The Book of Involutions, Volume 44
This monograph is an exposition of the theory of central simple algebras with involution, in relation to linear algebraic groups. It provides the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields. Involutions are viewed as twisted forms of (hermitian) quadrics, leading to new developments on the model of the algebraic theory of quadratic forms. In addition to classical groups, phenomena related to triality are also discussed, as well as groups of type $F_4$ or $G_2$ arising from exceptional Jordan or composition algebras. Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type $D_4$. This volume also contains a Bibliography and Index. Features: original material not in print elsewhere a comprehensive discussion of algebra-theoretic and group-theoretic aspects extensive notes that give historical perspective and a survey on the literature rational methods that allow possible generalization to more general base rings
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Invariants of Involutions
7 The Discriminant
14 Unitary Involutions
Algebras of Degree Four
17 Whitehead Groups
Algebras of Degree Three
23 Automorphism Groups of Algebras
24 Root Systems
Composition and Triality
35 Clifford Algebras and Triality
Cubic Jordan Algebras
Trialitarian Central Simple Algebras
44 Classification of Algebras and Groups of Type DA
action adjoint alternating arbitrary associated assume automorphism basis bilinear form called canonical central simple algebra char F characteristic claim Clifford algebra cohomology commutative composition condition connected consider construction contains Corollary corresponding cubic defined definition denote determined dimension dimp discriminant division element Endf(V equivalence étale F-algebra Example exists extension F-algebra factor field F finite follows Fsep functor Galois given group scheme hence hermitian homomorphism hyperbolic ideal identify induces invariant involution isomorphism Jordan kind Lemma linear Moreover multiplication natural nonsingular norm obtain orthogonal particular polynomial Proof Proposition prove quadratic form quadratic pair quadratic space quaternion algebra reduced relation respect restriction result satisfies scalar separable shows similitudes space split structure suffices Suppose Symd(A symmetric symplectic Theorem Tits trace trivial twisted uniquely unitary vector space yields
Page i - Interpolation and approximation by rational functions in the complex domain, 1935 19 REAC Paley and N. Wiener, Fourier transforms in the complex domain, 1934 18 M. Morse, The calculus of variations in the large, 1934 17 JM Wedderburn, Lectures on matrices, 1934 16 GA Bliss, Algebraic functions, 1933 15 MH Stone, Linear transformations in Hubert space and their applications to analysis...
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