By Brahmagupta

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The most naive definition of an algebraic deformation is as follows. k/. Let I0 be the maximal ideal corresponding to 0. 11. An algebraic deformation over B ´ kŒ† of an algebra A0 is a Balgebra A which is a free B-module, together with the identification Á0 W A=I0 A ! A0 of the zero-fiber of A with A0 as algebras. Any algebraic deformation defines a formal deformation. 1 A=I0n A). Then Ay is a topologically free kŒŒ„-module which is a deformation of A0 . 12. An algebraic quantization of a Poisson algebra A0 is an algebraic deformation A of A0 such that the completion Ay is a deformation quantization of A0 .

2 Quantum integrable systems 41 classical system is the quasiclassical limit of the quantum system. Also, if we have a quantum mechanical system defined by a Hamiltonian H which is included in a quantum integrable system H D H1 ; : : : ; Hn then we say that H1 ; : : : ; Hn are quantum integrals of H . Remark. It is obvious that if H1 ; : : : ; Hn form a quantum integrable system, then they are algebraically independent. X/Œ„. 2. M D T X, A0 D Cpol a quantum integrable system in A, upon evaluation „ !

E. the case when is 1 a nondegenerate bivector. , which is a closed, nondegenerate 2-form (D symplectic structure) on M . In this case, Kontsevich’s theorem is easier and was proved by De Wilde–Lecomte, and by later Deligne and Fedosov. Moreover, in this case there is the following additional result, also due to Kontsevich, [Ko1], [Ko2]. 9. „///. Remark. „//, so Hochschild cochains are, by definition, linear maps A˝n 0 ! „//. 10. B/ exists. M; C/, and t1 ; : : : ; tr are the coordinates on H corresponding to this basis.