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 𝔞k 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+𝔞k. This is the 𝔞-adic topology. If k𝔞k=(0) 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 Nk=𝔞kM as a basis of neighbourhoods in M containing 0. An element mM is an element of Nk if m=0 (mod 𝔞kM) . As above, there is a unique topology on M such that the module operations are continuous with basis given by the sets m+𝔞kM , where mM. This is the 𝔞k-adic topology on M.

1.3 Define a map d:M×M by d(x,y)=e-v(x-y), for all x,yM, where e is a real number e>1 and v(x) is the largest integer k such that x𝔞kM. 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[[h]], is a local ring with unique maximal ideal 𝔪=(h) generated by h. In this case the 𝔪-adic topology on a k[[h]]-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 bn 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 bn-b𝔞kM for all n>N. A sequence bn of elements in M converges to bM if for every positive integer k>0 there exists a positive integer N such that bn-b𝔞kM 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= pn and Q= qn in M are equivalent if pn-qn converges to 0 in the 𝔞-adic topology, i.e., P~Q  if for every k there exists an N  such that  pn-qn𝔞kM  for all n>N. The set of all equivalece classes of Caucy sequences in M is the completion Mˆ of M

1.7 The completion Mˆ is an A ˆ module with operations given by P+Q = pn+qn ,and, an P = anpn , where P= pn and Q= qn are Cauchy sequences with elements in M and an is a Cauchy sequence of elements in A.

1.8 Define a map φ:M Mˆ by φ(b)= [ ( b,b,b, ) ], i.e., φ(b) is the equivalence class of the sequence bn such that bn=b for all n. This map has kernel k𝔞kM. The map φ is injective if M is Haussdorff in the 𝔞-adic topology.

1.9 Define a basis Nk of neigbourhoods of 0 in the completion Mˆ by: PNk  if there exists an N such that  pn𝔞kM  for all n>N. The collection of sets P+Nk where P Mˆ is a basis for a topology on Mˆ . The module operations and the map φ are continuous.

1.10 Let k be a field. Then k[[h]] is a local ring with maximal ideal 𝔪=(h) generated by the element h. In this case the 𝔪-adic topology is called the h-adic toplogy. Let M be a k[[h]] -module. Then a sequence of elements bn in M is a Cauchy sequence if for every positive integer k>0 there exists a postive integer N such that bn-bm hkM for all m,n>N, i.e., bn-bm is "divisible" by hk for all m,n>N. A sequence bn of elements in k[[h]] converges to bM if for every positive integer k>0 there exists a positive integer N such that bn-bhkM  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[[h]] -module M in the h-adic topology. IF A is an algebra over a field k then Akk[[h]] is a k[[h]] -module in the h-adic topology and the completion of Akk[[h]] is A[[h]] , the ring of formal power series in h with coefficients in A. The ring A[[h]] is, in general, larger than Akk[[h]] .

1.12 If M is a complete k[[h]] -module in the h-adic toplogy then for each element x= j0xjhjM the element ehx= k0 (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[[h]] -module M is topologically free if M/hkM is a free k[[h]]/(hk) -module for all positive integers k>0.

Deformations and quantizations

2.1 A deformation of a commutative associative algebra A0 over k is an associative (not necessarily commutative) algebra A over k [[ h ]] such that

  1. A / h A = A 0 , and
  2. A is a topologically free k [[ h ]] -module.

2.2 Given a deformation A of a commutative algebra A0 we can define a new operation { , } on A0 by defining a mod h b mod h = [ a , b ] h mod h , where [ a , b ] = a b - b a . This makes A0 into a Poisson algebra. If A0 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 𝔤 φ be a Lie bialgebra and let δ : 𝔘𝔤 𝔘𝔤 𝔘𝔤 be the corresponding Poisson cobracket. A quantization of 𝔤 φ is a topological Hopf algebra A Δ over k [[ h ]] which is a topologically free k [[ h ]] -module and satisfies the following conditions:

  1. A / h A is identical with 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 A , where σ : A A A A is given by σ ( x y ) = y 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 A A . Let h be an indeterminate and let A [[ h ]] be the ring of formal power series in h with coefficients in A. This is a complete k [[ h ]] -module. A deformation of A is an associative k [[ h ]] -bilinear multiplication map m h : A [[ h ]] k [[ h ]] A [[ h ]] A [[ h ]] which can be written in the form m h = m + m 1 h + m 2 h 2 + , where the m i : A A A are k-linear maps which are extended to the completion A [[ h ]] .

If A is a bialgebra over k with multiplication m : A A A and comultiplication Δ : A A A then a deformation of A is a k [[ h ]] -linear multiplication and a k [[ h ]] -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 A A and Δ i : A A A are k-linear maps which are extended to the completion A [[ h ]] , and such that A [[ h ]] is a bialgebra under m h and Δ 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 μ h are equivalent if there is a k [[ h ]] -linear map f h : A [[ h ]] A [[ h ]] of the form f h = id + f 1 h + f 2 h 2 + , such that the f i : A A A are k-linear maps extended to the completion A [[ h ]] such that f h m h ( a b ) = μ h ( f h ( a ) f h ( b ) ) for all a , b A [[ h ]] .

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.

page history