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 Terry Wall (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).[1]

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.
Wall (1964, 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,[2] which has kernel I³, where I is the fundamental ideal of W(F).[3]

Examples

BW(C) is isomorphic to Z/2Z. This is an algebraic aspect of Bott periodicity 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 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

Lam (2005) pp.98–99
Lam (2005) p.113
Lam (2005) p.115

References

Deligne, Pierre (1999), "Notes on spinors", in Deligne, Pierre; Etingof, Pavel; Freed, Daniel S.; Jeffrey, Lisa C.; Kazhdan, David; Morgan, John W.; Morrison, David R.; Witten, Edward (eds.), Quantum fields and strings: a course for mathematicians, Vol. 1, Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997, Providence, R.I.: American Mathematical Society, pp. 99–135, ISBN 978-0-8218-1198-6, MR 1701598
Lam, Tsit-Yuen (2005), Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, 67, American Mathematical Society, ISBN 0-8218-1095-2, MR 2104929, Zbl 1068.11023
Wall, C. T. C. (1964), "Graded Brauer groups", Journal für die reine und angewandte Mathematik, 213: 187–199, ISSN 0075-4102, MR 0167498, Zbl 0125.01904

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[Lam9899-1] Lam (2005) pp.98–99

[Lam113-2] Lam (2005) p.113

[Lam115-3] Lam (2005) p.115