Holger P. Petersson - Böcker

Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.

Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.

Albert algebras provide key tools for understanding exceptional groups and related structures such as symmetric spaces. This self-contained book provides the first comprehensive reference on Albert algebras over fields without any restrictions on the characteristic of the field. As well as covering results in characteristic 2 and 3, many results are proven for Albert algebras over an arbitrary commutative ring, showing that they hold in this greater generality. The book extensively covers requisite knowledge, such as non-associative algebras over commutative rings, scalar extensions, projective modules, alternative algebras, and composition algebras over commutative rings, with a special focus on octonion algebras. It then goes into Jordan algebras, Lie algebras, and group schemes, providing exercises so readers can apply concepts. This centralized resource illuminates the interplay between results that use only the structure of Albert algebras and those that employ theorems about group schemes, and is ideal for mathematics and physics researchers.