In mathematics, the BrauerâÂÂWall 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 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 BrauerâÂÂWall 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.
- showed that there is an exact sequence
: 0 â B(F) â BW(F) â Q(F) â 0
where Q(F) is the group of graded quadratic extensions of F, defined as an extension of Z/2 by F<sup>*</sup>/F<sup>*2</sup> with multiplication (e,x)(f,y) = (e + f, (âÂÂ1)<sup>ef</sup>xy). 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.
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[ó] where ó 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[õ], C[õ], H[ô], H, H[õ], C[ô], R[ô] where ô and õ are odd elements of square âÂÂ1 and 1, such that conjugation by them on complex numbers is complex conjugation.
Notes
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