A Taste of Jordan Algebras (Universitext)
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This book describes the history of Jordan algebras and describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas. No knowledge is required beyond standard first-year graduate algebra courses.
From the Back Cover
In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras, though the modern quadratic methods are used throughout. Both the quadratic methods and the Zelmanov results go beyond the previous textbooks on Jordan theory, written in the 1960's and 1980's before the theory reached its final form.
This book is intended for graduate students and for individuals wishing to learn more about Jordan algebras. No previous knowledge is required beyond the standard first-year graduate algebra course. General students of algebra can profit from exposure to nonassociative algebras, and students or professional mathematicians working in areas such as Lie algebras, differential geometry, functional analysis, or exceptional groups and geometry can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings and can be applied to a variety of mathematical areas.
Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.
algebra is in a canonical way a homogeneous Riemannian manifold. The inversion map j : x → x−1 induces a diﬀeomorphism of J of period 2 leaving C invariant, and having there a unique ﬁxed point 1 [the ﬁxed points of the inversion map are the e − f for e + f = 1 supplementary orthogonal idempotents, and those with f = 0 lie in the other connected components of J−1 ], and provides a symmetry of the Riemannian manifold C at p = 1; here the exponential map is the ordinary algebraic exponential exp1
always always Thus, once a coordinate system has been chosen, the entire incidence structure is encoded algebraically in what is called a projective ternary system Tern(Π, χ), the set of aﬃne coordinates with the ternary product T (x, m, b) and distinguished elements 0, 1 (somewhat optimistically called a ternary ring in hopes that the product takes the form T (x, m, b) = xm + b for a+b = T (a, 1, b), xm = T (x, m, 0)). In general, the ternary system depends on the coordinate system (diﬀerent
sun in this region either, answering the division algebra part of (FAQ3). Zel’manov’s Division Theorem. The Jordan division algebras are precisely those of classical type: Quadratic Type: Jord(Q, c) for an anisotropic quadratic form Q over a ﬁeld ; Full Associative Type: ∆+ for an associative division algebra ∆; Hermitian Type: H(∆, ∗) for an associative division algebra ∆ with involution ∗; Albert Type: Jord(N, c) for an anisotropic Jordan cubic form N in 27 dimensions. As a coup de grˆ ace,
algebras evaporate in the blazing heat of a big algebraically closed ﬁeld. Division Evaporation Theorem. If J is a Jordan division algebra over a big algebraically closed ﬁeld Φ, then J = Φ1. As Roosevelt said, “We have nothing to fear but Φ itself.” This leads to the main structural result. Big Primitive Exceptional Theorem. A primitive i-exceptional Jordan algebra over a big algebraically closed ﬁeld Φ is a simple split Albert algebra Alb(Φ). Proof Sketch: The heart ♥(J) = i-Specializer(J) = 0
(a ⊕ bm) = (a ⊕ bm) + (a ⊕ −bm) = (a + a) ⊕ 0 = t(a)1, and using the Cayley–Dickson multiplication rule the new norm is (a⊕bm)(a ⊕ bm) = (a⊕bm)(a⊕(−b)m) = aa−µbb ⊕ −ba+ba m = n(a) − µn(b) 1. For commutativity of KD(A, µ), commutativity of the subalgebra A is clearly necessary, as is triviality of the involution by am − ma = (a − a)m; these also suﬃce to make the Cayley–Dickson product a1 a2 +µb1 b2 ⊕ a1 b2 + b1 a2 m symmetric in the indices 1 and 2. For associativity of KD(A, µ), associativity of