Hermitian forms and systems of quadratic forms
The author generalizes the method of Jacobson by which Hermitian forms \(h\) over a quaternion algebra \(D/F\) with canonical involution are reduced to quadratic forms \(q\) over \(F\): Let \((D,\sigma)\) be a finite-dimensional division algebra with involution \(\sigma\) of the first kind over the field \(F\), \(\lambda=\pm 1\), \[ S= \text{Symd}_\lambda (D,\sigma) = \{x+\lambda\sigma(x)\mid x\in D\},\;n= \dim_F(S). \] Finally let \((V,h)\) be a \(\lambda\)-Hermitian space over \((D,\sigma)\). Then \(h\) induces a quadratic map \(q=(q_1\cdots q_n): V\to F^n\) (depending on a base \(B\) of \(S/F\) which plays no role). The main results are the following: (1) \(h\) is isotropic iff \(q_h\) is isotropic; (2) If \(h\) is metabolic then \(q_h\) is metabolic. The converse is also true if \(h\) is regular and either char\(\,F\neq 2\) or \(D\neq F\). (3) Let \((V,h)\) and \((V',h')\) be two even \(\lambda\)-Hermitian spaces over \((D,\sigma)\). If \(q_h\approx q_{h'}\) then \((V,h)\approx(V',h')\) except in the case where char \(F\neq 2\), \(D=F\) and \(\lambda= -1\). (4) The result on anisotropic Hermitian forms over a quaternion division algebra in the case char \(F=2\), which is described in the summary.
scientific article; zbMATH DE number 6915987