugate transpose with ij-entry xji) is denoted rather (cid:3) anonymously by a star, X . Blackboardboldisusedforthestandardsystems N (naturalnum- (cid:15) bers 1;2;:::), I(the even-more-natural numbers 0;1;2;::: used as in- dices or cardinals), Z (the ring of integers), the (cid:12)elds Q (rational num- bers), R (realnumbers), C (complexnumbers), therealdivisionringsH (Hamilton’s quaternions), K (Cayley’s octonions), O (split octonions), A (the Albert algebra, a formally-real exceptional Jordan algebra). Contents 1. Introduction iii Part I. A Colloquial Survey of Jordan Theory 1 1. Origin of the Species 3 2. The Jordan River 7 3. Links with Lie Algebras and Groups 11 4. Links with Di(cid:11)erential Geometry 15 5. Links with the Real World 17 6. Links with the Complex World 24 7. Links with the In(cid:12)nitely Complex World 27 8. Links with Projective Geometry 30 9. Conclusion 38 Part II. The Historical Perspective: An Historical Survey of Jordan Structure Theory 39 Chapter 1. Jordan Algebras in Physical Antiquity 41 1. The matrix interpretation of quantum mechanics 41 2. The Jordan Program 42 3. The Jordan Operations 42 4. The Jordan Axioms 46 5. The Basic Examples 48 6. Special and Exceptional 51 7. Classi(cid:12)cation 51 Chapter 2. Jordan Algebras in the Algebraic Renaissance 55 1. Linear Algebras over General Scalars 56 2. Categorical Nonsense 57 3. Commutators and Associators 59 4. Lie and Jordan Algebras 61 5. The 3 Basic Examples Revisited 62 6. Jordan Matrix Algebras 63 7. Forms Permitting Composition 65 8. Composition Algebras 68 9. Split Composition Algebras 70 ix x CONTENTS 10. Classi(cid:12)cation 72 Chapter 3. Jordan Algebras in the Enlightenment 75 1. Forms of Algebras 75 2. Inverses and Isotopes 76 3. Twisted Hermitian Algebras 78 4. Spin and Quadratic Factors 80 5. Cubic Factors 83 6. Classi(cid:12)cation 85 Chapter 4. The Classical Theory 89 1. U-Operators 89 2. Quadratic Axioms 90 3. Inverses 92 4. Isotopes 94 5. Inner Ideals 94 6. Nondegeneracy 95 7. i-Special and i-exceptional 98 8. Artin-Wedderburn-Jacobson Structure Theorem 99 Chapter 5. The Final Classical Formulation 103 1. Capacity 103 2. Classi(cid:12)cation 104 Chapter 6. The Classical Methods 107 1. Peirce Decompositions 107 2. Coordinatization 108 3. Minimum Condition 111 Chapter 7. The Russian Revolution: 1977-1983 115 1. The Lull Before the Storm 115 2. The First Tremors 116 3. The Main Quake 118 4. Aftershocks 119 Chapter 8. Zel’manov’s Exceptional Methods 123 1. I-Finiteness 123 2. Absorbers 125 3. The Primitive Heart 126 4. The Big Primitive Classi(cid:12)cation 129 5. Semiprimitive Imbedding 133 6. The Ultraproduct Argument 134 Part III. The Classical Theory 139