The Book of Involutions, Volume 44Max-Albert Knus American Mathematical Soc., 30 juin 1998 - 593 pages 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 |
Table des matières
| 1 | |
Invariants of Involutions | 71 |
Similitudes | 153 |
Algebras of Degree Four | 205 |
Algebras of Degree Three | 279 |
Algebraic Groups | 323 |
Galois Cohomology | 383 |
Composition and Triality | 451 |
Cubic Jordan Algebras | 513 |
Trialitarian Central Simple Algebras | 547 |
| 573 | |
| 585 | |
Expressions et termes fréquents
A₁ adjoint involution algebra of degree algebra with involution algebraic group Alt(A alternative algebra assume automorphism B₁ bijection bilinear form Brauer-equivalent canonical involution central simple algebra char F characteristic Clifford algebra Co(V cohomology commutative composition algebra Corollary corresponding cubic étale defined denote division algebra element EndF EndƑ(V equivalence étale F-algebra exact sequence F-algebra field extension field F finite follows form h Fsep functor Galois GO(A group scheme groupoid H¹(F hence hermitian form homomorphism hyperbolic idempotent ind(A induces Int(u invariant involution isometry isomorphism isotropic Jordan algebras Lemma Let G Lie algebra linear multiplication norm orthogonal involution Pfister form polynomial Proof Proposition quadratic form quadratic pair quadratic space quaternion algebra resp restriction right ideal scalar extension second kind shows similitudes Skew Skew(A subalgebra subgroup subspace surjective Sym(A Sym(B symplectic involution Theorem Tits twisted composition unitary involution v₁ vector space
Fréquemment cités
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...
Page 1 - Let V be a finite dimensional vector space over a field F.
Page xxi - Two sequences s and s' are called "equivalent" if they can be transformed into each other by a sequence of SHIFT-operations and s is called "canonical" if no more SHIFToperation is applicable.