Nonassociative constructions from inverse property quasigroups
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The notion of a Hopf algebra has been generalized many times by weakening certain
properties; we introduce Hopf quasigroups which weaken the associativity of the
algebra. Hopf quasigroups are coalgebras with a nonassociative product satisfying
certain conditions with the antipode re
ecting the properties of classical inverse
property quasigroups.
The de nitions and properties of Hopf quasigroups are dualized to obtain a theory
of Hopf coquasigroups, or `algebraic quasigroups'. In this setting we are able to
study the coordinate algebra over a quasigroup, and in particular the 7-sphere.
One particular class of Hopf quasigroups is obtained by taking a bicrossproduct
of a subgroup and a set of coset representatives, in much the same way that Hopf
algebras are obtained from matched pairs of groups. Through this construction
the bicrossproduct can also be given the structure of a quasi-Hopf algebra.
We adapt the theory of Hopf algebras to Hopf (co)quasigroups, de ning integrals
and Fourier transformations on these objects. This leads to the expected properties
of separable and Frobenius Hopf coquasigroups and notions of (co)semisimplicity.
Authors
Klim, JenniferCollections
- Theses [4116]