## Abstract

This paper introduces the concept of n-valued groups and studies their algebraic and topological properties. We explore a number of examples. An important class consists of those that we call n-coset groups; they arise as orbit spaces of groups G modulo a group of automorphisms with n elements. However, there are many examples that do not arise from this construction. We see that the theory of n-valued groups is distinct from that of groups with a given automorphism group. There are natural concepts of the action of an n-valued group on a space and of a representation in an algebra of operators. We introduce the (purely algebraic) notion of an n-Hopf algebra and show that the ring of functions on an n-valued group and, in the topological case, the cohomology has an n-Hopf algebra structure. The cohomology algebra of the classifying space of a compact Lie group admits the structure of an n-Hopf algebra, where n is the order of the Weyl group; the homology with dual structure is also an n-Hopf algebra. In general the group ring of an n-valued group is not an n-Hopf algebra but it is for an n-coset group constructed from an abelian group. Using the properties of n-Hopf algebras we show that certain spaces do not admit the structure of an n-valued group and that certain commutative n-valued groups do not arise by applying the n-coset construction to any commutative group.

Original language | English |
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Pages (from-to) | 325-349 |

Number of pages | 25 |

Journal | Transformation Groups |

Volume | 2 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1997 |

Externally published | Yes |