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 W to BW 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
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, 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