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Quadratic Forms

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0002
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a ... More


Quadratic Forms over a ∈ Local Field

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0007
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary ... More


Semi-ramified Quadrangles of Type E6, E7 and E8

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0012
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang semi-ramified quadrangle of type E⁶, E₇ and E₈. The basic proposition is that Ξ‎ is a ... More


Quadratic Forms of Type F4

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0009
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and ... More


Quadratic Forms of Type E6, E7 and E8

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0008
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ ... More


Constructions with regular degenerate quadratic forms

Brian Conrad and Gopal Prasad

in Classification of Pseudo-reductive Groups (AM-191)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167923
eISBN:
9781400874026
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167923.003.0007
Subject:
Mathematics, Numerical Analysis

This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root ... More


Moufang Quadrangles

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0004
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter proves various results about Moufang quadrangles. It first considers the notions of a proper involutory set, a proper indifferent set, and a proper anisotropic pseudo-quadratic space. It ... More


Quadrangles of Type E6, E7 and E8: Summary

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0014
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ‎ is an unramified ... More


Existence

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0016
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the ... More


Residually Pseudo-Split Buildings

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0033
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents results about a residually pseudo-split Bruhat-Tits building Ξ‎L. It begins with a case for some quadratic space of type E⁶, E₇, and E₈ in order to identify an unramified ... More


Orthogonal Buildings

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0035
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents a few results about certain forms of orthogonal buildings. It begins with notations stating that V is a K-vector space of positive dimension, (K, V, q) is a quadratic space of ... More


Linked Tori, II

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0006
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = ... More


The Other Bruhat-Tits Buildings

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0018
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter summarizes the results about the residues of Bruhat-Tits buildings other than those associated with the exceptional Moufang quadrangles examined in previous chapters. It first considers ... More


Linked Tori, I

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0005
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates the consequences of the assumption that one Moufang set is weakly isomorphic to another. It first introduces some well-known facts about involutions which are assembled in a ... More


Totally Wild Quadratic Forms of Type E7

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0015
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over ... More


Residues

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0010
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter deals with the residues of a Bruhat-Tits building whose building at infinity is an exceptional quadrangle. It begins with the remark that if Λ‎ is an arbitrary quadratic space of type Eℓ ... More


Unramified Quadrangles of Type E6, E7 and E8

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0011
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a ... More


Quadrangles of Type F4

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0017
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter deals with the case that the building at infinity Λ‎ of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type F₄. It begins with the hypothesis stating that Λ‎ = (K, L, q) is a ... More


Linear Automorphisms

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0029
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter considers the notion of a linear automorphism of an arbitrary spherical building satisfying the Moufang property. It begins with the notation whereby Ω‎ = (U₊, U₁, ..., Uₙ) is the root ... More


Preliminaries

Jayme Vaz and Roldão da Rocha

in An Introduction to Clifford Algebras and Spinors

Published in print:
2016
Published Online:
August 2016
ISBN:
9780198782926
eISBN:
9780191826146
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198782926.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this chapter, a review of the main concepts, examples, and results from linear algebra are presented, as these are essential for developing the subsequent ideas in this book. As the concept of ... More


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