The Book of Involutions

  title={The Book of Involutions},
  author={Max A. Knus and Alexander S. Merkurjev and Markus Rost and Jean-Pierre Tignol and Jacques Tits},
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… 

The J-invariant and the Tits algebras of a linear algebraic group

In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group G and the degree one parameters of its J-invariant. Our main technical tool is

Critical varieties and motivic equivalence for algebras with involution.

Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the


We consider a central division algebra (over a field) endowed with a quadratic pair or with a symplectic involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the

On generic triality

It is well known that the study of central simple algebras with involution is closely related to that of classical simple adjoint algebraic groups, the latter occuring as automorphism groups of these

Involution Matrix Algebras – Identities and Growth

The paper is a survey on involutions (anti-automorphisms of order two) of different kinds. Starting with the first systematic investigations on involutions of central simple algebras due to Albert

Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)

Abstract:The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or

Linear algebraic groups with good reduction

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has

Orthogonal Involutions on Algebras of Degree 16 and the Killing Form of E 8

We exploit various inclusions of algebraic groups to give a new construction of groups of type E 8 , determine the Killing forms of the resulting E 8 's, and define an invariant of central simple

Separable algebras and coflasque resolutions

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between



On compositions and triality.

In this paper we develop a general theory of compositions for quadratic spaces of rank 8 with trivial Arf and Clifford invariants. Using this theory, and adapting a classical technique of C.

The multipliers of similitudes and the Brauer group of homogeneous varieties.

A classical theorem of Dieudonne [8], Theoreme 2 asserts that the multipliers of similitudes of a quadratic space of even dimension are norms from the discriminant extension. The present paper yields

The centers of generic division algebras with involution

The main goal of this paper is a study of the centers of the generic central simple algebras with involution. These centers are shown to be invariant fields under finite groups in a way analagous to

The Brauer group of a commutative ring

Introduction. This paper contains the foundations of a general theory of separable algebras over arbitrary commutative rings. Of the various equivalent conditions for separability in the classical

On division algebras

? 1. The object of this paper is to develop some of the simpler properties of division algebras, that is to say, linear associative algebras in which division is possible by any element except zero.

Algebras of odd degree with involution, trace forms and dihedral extensions

A 3-fold Pfister form is associated to every involution of the second kind on a central simple algebra of degree 3. This quadratic form is associated to the restriction of the reduced trace quadratic

A Variety Associated to an Algebra with Involution

Abstract A variety IV(A, *) is canonically associated to a central simple algebra A with orthogonal involution of the first kind *. The properties of this variety are the main topic of this paper.

On the cohomology groups of an associative algebra

The cohomology theory of associative algebras is concerned with the m-linear mappings of an algebra W into a two-sided W-module A. In this theory, the additive group (2(m):$) of the m-linear mappings

On a homomorphism property of certain Jordan algebras

This result has a number of important consequences. It may be seen to imply that no simple exceptional finite dimensional Jordan algebra of characteristic not two is a homomorphic image of a


The purpose of this paper is to show how the transfer (Verlagerung) of a group A into a subgroup B of finite index can be obtained and generalized in the framework of the cohomology theory of groups