Brauer–Wall group
In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field F is a group BW(F) classifying finite-dimensional graded central division algebras over the field. It was first defined by {{harvs|txt|authorlink=Terry Wall|first=Terry|last=Wall|year=1964}} as a generalization of the Brauer group.
The Brauer group of a field F is the set of the similarity classes of finite-dimensional central simple algebras over F under the operation of tensor product, where two algebras are called similar if the commutants of their simple modules are isomorphic. Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite-dimensional central division algebras. The analogous construction for Z/2Z-graded algebras defines the Brauer–Wall group BW(F).Lam (2005) pp.98–99
Properties
- The Brauer group B(F) injects into BW(F) by mapping a CSA A to the graded algebra which is A in grade zero.
- {{harvtxt|Wall|1964|loc=theorem 3}} showed that there is an exact sequence
:: 0 → B(F) → BW(F) → Q(F) → 0
:where Q(F) is the group of graded quadratic extensions of F, defined as an extension of Z/2 by F*/F*2 with multiplication (e,x)(f,y) = (e + f, (−1)efxy). The map from BW(F) to Q(F) is the Clifford invariant defined by mapping an algebra to the pair consisting of its grade and determinant.
- There is a map from the additive group of the Witt–Grothendieck ring to the Brauer–Wall group obtained by sending a quadratic space to its Clifford algebra. The map factors through the Witt group,Lam (2005) p.113 which has kernel I3, where I is the fundamental ideal of W(F).Lam (2005) p.115
Examples
- BW(C) is isomorphic to Z/2Z. This is an algebraic aspect of Bott periodicity {{cn|date=April 2024}} of period 2 for the unitary group. The 2 super division algebras are C, C[γ] where γ is an odd element of square 1 commuting with C.
- BW(R) is isomorphic to Z/8Z. This is an algebraic aspect of Bott periodicity {{cn|date=April 2024}} of period 8 for the orthogonal group. The 8 super division algebras are R, R[ε], C[ε], H[δ], H, H[ε], C[δ], R[δ] where δ and ε are odd elements of square −1 and 1, such that conjugation by them on complex numbers is complex conjugation.
Notes
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References
- {{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | editor1-last=Deligne | editor1-first=Pierre | editor1-link=Pierre Deligne | editor2-last=Etingof | editor2-first=Pavel | editor2-link=Pavel Etingof | editor3-last=Freed | editor3-first=Daniel S. | editor3-link=Dan Freed | editor4-last=Jeffrey | editor4-first=Lisa C. | editor4-link=Lisa Jeffrey | editor5-last=Kazhdan | editor5-first=David | editor5-link=David Kazhdan | editor6-last=Morgan | editor6-first=John W. | editor6-link=John Morgan (mathematician) | editor7-last=Morrison | editor7-link=David R. Morrison (mathematician)|editor7-first=David R. | editor8-last=Witten | editor8-first=Edward | editor8-link=Edward Witten | title=Quantum fields and strings: a course for mathematicians, Vol. 1 | url=http://www.math.ias.edu/QFT | publisher=American Mathematical Society | location=Providence, R.I. | series=Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997 | isbn=978-0-8218-1198-6 | mr=1701598 | year=1999 | chapter=Notes on spinors | pages=99–135}}
- {{citation | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
- {{Citation | last=Wall | first=C. T. C. | authorlink=C. T. C. Wall | title=Graded Brauer groups | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002180359 | mr=0167498 | zbl=0125.01904 | year=1964 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=1964 | issue=213 | pages=187–199| doi=10.1515/crll.1964.213.187 | s2cid=115679955 }}
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