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On a Jordan subalgebra of commutative algebras. (English) Zbl 0821.17027
Let $$V$$ be a finite dimensional vector space over an infinite field $$K$$ of characteristic not 2 or 3. A commutative product on $$V$$ is given by an homogeneous quadratic polynomial map $$p$$ from $$V$$ to $$V$$. In section 1 the authors show that if $$A = (V,p)$$ is commutative unital and for any $$a \in V$$ the quadratic polynomial map $$q_ a$$ is given by $$q_ a(x) = 2x (xa) - x^ 2 a$$ then the set $${\mathcal J} (A) = \{x \in V : [p,q_ a] = 0$$ for all $$a \in V\}$$ is a Jordan subalgebra of $$A$$. Here $$[p \cdot q_ a] (x) = (Dq_ a(x)) p(x) - (Dp(x)) q_ a(x)$$. A subalgebra as this was previously considered for arbitrary scalar rings without 2- and 3-torsion by M. Koecher [J. Algebra 62, 479-493 (1980; Zbl 0423.17001)] who proved its Jordan character by different methods. The authors prove that some identities satisfied by each Jordan algebra still hold in $$A$$ provided that some of the entries belong to $${\mathcal J} (A)$$. For instance, $$a^ 2 (ax) = a(a^ 2 x)$$ and $$[L(a^ k), L(a^ m)] (x) = 0$$ for all $$x \in V$$ and $$a \in {\mathcal J}(A)$$.
On the other hand, it was proved in [T. A. Springer, Jordan algebras and algebraic groups (Springer 1973; Zbl 0259.17003)] that inversion maps of finite dimensional unital Jordan algebras can be characterized by the axioms (J.1), (J.2) and (J.3) of Springer’s book. It was shown by R. Niemczyk and S. Walcher [Commun. Algebra 19, 2169-2194 (1991; Zbl 0786.17001)] that if $$K$$ is of characteristic zero then each birational map of the finite dimensional vector space $$V$$ satisfying (J.1) and (J.2) arises from the general solution of the formal differential equation $$\dot {y} = y^ 2$$ in a unital $$R$$-algebra and these rational maps are in 1-1-correspondence with the unital $$R$$- algebras. The main result of section 2 asserts a weak form of (J.3) for these algebras and also for other real and complex commutative algebras given by a slight modification of (J.1). In the last case the proof uses tools of the theory of differential equations. This and the Lefschetz principle allow the proof in the first one.

##### MSC:
 17C50 Jordan structures associated with other structures 17C55 Finite-dimensional structures of Jordan algebras
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