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Dirac’s algebra and Brauer-Wall groups. (English) Zbl 0895.11018

Nencka, Hanna (ed.) et al., Geometry and nature: in memory of W. K. Clifford. A conference on new trends in geometrical and topological methods, July 30–August 5, 1995, Madeira, Portugal. Providence, RI: American Mathematical Society. Contemp. Math. 203, 25-36 (1997).
Dirac’s construction of the Clifford algebra \(\mathbb{C}\otimes L_{\mathbb{R}} (-3,1)\) is revisited by introducing the “Dirac”-ring-morphism \(A\mapsto D(A) \subset A\oplus A\). This morphism is used to construct Clifford algebras and to study the related Brauer-Wall groups. After discussing mod 2 graded algebras, it is shown that the Dirac construction induces an operator on the (graded) Brauer-Wall group, by \(D(D(A))\cong A\widehat{\otimes} M_{1,1}(k)\). Furthermore, this result is then connected to quadratic algebras with standard involution which are used later on to construct the celebrated mod 8 periodicity of \(BW(\mathbb{R})\). Moreover, a structure theorem for \(BW(k)\) is given. This should be compared with the results of A. J. Hahn [Quadratic algebras. Clifford algebras and arithmetic Witt groups, Springer Verlag (New York 1994; Zbl 0791.11021)].
The proofs rely on the notion of a (graded) anti-center. Suitable properties of it are established, and the connection to the graded Arf-invariant, which can be seen to be a graded quadratic form, is emphasized. The paper closes with an example, a quaternion model for the hyperbolic space \(H^5\). The projective character of the involved spin groups is clearly worked out. This provides the main motivation for the framework given.
For the entire collection see [Zbl 0864.00055].

MSC:

11E88 Quadratic spaces; Clifford algebras
15A66 Clifford algebras, spinors
11E81 Algebraic theory of quadratic forms; Witt groups and rings

Citations:

Zbl 0791.11021
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