Last updates: 22 January 2011
1.1 Let be a ring. Let be an ideal in . We view the powers of the ideal as a basis of neighbourhoods in containing . There is a unique topology on such that the ring operations are continuous with a basis given by the sets . This is the -adic topology. If then this topology is Hausdorff.
1.2 Let be an -module. We can transfer the -adic topology on to a topology on . We view the sets as a basis of neighbourhoods in containing . An element is an element of if . As above, there is a unique topology on such that the module operations are continuous with basis given by the sets , where . This is the -adic topology on .
1.3 Define a map by where is a real number and is the largest integer such that . If the -adic toplogy on is Hausdorff the is a metric on which generates the -adic topology.
1.4 If is a local ring then it is natural to take where is the unique maximal ideal in . If is a field and is an indeterminate then the ring of formal power series in , , is a local ring with unique maximal ideal generated by . In this case the -adic topology on a -module is called the -adic topology on .
1.5 Let be a ring and be an ideal of . Let be an -module. A sequence of elements in is a Cauchy sequence in the -adic topology if for every positive integer there exists a positive integer such that A sequence of elements in converges to if for every positive integer there exists a positive integer such that The module is complete in the -adic topology if every Cauchy sequenc in converges. A ring is complete in the -adic topology if when viewed as an -module it is completd in the -adic topology. If the -adic topology is Hausdorff then this definition of completeness is the same as the ordinary defingion of completenes when we view that is a metric space as in (1.1).
1.6 Two Cauchy sequences and in are equivalent if converges to in the -adic topology, i.e., The set of all equivalece classes of Caucy sequences in is the completion of
1.7 The completion is an module with operations given by where and are Cauchy sequences with elements in and is a Cauchy sequence of elements in .
1.8 Define a map by i.e., is the equivalence class of the sequence such that for all . This map has kernel The map is injective if is Haussdorff in the -adic topology.
1.9 Define a basis of neigbourhoods of in the completion by: The collection of sets where is a basis for a topology on . The module operations and the map are continuous.
1.10 Let be a field. Then is a local ring with maximal ideal generated by the element . In this case the -adic topology is called the -adic toplogy. Let be a -module. Then a sequence of elements in is a Cauchy sequence if for every positive integer there exists a postive integer such that i.e., is "divisible" by for all . A sequence of elements in converges to if for every positive integer there exists a positive integer such that The module is complete in the -adic topology if every Cauchy sequence in converges.
1.11 As in (1.2) we can define the completion of a -module in the -adic topology. IF is an algebra over a field then is a -module in the -adic topology and the completion of is , the ring of formal power series in with coefficients in . The ring is, in general, larger than .
1.12 If is a complete -module in the -adic toplogy then for each element the element is a well defined element of .
1.13 A -module is topologically free if is a free -module for all positive integers .
2.1 A deformation of a commutative associative algebra over is an associative (not necessarily commutative) algebra over such that
2.2 Given a deformation of a commutative algebra we can define a new operation on by defining where This makes into a Poisson algebra. If was a Poisson algebra to start with then we would like this new Poisson structure to be the same as the old one.
2.3 Dualize the above definitions to define a deformation of a cocommutative Poisson algebra. Then extend the picture to co-Poisson-Hopf algebras. This is the motivation for the following definition.
Let be a Lie bialgebra and let be the corresponding Poisson cobracket. A quantization of is a topological Hopf algebra over which is a topologically free module and satisfies the following conditions:
2.4 The definition of deformations given in (2.1) is too general for some purposes. Assume that is an algebra over a field with multiplication Let be an indeterminate and let be the ring of formal power series in with coefficients in . This is a complete module. A deformation of is an associative bilinear multiplication map which can be written in the form where the are linear maps which are extended to the completion
If is a bialgebra over with multiplication and comultiplication then a deformation of is a linear multiplication and a linear comultiplication such that and are linear maps which are extended to the completion and such that is a bialgebra under and
2.5 Suppose that are both deformations of an algebra over with multiplication . The two deformations and are equivalent if there is a linear map of the form such that the are linear maps extended to the completion such that for all
Drinfel'd has completely formalized the quantization process in the following paper in which he also introduced the object which is now called the Drinfel'd-Jimbo quantum group.[D1] V.G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985) 254-258.
Concerning open problems in the theory of quantization:[D3] V.G. Drinfel'd, On some unsolved problems in quantum group theory, in "Quantum Groups" Proceedings of the Euler International Mathematical Institute, Leningrad, Springer Lect. Notes No. 1510, P. Kulish Ed., (1991) 1-8.
The following books have discussions of the -adic toplogy and completions. The definitions of completion for a metric space are found in Rudin's elementary analysis book Chapt. 3 Exercise 23-24.
[AM] M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley 1969. [ZS] O. Zariski and P. Samuel, Commutative algebra, Vol. II Van-Nostrand 1960.
Deformation theory was developed by Gerstenhaber in a series of papers in Annals of Mathematics 1964-74. More recently this theory has been developing in the context of quantum groups, in particular see [GGS] and the references there.[GGS] M. Gerstenhaber, A. Giaquinto, and S. Schack, Quantum symmetry, Proceedings of Workshops in the Euler Int. Math. Inst., Leningrad 1990, Springer Lecture Notes No. 1510, P. Kulish Ed., (1991) 9-46. [Sn] S. Shnider, Deformation cohomology for bialgebras and quasi-bialgebras, Contemporary Mathematics 134 Amer. Math. Soc. (1992) 259-296.