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Semiquantum geometry. (English) Zbl 0896.17004

The theme of the present paper is noncommutative geometry close to the usual one. A first variation is the notion of a Poisson filtered algebra; this is an associative algebra \(A\) provided with a central Poisson subalgebra \(Z\) and with a biderivation \(\{ , \}: Z\times A \to A\) extending the Poisson bracket on \(Z\). There is also a corresponding notion of Poisson bimodule over a Poisson filtered algebra; both concepts are interpreted geometrically and in the context of formal deformation theory. Several examples are discussed: quantum groups at roots of 1, Manin’s quantum abelian varieties, etc. A second variation is the notion of a noncommutative space of finite type; this is a scheme provided with a coherent sheaf of algebras. Among several examples, quantum \(SL(2)\) at a root of 1 is discussed in detail. In this case, the authors are able to derive from a calculation of usual cohomology of sheaves the quantum Bott-Borel-Weil theorem.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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References:

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