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
Algebras of Degree Four
Algebras of Degree Three
adjoint involution algebra of degree algebra over F algebraic group algebras with involution alternative algebra arbitrary assume automorphism bijection bilinear form canonical involution central simple algebra char F Clifford algebra cohomology commutative composition algebra Corollary corresponding cubic étale defined denote division algebra element equivalent étale F-algebra exact sequence F x F field extension field F finite follows form h Fsep functor Galois group scheme groupoid hence hermitian form homomorphism hyperbolic idempotent induces Int(u invariant involution isometry isotropic Jordan algebras Lemma Let G Lie algebra linear morphisms multiplication nontrivial norm orthogonal involution Pfister form polynomial Proof Proposition quadratic étale quadratic form quadratic pair quadratic space quaternion algebra resp restriction right ideal scalar extension second kind shows similitudes simply connected subalgebra subgroup subspace surjective symplectic involution Theorem trivial twisted composition uniquely determined unitary involution vector space ст тг
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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