## 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.

### MSC:

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

### Keywords:

$$n$$-inner product space; simple; simplicity

Zbl 0673.46012