## The $h$-adic topology

1.1 Let $A$ be a ring. Let $𝔞$ be an ideal in $A$. We view the powers ${𝔞}^{k}$ of the ideal $𝔞$ as a basis of neighbourhoods in $A$ containing $0$. There is a unique topology on $A$ such that the ring operations are continuous with a basis given by the sets $a+{𝔞}^{k}$. This is the $𝔞$-adic topology. If ${\cap }_{k}{𝔞}^{k}=\left(0\right)$ then this topology is Hausdorff.

1.2 Let $M$ be an $A$-module. We can transfer the $𝔞$-adic topology on $A$ to a topology on $M$. We view the sets ${N}_{k}={𝔞}^{k}M$ as a basis of neighbourhoods in $M$ containing $0$. An element $m\in M$ is an element of ${N}_{k}$ if . As above, there is a unique topology on $M$ such that the module operations are continuous with basis given by the sets $m+{𝔞}^{k}M$, where $m\in M$. This is the ${𝔞}^{k}$-adic topology on $M$.

1.3 Define a map $d:M × M\to ℝ$ by where $e$ is a real number $e>1$ and $v\left(x\right)$ is the largest integer $k$ such that $x\in {𝔞}^{k}M$. If the $𝔞$-adic toplogy on $M$ is Hausdorff the $d$ is a metric on $M$ which generates the $𝔞$-adic topology.

1.4 If $A$ is a local ring then it is natural to take $I=𝔪$ where $𝔪$ is the unique maximal ideal in $A$. If $k$ is a field and $h$ is an indeterminate then the ring of formal power series in $h$, $k\left[\left[h\right]\right]$, is a local ring with unique maximal ideal $𝔪=\left(h\right)$ generated by $h$. In this case the $𝔪$-adic topology on a $k\left[\left[h\right]\right]$-module $M$ is called the $h$-adic topology on $M$.

1.5 Let $A$ be a ring and $𝔞$ be an ideal of $A$. Let $M$ be an $A$-module. A sequence of elements $\left\{{b}_{n}\right\}$ in $M$ is a Cauchy sequence in the $𝔞$-adic topology if for every positive integer $k>0$ there exists a positive integer $N$ such that A sequence $\left\{{b}_{n}\right\}$ of elements in $M$ converges to $b\in M$ if for every positive integer $k>0$ there exists a positive integer $N$ such that The module $M$ is complete in the $𝔞$-adic topology if every Cauchy sequenc in $M$ converges. A ring $A$ is complete in the $𝔞$-adic topology if when viewed as an $A$-module it is completd in the $𝔞$-adic topology. If the $𝔞$-adic topology is Hausdorff then this definition of completeness is the same as the ordinary defingion of completenes when we view that $M$ is a metric space as in (1.1).

1.6 Two Cauchy sequences $P=\left\{{p}_{n}\right\}$ and $Q=\left\{{q}_{n}\right\}$ in $M$ are equivalent if $\left\{{p}_{n}-{q}_{n}\right\}$ converges to $0$ in the $𝔞$-adic topology, i.e., The set of all equivalece classes of Caucy sequences in $M$ is the completion $\stackrel{ˆ}{M}$ of $M$

1.7 The completion $\stackrel{ˆ}{M}$ is an $\stackrel{ˆ}{A}$ module with operations given by $P+Q = pn+qn ,and, an P = anpn ,$ where $P=\left\{{p}_{n}\right\}$ and $Q=\left\{{q}_{n}\right\}$ are Cauchy sequences with elements in $M$ and $\left\{{a}_{n}\right\}$ is a Cauchy sequence of elements in $A$.

1.8 Define a map $\phi :M\to \stackrel{ˆ}{M}$ by $φ(b)= [ ( b,b,b,… ) ],$ i.e., $\phi \left(b\right)$ is the equivalence class of the sequence $\left\{{b}_{n}\right\}$ such that ${b}_{n}=b$ for all $n$. This map has kernel ${\cap }_{k}{𝔞}^{k}M.$ The map $\phi$ is injective if $M$ is Haussdorff in the $𝔞$-adic topology.

1.9 Define a basis ${N}_{k}$ of neigbourhoods of $0$ in the completion $\stackrel{ˆ}{M}$ by: The collection of sets $P+{N}_{k}$ where $P\in \stackrel{ˆ}{M}$ is a basis for a topology on $\stackrel{ˆ}{M}$. The module operations and the map $\phi$ are continuous.

1.10 Let $k$ be a field. Then $k\left[\left[h\right]\right]$ is a local ring with maximal ideal $𝔪=\left(h\right)$ generated by the element $h$. In this case the $𝔪$-adic topology is called the $h$-adic toplogy. Let $M$ be a $k\left[\left[h\right]\right]$-module. Then a sequence of elements $\left\{{b}_{n}\right\}$ in $M$ is a Cauchy sequence if for every positive integer $k>0$ there exists a postive integer $N$ such that i.e., ${b}_{n}-{b}_{m}$ is "divisible" by ${h}^{k}$ for all $m,n>N$. A sequence $\left\{{b}_{n}\right\}$ of elements in $k\left[\left[h\right]\right]$ converges to $b\in M$ if for every positive integer $k>0$ there exists a positive integer $N$ such that The module $M$ is complete in the $h$-adic topology if every Cauchy sequence in $M$ converges.

1.11 As in (1.2) we can define the completion of a $k\left[\left[h\right]\right]$-module $M$ in the $h$-adic topology. IF $A$ is an algebra over a field $k$ then $A{\otimes }_{k}k\left[\left[h\right]\right]$ is a $k\left[\left[h\right]\right]$-module in the $h$-adic topology and the completion of $A{\otimes }_{k}k\left[\left[h\right]\right]$ is $A\left[\left[h\right]\right]$, the ring of formal power series in $h$ with coefficients in $A$. The ring $A\left[\left[h\right]\right]$ is, in general, larger than $A{\otimes }_{k}k\left[\left[h\right]\right]$.

1.12 If $M$ is a complete $k\left[\left[h\right]\right]$-module in the $h$-adic toplogy then for each element $x={\sum }_{j\ge 0}{x}_{j}{h}^{j}\in M$ the element $ehx= ∑ k≥0 (hx)k k! = 1+x0h+ (x0h+2x1) ( h2 2 )+ ( x03 +3 ( x0x1+x1x0 ) +6x2 ) ( h3 3! )+⋯$ is a well defined element of $M$.

1.13 A $k\left[\left[h\right]\right]$-module $M$ is topologically free if $M/{h}^{k}M$ is a free $k\left[\left[h\right]\right]/\left({h}^{k}\right)$-module for all positive integers $k>0$.

## Deformations and quantizations

2.1 A deformation of a commutative associative algebra ${A}_{0}$ over $k$ is an associative (not necessarily commutative) algebra $A$ over $k\left[\left[h\right]\right]$ such that

1. $A/hA={A}_{0},$ and
2. $A$ is a topologically free $k\left[\left[h\right]\right]-$module.

2.2 Given a deformation $A$ of a commutative algebra ${A}_{0}$ we can define a new operation $\left\{\phantom{\rule{.5em}{0ex}},\phantom{\rule{.5em}{0ex}}\right\}$ on ${A}_{0}$ by defining $a mod h b mod h = [ a , b ] h mod h ,$ where $\left[a,b\right]=ab-ba.$ This makes ${A}_{0}$ into a Poisson algebra. If ${A}_{0}$ 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 $\left(𝔤,\phi \right)$ be a Lie bialgebra and let $\delta :\mathrm{𝔘𝔤}\to \mathrm{𝔘𝔤}\otimes \mathrm{𝔘𝔤}$ be the corresponding Poisson cobracket. A quantization of $\left(𝔤,\phi \right)$ is a topological Hopf algebra $\left(A,\Delta \right)$ over $k\left[\left[h\right]\right]$ which is a topologically free $k\left[\left[h\right]\right]-$module and satisfies the following conditions:

1. $A/hA$ is identical with $\mathrm{𝔘𝔤}$ as a Hopf algebra, and
2. (Co-Poisson compatibility) $h - 1 ( Δ ( a ) - σ ( Δ ( a ) ) ) mod h = δ ( a mod h )$ for $a\in A,$ where $\sigma :A\otimes A\to A\otimes A$ is given by $\sigma \left(x\otimes y\right)=y\otimes x.$

2.4 The definition of deformations given in (2.1) is too general for some purposes. Assume that $A$ is an algebra over a field $k$ with multiplication $m:A\otimes A\to A.$ Let $h$ be an indeterminate and let $A\left[\left[h\right]\right]$ be the ring of formal power series in $h$ with coefficients in $A$. This is a complete $k\left[\left[h\right]\right]-$module. A deformation of $A$ is an associative $k\left[\left[h\right]\right]-$bilinear multiplication map ${m}_{h}:A\left[\left[h\right]\right]{\otimes }_{k\left[\left[h\right]\right]}A\left[\left[h\right]\right]\to A\left[\left[h\right]\right]$ which can be written in the form $m h = m + m 1 h + m 2 h 2 + ⋯ ,$ where the ${m}_{i}:A\otimes A\to A$ are $k-$linear maps which are extended to the completion $A\left[\left[h\right]\right].$

If $A$ is a bialgebra over $k$ with multiplication $m:A\otimes A\to A$ and comultiplication $\Delta :A\to A\otimes A$ then a deformation of $A$ is a $k\left[\left[h\right]\right]-$linear multiplication and a $k\left[\left[h\right]\right]-$linear comultiplication $m h = m + m 1 h + m 2 h 2 + ⋯ , Δ h = Δ + Δ 1 h + Δ 2 h 2 + ⋯ ,$ such that ${m}_{i}:A\otimes A\to A$ and ${\Delta }_{i}:A\to A\otimes A$ are $k-$linear maps which are extended to the completion $A\left[\left[h\right]\right],$ and such that $A\left[\left[h\right]\right]$ is a bialgebra under ${m}_{h}$ and ${\Delta }_{h}.$

2.5 Suppose that $m h = m + m 1 h + m 2 h 2 + ⋯ , and μ h = μ + μ 1 h + μ 2 h 2 + ⋯ ,$ are both deformations of an algebra $A$ over $k$ with multiplication $m$. The two deformations ${m}_{h}$ and ${\mu }_{h}$ are equivalent if there is a $k\left[\left[h\right]\right]-$linear map ${f}_{h}:A\left[\left[h\right]\right]\to A\left[\left[h\right]\right]$ of the form ${f}_{h}=\mathrm{id}+{f}_{1}h+{f}_{2}{h}^{2}+\cdots ,$ such that the ${f}_{i}:A\otimes A\to A$ are $k-$linear maps extended to the completion $A\left[\left[h\right]\right]$ such that $f h ∘ m h ( a ⊗ b ) = μ h ( f h ( a ) ⊗ f h ( b ) )$ for all $a,b\in A\left[\left[h\right]\right].$

## References

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 $h$-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.