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Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. (English) Zbl 0694.16008
The “Physics for algebraists” is in the context of quantum mechanics combined with gravity, describing Yang-Baxter equations, position observables, momentum space, momentum and position quantization, etc. This physics leads to the search for self-dual algebraic structures, and finally to a bicrossproduct construction of non-commutative and non- cocommutative Hopf algebras. There are numerous examples, including a modification of the Weyl algebra. In the appropriate context, the compatibility conditions on the structure maps reduce to the classical Yang-Baxter equations. An example is given related to the double Hopf algebra D(H) of Drinfeld.
Reviewer: E.J.Taft

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81S05 Commutation relations and statistics as related to quantum mechanics (general)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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