Simple \(n\)-inner product spaces. (English) Zbl 0754.15027

Let \(n\) be a natural number, \(L\) be a linear space with \(\dim(L)\geq n\) and \((.,.\mid .,\ldots,.)\) be a real function on \(L^{n+1}\) with the properties:
1. \((a,a\mid a_ 2,\ldots,a_ n)\geq 0\); \((a,a\mid a_ 2,\ldots,a_ n)=0\) iff \(a,a_ 2,\ldots,a_ n\) are linearly dependent; 2. \((a,b\mid a_ 2,\ldots,a_ n)=(a,b\mid a_{i_ 2},\ldots,a_{i_ n})\) for every permutation \((i_ 2,\ldots,i_ n)\) of \((2,\ldots,n)\); 3. \((a,b\mid a_ 2,\ldots,a_ n)=(b,a\mid a_ 2,\ldots,a_ n)\); 4. If \(n>1\), \((a,a\mid a_ 2,a_ 3,\ldots,a_ n)=(a_ 2,a_ 2\mid a,a_ 3,\ldots,a_ n)\); 5. \((.,.\mid .,\ldots,.)\) is linear in the first variable.
Then \((L;(.,.\mid .,\ldots,.))\) will be called \(n\)-inner product space [cf. the author, Math. Nachr. 140, 299-319 (1989; Zbl 0673.46012)]. An \(n\)-inner product space \((n>1)\) is called simple if there exists an inner product \((.,.)\) on \(L\) such that: \[ (a,b\mid a_ 2,\ldots,a_ n)=\left|\begin{matrix} (a,b) & (a,a_ 2) & \ldots & (a,a_ n) \\ (a_ 2,b) & (a_ 2,a_ 2) & \ldots & (a_ 2,a_ n) \\ \vdots & \vdots & \ddots & \vdots \\ (a_ n,b) & (a_ n,a_ 2) & \ldots & (a_ n,a_ n) \end{matrix} \right| \] holds. Among others, the author gives necessary and sufficient conditions for the simplicity of an \(n\)-inner product space.


15A63 Quadratic and bilinear forms, inner products
46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)


Zbl 0673.46012