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Rechtsdistributive Multiplikationen auf homogenen Räumen. (Right distributive multiplications on homogeneous spaces). (German) Zbl 0603.22001

The paper is concerned with locally compact topological right distributive quasigroups, and specifically with those on which a Lie group acts continuously as a transitive group of automorphisms containing all right translations. If D is such a topological right distributive quasigroup and \(\Sigma\) the closure of the group generated by the right translations, then the quasigroup structure of D can be retrieved within the Lie group \(\Sigma\) in the following way: D corresponds to a conjugacy class \({\mathfrak C}\) in \(\Sigma\) (namely of a right translation), and the quasigroup multiplication and the action of \(\Sigma\) correspond to conjugation. \({\mathfrak C}\) generates a dense subgroup of \(\Sigma\) and constitutes a global cross section of the factor projection of \(\Sigma\) onto the homogeneous space \(\Sigma /\Sigma_{\alpha}\) where \(\Sigma_{\alpha}\) denotes the centralizer of \(\alpha\in {\mathfrak C}\). Conversely under these circumstances this construction always leads to a quasigroup. It is one of the aims of this paper to explore systematically the possibilities which this construction principle offers under various additional restrictions. A complete classification is obtained of the following situations: if the connected component \(\Delta\) of \(\Sigma\) is compact, if it is quasisimple, if \(D\cong \Sigma /\Sigma_{\alpha}\) is a two-dimensional manifold, and if D is minimal (i.e. the subgroup generated by any two right translations \(\hat=\) elements of \({\mathfrak C}\) is dense in \(\Sigma)\). Note that all these results operate under assumptions which directly or indirectly assure that one is working within a Lie group.
[There is a partial overlap with generalized symmetric spaces, with the right translations acting as symmetries, namely whenever the derivative at the unique fixed point of a right translation has no nonzero fixed vector in the tangent space. On the other hand, for a generalized symmetric space, the condition that \({\mathfrak C}\) is a global cross section is rather special. In the present paper, the theory of generalized symmetric spaces and the pertaining classification results are not used; it would be interesting to know if they could be of any help.]
§§ 1-5 of the paper are concerned with general information about topological right distributive quasigroups and with preparations for the study of quasigroups which can be found within Lie groups.
In § 6 such right distributive quasigroups are constructed in the semidirect product \(\Sigma\) of a nonabelian connected Lie group \(\Delta\) with a cyclic group whose generator \(\alpha\) acts on \(\Delta\setminus \{1\}\) without fixed element (\(\Delta\) is then necessarily solvable). The immense wealth of right distributive quasigroups which can be constructed in this way demonstrates that in order to obtain complete classifications it is necessary to impose drastic restrictions. The various classification results are the subject of §§ 7-10:
§ 7 classifies all right distributive quasigroups which can be constructed from a Lie group \(\Sigma\) whose connected component \(\Delta\) is compact. It is shown that these quasigroups are obtained by the construction principle of § 6 just described, with \(\Delta\) being a torus. They coincide with the right distributive quasigroups which are themselves homeomorphic to a torus. A complete classification up to isomorphism is obtained.
§ 8 is concerned with the case that the connected component \(\Delta\) of \(\Sigma\) is quasisimple. \(\Delta\) is then even center free, the center Z of a maximal compact subgroup K is a one-dimensional torus, and \(\alpha\in Z\setminus \{1\}\). Using the classification of simple Lie groups, all these situations can be enumerated; comparison then shows that they correspond to the non-compact irreducible Hermitian symmetric spaces.
§ 9 classifies the minimal right distributive quasigroups obtained from Lie groups. For these, D is homeomorphic to \({\mathbb{R}}\) or \({\mathbb{R}}^ 2\). If \(D\cong {\mathbb{R}}\), then \(\Sigma\) is the connected component of \(L_ 2\). For \(D\cong {\mathbb{R}}^ 2\) there are the following possibilities: \(\Sigma =\Delta =PSL_ 2({\mathbb{R}})\) and \(\alpha\) a rotation, \(\Sigma = the\) Euclidean motion group and \(\alpha\) a rotation (of order \(\neq\) 2, 3, 4, 6 in view of minimality), \(\Sigma = the\) semidirect product of \({\mathbb{R}}^ 2\) with a spiral one-parameter subgroup S of \(SO_ 2({\mathbb{R}})\times dilatation\) group with \(\alpha\in S\), \(\alpha\) not a dilatation. Among these, the case \(\Sigma =PSL_ 2({\mathbb{R}})\) can be singled out by the property that the resulting quasigroups do not satisfy two-sided distributivity.
In § 10 all topological right distributive quasigroups whose point set is a two-dimensional manifold are obtained. From previous work of the author if follows that the only possible surfaces are the torus and \({\mathbb{R}}^ 2\). The torus case is taken care of by § 7. For \({\mathbb{R}}^ 2\), apart from direct products of right distributive quasigroups on \({\mathbb{R}}\) (which have been determind previously by the author), or the minimal right distributive quasigroups (described in § 9), there are the following additional possibilities: \(\Sigma\) is the semidirect product of \({\mathbb{R}}^ 2\) by a rotation of order 2, 3, 4, 6 or \(\Delta\) is the semidirect product of \({\mathbb{R}}^ 2\) with the group of linear transformations generated by \((x,y)\to (tx,t^{-1}y)\) for all \(t>0\) and the involution \(\alpha\) : (x,y)\(\to (y,x).\)
Besides the classification of simple Lie groups and the classification of Lie groups acting transitively on spheres, the proofs use information about quotients of compact Lie groups with global section [H. Scheerer, Math. Ann. 206, 149-155 (1973; Zbl 0247.22010)] and about the (algebraic) topology of manifolds admitting an idempotent multiplication [the author, Math. Z. 145, 43-62 (1975; Zbl 0322.57008); the author and H. Scheerer, ibid. 182, 95-119 (1983; Zbl 0505.55011)]. Previous papers of the author about quasigroups on 1-dimensional manifolds and about differentiable quasigroups also play an important role [ibid. 145, 63-68 (1975; Zbl 0322.57009); Arch. Math. 35, 121-126 (1980; Zbl 0435.22007)].
In many instances, the information that the quasigroup D can be retrieved within a Lie group is not just imposed as an assumption (as was done for the purpose of this review) but deduced from more general assumptions. The reader should be warned that in these cases it is often an intricate question which topology on the homeomorphism group of D should be used in order to make the closure of the set of right translations of D a Lie group (whose Lie topology might be still another, finer topology). Standard topologies which are used for this purpose are the compact-open topology, the g-topology of Arens and the modified compact-open topology of Gleason-Palais, depending on the situation. Also, in the heat of battle, the author does not always tell very clearly which topology he is using for a specific argument.
Reviewer: H.Hähl

MSC:

22A30 Other topological algebraic systems and their representations
20N05 Loops, quasigroups
57S20 Noncompact Lie groups of transformations
22E25 Nilpotent and solvable Lie groups
22E46 Semisimple Lie groups and their representations
57S15 Compact Lie groups of differentiable transformations
53C35 Differential geometry of symmetric spaces
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