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\(R\)-commutative geometry and quantization of Poisson algebras. (English) Zbl 0785.46057

The author extends the basic notions of differential geometry from commutative (or super-commutative) algebras to \(r\)-commutative algebras. By the definition the last one is an algebra \(A\) equipped with a Yang- Baxter operator \(R: A\otimes A\to A\otimes A\) satisfying \(m=mR\), where \(m: A\otimes A\to A\) is the multiplication map, together with the compatibility conditions \(R(a\otimes 1)=1\otimes a\), \(R(1\otimes a)=a\otimes 1\), \(R(\text{id}\otimes m)= (m\otimes\text{id}) R_ 2 R_ 1\), \(R(m\otimes\text{id})= (\text{id}\otimes m)R_ 1 R_ 2\).
The \(r\)-symmetric and \(r\)-Weyl algebras are described. The relation between \(r\)-commutative geometry and the quantization of Poisson algebras, as well as noncommutative tori, quantum groups, quantum matrix algebras and quantum vector spaces are considered.

MSC:

46L87 Noncommutative differential geometry
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