This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms Besides 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 For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras

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The Book of Involutions

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

Projective modules and vector bundles The Grothendieck group $K_0$ $K_1$ and $K_2$ of a ring Definitions of higher $K$-theory The fundamental theorems of higher $K$-theory The higher $K$-theory of fields Nomenclature Bibliography Index

The K-Book: An Introduction to Algebraic K-theory

Introduction Summary Part I. Algebra: 1. Algebraic Poincare complexes 2. Algebraic normal complexes 3. Algebraic bordism categories 4. Categories over complexes 5. Duality 6. Simply connected assembly 7. Derived product and Hom 8. Local Poincare duality 9. Universal assembly 10. The algebraic pi-pi theorem 11. -sets 12. Generalized homology theory 13. Algebraic L-spectra 14. The algebraic surgery exact sequence 15. Connective L-theory Part II. Topology: 16. The L-theory orientation of topology 17. The total surgery obstruction 18. The structure set 19. Geometric Poincare complexes 20. The simply connected case 21. Transfer 22. Finite fundamental group 23. Splitting 24. Higher signatures 25. The 4-periodic theory 26. Surgery with coefficients Appendices Bibliography Index.

Algebraic L-theory and Topological Manifolds

We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G.

The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory

K-Theory of Forms.

Nonabelian K-theory: The nilpotent class of K1 and general stability

This is the first in a series of papers dedicated to the structure of hyperbolic unitary groups over form rings and their subgroups. In this part, we recall foundations of the theory and study the elementary subgroup of the hyperbolic unitary group over a form ring. In particular, using a variant of Suslin’s patching method, we prove standard commutator formulae for relative elementary subgroups.

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Structure of hyperbolic unitary groups i. elementary subgroups

On modules with quadratic forms

We define a notion of group functor G on categories of graded modules, which unifies previous concepts of a group functor G possessing a notion of elementary subfunctor E. We show under a general condition which is easily checked in practice that the elementary subgroup E(M) of G(M) is normal for all quasi-weak Noetherian objects M in the source category of G. This result includes all previous ones on Chevalley and classical groups G of rank ≥ 2 over a commutative or module finite ring M (since such rings are quasi-weak Noetherian) and settles positively unanswered cases of normality for these group functors.

Anthony Bak

Papers

K-Theory of Forms.

Nonabelian K-theory: The nilpotent class of K1 and general stability

Structure of hyperbolic unitary groups i. elementary subgroups

On modules with quadratic forms

Normality for elementary subgroup functors