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 classifying finite-dimensional graded central division algebras over the field. It was first defined by 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.
Properties
- The Brauer group B injects into BW by mapping a CSA A to the graded algebra which is A in grade zero.
- showed that there is an exact sequence
- 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.
Examples
- BW 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 where γ is an odd element of square 1 commuting with C.
- BW 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, C, H, H, H, C, R where δ and ε are odd elements of square −1 and 1, such that conjugation by them on complex numbers is complex conjugation.