R-commutative geometry and quantization of Poisson algebras

Abstract

An r-commutative algebra is an algebra A equipped with a Yang-Baxter operator R: A ⊗ A → A ⊗ A satisfying m = mR, where m: A ⊗ A → A is the multiplication map, together with the compatibility conditions R( a ⊗ 1) = 1 ⊗ a, R(1 ⊗ a) = a ⊗ 1, R(id ⊗ m) = ( m ⊗ id) R 2 R 1, and R( m ⊗ id) = (id ⊗ m) R 1 R 2. The basic notions of differential geometry extend from commutative (or supercommutative) algebras to r-commutative algebras. Examples of r-commutative algebras obtained by quantization of Poisson algebras include the Weyl algebra, noncommutative tori, quantum groups, and certain quantum vector spaces. In many of these cases the r-commutative de Rham cohomology is stable under quantization.