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).

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, which has kernel I³, where I is the fundamental ideal of W(F).

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.

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, vol. 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, 1964 (213): 187–199, doi:10.1515/crll.1964.213.187, ISSN 0075-4102, MR 0167498, S2CID 115679955, Zbl 0125.01904