Normed Bilinear Pairings for Semi-Euclidean Spaces near the Hurwitz-Radon Range

Abstract

A codimension c(≥ 0) orthogonal multiplication of type (k,l;m;m+ c) for semi-Euclidean spaces is a normed bilinear map F: R^k,l × R^m → R^m+C, k + l ≤ m, k > 0, where the signatures of the semi-Euclidean spaces R^m and R^m+C are suppressed. For the Hurwitz-Radon range, i.e. c = 0, F gives rise to (and is determined by) a module over the Clifford algebra C _l,k- 1 whose generators possess certain invariance properties with respect to the semi-Euclidean structure on the module. For c = 1, we prove an Adem-type restriction-extension theorem to the effect that F (up to isometries on the source and the range) restricts to an orthogonal multiplication of type (k, l; m; m) if m is even, and extends to an orthogonal multiplication of type (k, l; m + 1; m + 1) if m is odd. The resulting types are in the Hurwitz-Radon range, thereby classified. The main results of the paper give a full description of codimension two full orthogonal multiplications F of type (k, l; m; m + 2). We show that, for m even, F extends to an orthogonal multiplication of type (k, l; m + 2; m + 2). For m odd, we have k + l = 3 and F restricts (again up to isometries on the source and the range) to an orthogonal multiplication of type (k, l; m — 1; m — 1) which is a direct summand of F. AMS Subject Classification: Primary 15A63, Secondary 11E25.