Quantization

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 22 January 2011

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 $m=0 (mod {𝔞}^{k}M)$. As above, there is a unique topology on $M$ such that the module operations are continuous with basis given by the sets $m+{𝔞}^{k}M$, where $m\in M$. This is the ${𝔞}^k$-adic topology on $M$.

1.3 Define a map $d:M\hspace{0.17em}\times \hspace{0.17em}M\to ℝ$ by $$d\left(x,y\right)=e^{-v\left(x-y\right)},\text{for all} x,y\in M,$$ where $e$ is a real number $e>1$ and $v\left(x\right)$ is the largest integer $k$ such that $x\in {𝔞}^{k}M$. 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 $$b_n-b\in {𝔞}^{k}M\text{for all} n>N.$$ 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 $b_n-b\in {𝔞}^{k}M\text{for all} n>N.$ 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., $$P~Q \text{if for every} k \text{there exists an} N \text{such that} p_n-q_n\in {𝔞}^{k}M \text{for all} n>N.$$ The set of all equivalece classes of Caucy sequences in $M$ is the completion $\hat{M}$ of $M$

1.7 The completion $\hat{M}$ is an $\hat{A}$ module with operations given by $$\begin{array}{rcl}P+Q& =& \left\{p_n+q_n\right\},\text{and,}\\ \left\{a_n\right\}P& =& \left\{a_n{p}_n\right\},\end{array}$$ 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 $\phi :M\to \hat{M}$ 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 ${\cap }_k{𝔞}^{k}M.$ 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 $\hat{M}$ by: $$P\in N_k \text{if there exists an} N \text{such that} p_n\in {𝔞}^{k}M \text{for all} n>N.$$ The collection of sets $P+N_k$ where $P\in \hat{M}$ is a basis for a topology on $\hat{M}$. 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 $$b_n-b_m\in h^{k}M \text{for all} m,n>N,$$ 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 $$b_n-b\in h^{k}M \text{for all} n>N.$$ 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 $$e_{hx}=\sum _{k\ge 0}\frac{{\left(hx\right)}^k}{k!}=1+x_{0}h+\left(x_{0}h+2x_1\right)\left(\frac{h^2}{2}\right)+\left(x_0^3+3\left(x_0{x}_1+x_1{x}_0\right)+6x_2\right)\left(\frac{h^3}{3!}\right)+\cdots$$ 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 $k>0$.

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 $k\left[\right[h\left]\right]$ such that

1. $A/hA=A_0,$ and
2. $A$ 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 $\{,\}$ 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 $\left(𝔤,\phi \right)$ 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 $k\left[\right[h\left]\right]$ which is a topologically free $k\left[\right[h\left]\right]-$module and satisfies the following conditions:

1. $A/hA$ is identical with $\mathrm{𝔘𝔤}$ as a Hopf algebra, and
2. (Co-Poisson compatibility) $$h^{-1}\left(\Delta \right(a)-\sigma (\Delta \left(a\right)\left)\right)\mathrm{mod}h=\delta \left(a\mathrm{mod}h\right)$$ 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 $$\begin{array}{rcl}{m}_h& =& m+m_{1}h+m_2{h}^2+\cdots ,and\\ {\mu }_h& =& \mu +{\mu }_{1}h+{\mu }_2{h}^2+\cdots ,\end{array}$$ 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 $a,b\in A\left[\right[h\left]\right].$

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.

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