Markus Rost

The Book of Involutions

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.

On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces

The Heegaard decompositions of genus 2 oftorus knot exteriors are classified with respect to homeomorphisms. It turns out that in general there are three different classes which are also the isotopy classes. A similar result is obtained for Seifert fibre spaces, having a disk as base and two exceptional fibres such as the exteriors of torus knots have or having a sphere as base and three exceptional fibres.

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 form to the space of symmetric elements; it is shown to classify involutions up to conjugation. Subfields with dihedral Galois group in central simple algebras of arbitrary odd degree with involution of the second kind are investigated. A complete set of cohomological invariants for algebras of degree 3 with involution of the second kind is given.

The chain lemma for Kummer elements of degree 3

Abstract Let A be a skew field of degree 3 over a field containing the 3rd roots of unity. We prove a sort of chain equivalence for Kummer elements in A. As a consequence one obtains a common slot lemma for presentations of A as a cyclic algebra.

On the dimension and other numerical invariants of algebras and vector products

Tensor categorical and diagrammatic techniques used in the theory of knot invariants can be applied to compute the dimension and other numerical invariants for certain algebraic structures defined by tensor identities. These techniques are described, and applied to symmetric composition algebras and 3-vector products.