Gyrosemidirect product structure of bounded symmetric domains
Let \((P,+)\) be a loop, for \(a,b\in P\) let \(a^+: P \to P\); \(x \to a+x\), \(\delta_{a,b}=((a+b)^+)^{-1} \circ a^+\circ b^+\) and \(\Delta :=\langle \{\delta_{a,b} \mid a,b \in P\}\rangle\) the subgroup of \(\text{Sym }P\) generated by the permutations \(\delta_{a,b}\). If the loop \((P,+)\) has the property \(\Delta \leq \text{Aut}(P,+)\), then for any group \(K\) with \(\Delta \leq K \leq \text{Aut}(P,+)\) the set of permutations \(P^+\rtimes_QK:=\{a^+\circ \alpha \mid a \in P,\;\alpha \in K\}\) forms a subgroup of \(\text{Sym }P\) (with \((a^+\circ \alpha) \circ (b^+\circ \beta)=(a+\alpha(b))^+\circ (\delta_{a,\alpha(b)} \circ \alpha \circ \beta)\)), the quasi-direct product (here called gyrosemidirect product). In this paper there is considered a bounded symmetric domain \(D\) in a Banach space and \(\text{Aut}(D)\), the group of all holomorphic automorphisms of \(D\). With the exponential map, there is defined for any \(a \in D\) an application \(\Phi_a \in \text{Aut}(D)\) and it is proved (Theorem 3.1) that \((D,\oplus)\) with \(a\oplus b:=\Phi_a(b)\) is a gyrogroup (=\(K\)-loop=Bruck-loop, [\textit{Y. You} and the second author, Result. Math. 28, No. 3-4, 359-371 (1995; Zbl 0840.20071)] i.e. besides \(\Delta \leq \text{Aut}(D,\oplus)\), \(\ominus(a \oplus b)=\ominus a \oplus (\ominus b)\) and \(\delta_{a,b}=\delta_{a,b\oplus a}\)). Furthermore, for \(K :=\{\varphi \in \text{Aut}(D)\mid \varphi(0)=0\}\) it is shown that \(\Delta \leq K \leq \text{Aut}(D,\oplus)\) and \(\text{Aut}(D)=D^{\oplus}\rtimes_QK\) (Theorem 3.2).
scientific article; zbMATH DE number 687808
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