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Hurwitz pairs equipped with complex structures. (English) Zbl 0591.15019
Deformations, Proc. Semin., Łódź-Warsaw/Pol. 1982-84, Lect. Notes Math. 1165, 184-195 (1985).
[For the entire collection see Zbl 0568.00006.]
The authors are concerned with a reformulation of a classic problem of A. Hurwitz (1923), that of finding all sets of real constants \(c^ k_{j\alpha}\), \(j,k=1,...,n\); \(\alpha =1,...,p\), \(p\leq n\), such that the bilinear forms \(\eta_ j=x_{\alpha}c^ k_{j\alpha}y_ k\) (summation convention) satisfy the condition \(\Sigma_ j\eta^ 2_ j=\Sigma_{\alpha}x^ 2_{\alpha}\Sigma_ ky^ 2_ k\). The reformulation introduces the notion of a ”Hurwitz pair” (V,S), a pair of real spaces of bidimension (n,p) with a notion of scalar multiplication of S on V. This allows the use of the language of Clifford algebras. The classification of such pairs is given and their symplectic geometry is investigated.
Reviewer: M.E.Keating

MSC:
15A66 Clifford algebras, spinors
15A63 Quadratic and bilinear forms, inner products
15A90 Applications of matrix theory to physics (MSC2000)
Citations:
Zbl 0568.00006