III. Deformations of Hopf algebras

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

Last update: 22 October 2012

The basic material on completions given in §1 can be found in many books, in particular, [AMa1969] Chapt 10. The book [SSt1993] has a comprehensive treatment of deformation theory. Theorem (2.6) is stated and proved in [SSt1993] Prop. 11.3.1.

h-adic completions

Morivation for h-adic completions

We will be working with algebras over [[h]], the ring of formal power series in a variable h with coefficients in . A typical element of [[h]] which is not in [h] is the element

eh=1+h+ h22!+ h33!+.

The ring [[h]] is just [h] extended a little bit so that some nice elements that we want to write down, like eh, are in [[h]].

An algebra over [[h]] is a vector space over [[h]], i.e. a free [[h]]-module, which has a multiplication and an identity which satisfy the conditions in I (1.1) If A is an algebra over then we can extend coefficients and get a new algebra A[[h]] which is over [[h]]. But sometimes this new algebra is not quite big enough so we need to extend it a little bit and work with the h-adic completion A[[h]] which contains all the nice elements that we want to write down.

Continuing in thie vein we will want to consider the tensor product A[[h]] A[[h]]. Again, this algebra is not quite big enough and we extend it to get a slightly bigger object A[[h]] A[[h]] so that all the elements we are are available.

The algebra A[[h]], an example of an h-adic completion

If A is an algebra over k then set

A[[h]]= { a0+a1h+ a2h2+ aiA }

of formal power series with coefficients in A is the completion of the the k[[h]]-module k[[h]]kA in the h-adic topology. The k[[h]]-linear extension of the multiplication in A gives A[[h]] the structure of a k[[h]]-algebra. The ring A[[h]] is, in general larger than k[[h]]kA. For each element a=j0 ajhjA [[h]] the element

eha=0 (ha) ! =1+a0h+ (a02+2a1) (h22)+ ( a03+3 (a0a1+a1a0) +6a2 ) (h33!)+

is a well defined element of A[[h]].

Definition of the h-adic topology

Let k be a field and let h be an indeterminate. The ring k[[h]] is a local ring with unique maximal ideal (h). Let M be a k[[h]]-module. The sets

m+hnM,mM, n,

form a basis for a topology on M called the h-adic topology. Define a map d:M×M by

d(x,y)= e-v(x-y), for allx,yM,

where e is a real number e>1 and v(x) is the largest nonnegative integer n such that xhnM. Then d is a metric on M which generates the h-adic topology.

Definition of an h-adic completion

Let M be a k[[h]]-module. The completion of the metric space M is a metric space M^ which contains M in a natural way and which has a natural k[[h]]-module structure. The completion M^ of M is defined in the usual way, as a set of equivalence classes of Cauchy sequences of elements of M. Let us review this construction.

A sequence of elements {pn} in M is a Cauchy sequence in the h-adic topology if for every positive integer >0 there exists a positive integer N such that

pn-pm hM,for allm ,n>N,

i.e. pn-pm is "divisible" by h for all n,m>N. Two Cauchy sequences P={pn} and Q={qn} are equivalent if the sequence {pn-qn} converges to 0, i.e.

P~Qif for every there exists anNsuch that pn-qnhM for alln>N.

The set of all equivalence classes of Cauchy sequences in M is the completion M^ of M.

The completion M^ is a k[[h]]-module where the operations are determined by

P+Q={pn+qn}, andaP= {apn},

where P={pn} and Q={qn} are Cauchy sequences with elements in M and ak[[h]]. Define a map

ϕ: M M^ m [(m,m,m,)],

i.e. ϕ(m) is the equivalence class of the sequence {pn} such that pn=m for all n. This map is injective and thus we can view M as a submodule of M^.

Deformations

Motivation for deformations

We are going to make the quantum group by deforming the enveloping algebra 𝔘𝔤 of a complex simple Lie algebra 𝔤 as a Hopf algebra. This last condition is important because the enveloping algebra 𝔘𝔤 does not have any deformations as an algebra.

Deformation as a Hopf algebra

Assume that (A,m,ι,Δ,ε,S) is a Hopf algebra over k. Let A[[h]] A[[h]] denote the completion of A[[h]] k[[h]] A[[h]] in the h-adic topology. A deformation of A as a Hopf algebra is a tuple ( A[[h]], mh,ιh, Δh, εh,Sh ) where

mh: A[[h]] A[[h]] A[[h]], Δh: A[[h]] A[[h]] A[[h]], ιh:k [[h]]A [[h]], εh:A [[h]]k [[h]],and Sh:A [[h]]A [[h]],

are k[[h]]-linear maps which are continuous in the h-adic topology, satisfy axioms (1) - (7) in the definition of a Hopf algebra, and can be written in the form

mh = m+m1h+m2h2 + Δh = Δ+Δ1h+ Δ2h2+ ιh = ι+ι1h+ι2 h2+ εh = ε+ε1h+ε2 h2+ Sh = S+S1h+S2 h2+

where, for each positive integer i,

mi:AAA, Δi:A AA, ιi:kA, εi:Ak, andSi:A A,

are k-linear maps which are extended first k[[h]]-linearly and then to the h-adic completion. We shall abuse language (only slightly) and call ( A[[h]], mh,ιh, εh,Δh ,Sh ) a Hopf algebra over k[[h]].

Definition of equivalent deformations

Two Hopf algebra deformations ( A[[h]], mh,ιh, εh,Δh ,Sh ) and ( A[[h]], mh, ιh, Δh, Sh ) of a Hopf algebra (A,m,ι,Δ,ε,S) are equivalent if there is an isomorphism

fh: ( A[[h]], mh,ιh, εh,Δh ,Sh ) ( A[[h]], mh, ιh, Δh, Sh )

of h-adically complete Hopf algebras over k[[h]] which can be written in the form

fh=idA+ f1h+f2h2 +

such that, for each positive integer i, fi:AA is a k-linear map which is extended k[[h]]-linearly to k[[h]]kA and then to the h-adic completion A[[h]].

Definition of the trivial deformation as a Hopf algebra

Let (A,m,ι,Δ,ε,S) be a Hopf algebra. The trivial deformation of A as a Hopf algebra is the Hopf algebra ( A[[h]], mh,ιh, εh,Δh ,Sh ) over k[[h]] such that mh=m, ιh=ι, Δh=Δ, εh=ε and Sh=S (extended to A[[h]]).

Deformation as an algebra

Assume that (A,m,ι) is an algebra over k. Let A[[h]] A[[h]] denote the completion of A[[h]] k[[h]] A[[h]] in the h-adic topology. A deformation of A as an algebra is a tuple ( A[[h]], mh,ιh ) where

mh: A[[h]] A[[h]] A[[h]], ιh: k[[h]] A[[h]],

are k[[h]]-linear maps which are continuous in the h-adic topology, satisfy the axioms the definition of an algeebra (see I (1.1)) and can be written in the form

mh = m+m1h+m2h2+ ιh = ι+ι1h+ι2h2 +

where, for each positive integer i,

mi:AAA, ιi:kA,

are k-linear maps which are extended first k[[h]]-linearly and then to the h-adic completion. We shall abuse language (only slightly) and call ( A[[h]], mh,ιh ) an algebra over k[[h]].

This definition is exactly like the definition of a deformation as a Hopf algebra in (2.2) above except that we only need to start with an algebra and we only require the result to be an algebra. We can define equivalence of deformations as algebras in exactly the same way that we defined them for deformations as Hopf algebras except that we only require the isomorphism fh to be an algebra isomorphism instead of a Hopf algebra isomorphism.

The trivial deformation as an algebra

Let (A,m,ι) be an algebra. The trivial deformation of A as an algebra is the algebra ( A[[h]], mh,ιh ) over k[[h]] such that mh=m and ιh=ι (extended to A[[h]]). The deformation of the quantum group given in V (1.3) is even more incredible if one keeps the following theorem in mind.

Let 𝔤 be a finite dimensional complex simple Lie algebra and let 𝔘𝔤 be the enveloping algebra of 𝔤. Then 𝔘𝔤 has no deformations as an algebra (up to equivalence of deformations).

In other words, all deformations of 𝔘𝔤 as an algebra are equivalent to the trivial deformation of 𝔘𝔤.

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

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