Last updates: 22 January 2011

**1.1** Let $A$ be a ring. Let $\U0001d51e$ be an ideal in $A$. We view the powers ${\U0001d51e}^{k}$ of the ideal $\U0001d51e$ 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 $a+{\U0001d51e}^{k}$. This is the $\U0001d51e$-**adic topology**. If
${\cap}_{k}{\U0001d51e}^{k}=\left(0\right)$
then this topology is Hausdorff.

**1.2** Let $M$ be an $A$-module. We can transfer the $\U0001d51e$-adic topology on $A$ to a topology on $M$. We view the sets
${N}_{k}={\U0001d51e}^{k}M$
as a basis of neighbourhoods in $M$ containing $0$. An element $m\in M$ is an element of ${N}_{k}$ if
$m=0(mod{\U0001d51e}^{k}M)$.
As above, there is a unique topology on $M$ such that the module operations are continuous with basis given by the sets
$m+{\U0001d51e}^{k}M$,
where $m\in M$. This is the ${\U0001d51e}^{k}$**-adic topology** on $M$.

**1.3** Define a map
$d:M\hspace{0.17em}\times \hspace{0.17em}M\to \mathbb{R}$
by
$$d\left(x,y\right)={e}^{-v\left(x-y\right)},\phantom{\rule{2em}{0ex}}\text{for all}x,y\in M,$$
where $e$ is a real number $e>1$ and $v\left(x\right)$ is the largest integer $k$ such that $x\in {\U0001d51e}^{k}M$. If the $\U0001d51e$-adic toplogy on $M$ is Hausdorff the $d$ is a metric on $M$ which generates the $\U0001d51e$-adic topology.

**1.4** If $A$ is a local ring then it is natural to take $I=\U0001d52a$ where $\U0001d52a$ 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 $\U0001d52a=\left(h\right)$ generated by $h$. In this case the $\U0001d52a$**-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 $\U0001d51e$ 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 $\U0001d51e$-adic topology if for every positive integer $k>0$ there exists a positive integer $N$ such that
$${b}_{n}-b\in {\U0001d51e}^{k}M\phantom{\rule{1em}{0ex}}\text{for all}nN.$$
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
${b}_{n}-b\in {\U0001d51e}^{k}M\phantom{\rule{1em}{0ex}}\text{for all}nN.$
The module $M$ is **complete** in the $\U0001d51e$-adic topology if every Cauchy sequenc in $M$ converges. A ring $A$ is complete in the $\U0001d51e$-adic topology if when viewed as an $A$-module it is completd in the $\U0001d51e$-adic topology. If the $\U0001d51e$-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 $\U0001d51e$-adic topology, i.e.,
$$P~Q\text{if for every}k\text{there exists an}N\text{such that}{p}_{n}-{q}_{n}\in {\U0001d51e}^{k}M\text{for all}nN.$$
The set of all equivalece classes of Caucy sequences in $M$ is the **completion**
$\stackrel{\u02c6}{M}$
of $M$

**1.7** The completion
$\stackrel{\u02c6}{M}$
is an
$\stackrel{\u02c6}{A}$
module with operations given by
$$\begin{array}{rcl}P+Q& =& \left\{{p}_{n}+{q}_{n}\right\},\phantom{\rule{3em}{0ex}}\text{and,}\\ \left\{{a}_{n}\right\}P& =& \left\{{a}_{n}{p}_{n}\right\},\end{array}$$
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
$\phi :M\to \stackrel{\u02c6}{M}$
by
$$\phi \left(b\right)=\left[\left(b,b,b,\dots \right)\right],$$
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
${\cap}_{k}{\U0001d51e}^{k}M.$
The map $\phi $ is injective if $M$ is Haussdorff in the $\U0001d51e$-adic topology.

**1.9** Define a basis ${N}_{k}$ of neigbourhoods of $0$ in the completion
$\stackrel{\u02c6}{M}$
by:
$$P\in {N}_{k}\text{if there exists an}N\text{such that}{p}_{n}\in {\U0001d51e}^{k}M\text{for all}nN.$$
The collection of sets
$P+{N}_{k}$
where
$P\in \stackrel{\u02c6}{M}$
is a basis for a topology on
$\stackrel{\u02c6}{M}$.
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 $\U0001d52a=\left(h\right)$ generated by the element $h$. In this case the $\U0001d52a$-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
$${b}_{n}-{b}_{m}\in {h}^{k}M\text{for all}m,nN,$$
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
$${b}_{n}-b\in {h}^{k}M\text{for all}nN.$$
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
$${e}_{hx}=\sum _{k\ge 0}\frac{{\left(hx\right)}^{k}}{k!}=1+{x}_{0}h+\left({x}_{0}h+2{x}_{1}\right)\left(\frac{{h}^{2}}{2}\right)+\left({x}_{0}^{3}+3\left({x}_{0}{x}_{1}+{x}_{1}{x}_{0}\right)+6{x}_{2}\right)\left(\frac{{h}^{3}}{3!}\right)+\cdots $$
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 $k>0$.

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

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

**2.2** Given a deformation $A$ of a commutative algebra ${A}_{0}$ we can define a new operation
$\{\phantom{\rule{.5em}{0ex}},\phantom{\rule{.5em}{0ex}}\}$
on ${A}_{0}$ by defining
$$ {a\mathrm{mod}h,b\mathrm{mod}h} =\frac{[a,b]}{h}\mathrm{mod}h,$$
where
$[a,b]=ab-ba.$
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
$\left(\U0001d524,\phi \right)$
be a Lie bialgebra and let
$\delta :\mathrm{\U0001d518\U0001d524}\to \mathrm{\U0001d518\U0001d524}\otimes \mathrm{\U0001d518\U0001d524}$
be the corresponding Poisson cobracket. A **quantization** of
$\left(\U0001d524,\phi \right)$
is a topological Hopf algebra
$\left(A,\Delta \right)$
over
$k\left[\right[h\left]\right]$
which is a topologically free
$k\left[\right[h\left]\right]-$module and satisfies the following conditions:

- $A/hA$ is identical with $\mathrm{\U0001d518\U0001d524}$ as a Hopf algebra, and
- (Co-Poisson compatibility) $${h}^{-1}\left(\Delta \right(a)-\sigma (\Delta \left(a\right)\left)\right)\mathrm{mod}h=\delta \left(a\mathrm{mod}h\right)$$ for $a\in A,$ where $\sigma :A\otimes A\to A\otimes A$ is given by $\sigma (x\otimes y)=y\otimes x.$

**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
$A\left[\right[h\left]\right]$
be the ring of formal power series in $h$ with coefficients in $A$. This is a complete
$k\left[\right[h\left]\right]-$module. A **deformation** of $A$ is an associative
$k\left[\right[h\left]\right]-$bilinear multiplication map
${m}_{h}:A\left[\right[h\left]\right]{\otimes}_{k\left[\right[h\left]\right]}A\left[\right[h\left]\right]\to A\left[\right[h\left]\right]$
which can be written in the form
$${m}_{h}=m+{m}_{1}h+{m}_{2}{h}^{2}+\cdots ,$$
where the
${m}_{i}:A\otimes A\to A$
are $k-$linear maps which are extended to the completion
$A\left[\right[h\left]\right].$

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
$k\left[\right[h\left]\right]-$linear multiplication and a
$k\left[\right[h\left]\right]-$linear comultiplication
$$\begin{array}{rcl}{m}_{h}& =& m+{m}_{1}h+{m}_{2}{h}^{2}+\cdots ,\\ {\Delta}_{h}& =& \Delta +{\Delta}_{1}h+{\Delta}_{2}{h}^{2}+\cdots ,\end{array}$$
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
$A\left[\right[h\left]\right],$
and such that
$A\left[\right[h\left]\right]$
is a bialgebra under
${m}_{h}$
and
${\Delta}_{h}.$

**2.5** Suppose that
$$\begin{array}{rcl}{m}_{h}& =& m+{m}_{1}h+{m}_{2}{h}^{2}+\cdots ,\phantom{\rule{.5em}{0ex}}and\\ {\mu}_{h}& =& \mu +{\mu}_{1}h+{\mu}_{2}{h}^{2}+\cdots ,\end{array}$$
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
$k\left[\right[h\left]\right]-$linear map
${f}_{h}:A\left[\right[h\left]\right]\to A\left[\right[h\left]\right]$
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
$A\left[\right[h\left]\right]$
such that
$${f}_{h}\circ {m}_{h}(a\otimes b)={\mu}_{h}\left({f}_{h}\right(a)\otimes {f}_{h}(b\left)\right)$$
for all
$a,b\in A\left[\right[h\left]\right].$

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,Concerning open problems in the theory of quantization:

[D3] V.G. Drinfel'd,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,

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,