The initial part of this paper presents "Physics for Algebraists" in the context of quantum mechanics combined with gravity. Such physical notions as the Yang-Baxter Equations, position observables, momentum space, momentum and position quantization, etc., are described. Many readers may wish to just read this initial part of the paper. The physics leads to the search for self-dual algebraic structures and finally to non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. The entire paper contains numerous examples. The non-commutative and non-cocommutative Hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted H1 {bowtie} H2. Among the examples is one obtained by modifying the Weyl algebra. We also give the context in which the compatibility requirements on the structure maps reduce to the Classical Yang-Baxter Equations, and an example related to Drinfel'd's double Hopf algebra D(H). © 1990.
