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.

$h\text{-adic}$ completions

Morivation for $h\text{-adic}$ completions

We will be working with algebras over $ℂ\left[\left[h\right]\right],$ the ring of formal power series in a variable $h$ with coefficients in $ℂ\text{.}$ A typical element of $ℂ\left[\left[h\right]\right]$ which is not in $ℂ\left[h\right]$ is the element

$$e^h=1+h+\frac{h^2}{2!}+\frac{h^3}{3!}+\dots \text{.}$$

The ring $ℂ\left[\left[h\right]\right]$ is just $ℂ\left[h\right]$ extended a little bit so that some nice elements that we want to write down, like $e^h,$ are in $ℂ\left[\left[h\right]\right]\text{.}$

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

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

The algebra $A\left[\left[h\right]\right],$ an example of an $h\text{-adic}$ completion

If $A$ is an algebra over $k$ then set

$$A\left[\left[h\right]\right]=\{a_0+a_{1}h+a_2{h}^2+\dots \mid a_i\in A\}$$

of formal power series with coefficients in $A$ is the completion of the the $k\left[\left[h\right]\right]\text{-module}$ $k\left[\left[h\right]\right]{\otimes }_{k}A$ in the $h\text{-adic}$ topology. The $k\left[\left[h\right]\right]\text{-linear}$ extension of the multiplication in $A$ gives $A\left[\left[h\right]\right]$ the structure of a $k\left[\left[h\right]\right]\text{-algebra.}$ The ring $A\left[\left[h\right]\right]$ is, in general larger than $k\left[\left[h\right]\right]{\otimes }_{k}A\text{.}$ For each element $a=\sum _{j\ge 0}{a}_j{h}^j\in A\left[\left[h\right]\right]$ the element

$$e^{ha}=\sum _{\ell \ge 0}\frac{{\left(ha\right)}^{\ell }}{\ell !}=1+a_{0}h+(a_0^2+2a_1)\left(\frac{h^2}{2}\right)+(a_0^3+3(a_0{a}_1+a_1{a}_0)+6a_2)\left(\frac{h^3}{3!}\right)+\dots$$

is a well defined element of $A\left[\left[h\right]\right]\text{.}$

Definition of the $h\text{-adic}$ topology

Let $k$ be a field and let $h$ be an indeterminate. The ring $k\left[\left[h\right]\right]$ is a local ring with unique maximal ideal $\left(h\right)\text{.}$ Let $M$ be a $k\left[\left[h\right]\right]\text{-module.}$ The sets

$$m+h^{n}M,m\in M,n\in \mathbb{N},$$

form a basis for a topology on $M$ called the $h\text{-adic}$ topology. Define a map $d:M\times M\to ℝ$ by

$$d(x,y)=e^{-v(x-y)},\text{for all}x,y\in M,$$

where $e$ is a real number $e>1$ and $v\left(x\right)$ is the largest nonnegative integer $n$ such that $x\in h^{n}M\text{.}$ Then $d$ is a metric on $M$ which generates the $h\text{-adic}$ topology.

Definition of an $h\text{-adic}$ completion

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

A sequence of elements $\left\{p_n\right\}$ in $M$ is a Cauchy sequence in the $h\text{-adic}$ topology if for every positive integer $\ell >0$ there exists a positive integer $N$ such that

$$p_n-p_m\in h^{\ell }M,\text{for all}m,n>N,$$

i.e. $p_n-p_m$ is "divisible" by $h^{\ell }$ for all $n,m>N\text{.}$ Two Cauchy sequences $P=\left\{p_n\right\}$ and $Q=\left\{q_n\right\}$ are equivalent if the sequence $\{p_n-q_n\}$ converges to 0, i.e.

$$P~Q\text{if for every}\ell \text{there exists an}N\text{such that}{p}_n-q_n\in h^{\ell }M\text{for all}n>N\text{.}$$

The set of all equivalence classes of Cauchy sequences in $M$ is the completion $\hat{M}$ of $M\text{.}$

The completion $\hat{M}$ is a $k\left[\left[h\right]\right]\text{-module}$ where the operations are determined by

$$P+Q=\{p_n+q_n\},\text{and}aP=\left\{a{p}_n\right\},$$

where $P=\left\{p_n\right\}$ and $Q=\left\{q_n\right\}$ are Cauchy sequences with elements in $M$ and $a\in k\left[\left[h\right]\right]\text{.}$ Define a map

$$\begin{array}{cccc}\varphi :& M& ⟶& \hat{M}\\ & m& ⟼& \left[(m,m,m,\dots )\right],\end{array}$$

i.e. $\varphi \left(m\right)$ is the equivalence class of the sequence $\left\{p_n\right\}$ such that $p_n=m$ for all $n\text{.}$ This map is injective and thus we can view $M$ as a submodule of $\hat{M}\text{.}$

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 $(A,m,\iota ,\Delta ,\epsilon ,S)$ is a Hopf algebra over $k\text{.}$ Let $A\left[\left[h\right]\right]⨶A\left[\left[h\right]\right]$ denote the completion of $A\left[\left[h\right]\right]{\otimes }_{k\left[\left[h\right]\right]}A\left[\left[h\right]\right]$ in the $h\text{-adic}$ topology. A deformation of $A$ as a Hopf algebra is a tuple $(A\left[\left[h\right]\right],m_h,{\iota }_h,{\Delta }_h,{\epsilon }_h,S_h)$ where

$$\begin{array}{c}{m}_h:A\left[\left[h\right]\right]⨶A\left[\left[h\right]\right]⟶A\left[\left[h\right]\right],{\Delta }_h:A\left[\left[h\right]\right]⟶A\left[\left[h\right]\right]⨶A\left[\left[h\right]\right],\\ {\iota }_h:k\left[\left[h\right]\right]⟶A\left[\left[h\right]\right],{\epsilon }_h:A\left[\left[h\right]\right]⟶k\left[\left[h\right]\right],\text{and}{S}_h:A\left[\left[h\right]\right]⟶A\left[\left[h\right]\right],\end{array}$$

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

$$\begin{array}{ccc}{m}_h& =& m+m_{1}h+m_2{h}^2+\dots \\ {\Delta }_h& =& \Delta +{\Delta }_{1}h+{\Delta }_2{h}^2+\dots \\ {\iota }_h& =& \iota +{\iota }_{1}h+{\iota }_2{h}^2+\dots \\ {\epsilon }_h& =& \epsilon +{\epsilon }_{1}h+{\epsilon }_2{h}^2+\dots \\ S_h& =& S+S_{1}h+S_2{h}^2+\dots \end{array}$$

where, for each positive integer $i,$

$$\begin{array}{c}{m}_i:A\otimes A⟶A,{\Delta }_i:A⟶A\otimes A,\\ {\iota }_i:k⟶A,{\epsilon }_i:A⟶k,\text{and}{S}_i:A⟶A,\end{array}$$

are $k\text{-linear}$ maps which are extended first $k\left[\left[h\right]\right]\text{-linearly}$ and then to the $h\text{-adic}$ completion. We shall abuse language (only slightly) and call $(A\left[\left[h\right]\right],m_h,{\iota }_h,{\epsilon }_h,{\Delta }_h,S_h)$ a Hopf algebra over $k\left[\left[h\right]\right]\text{.}$

Definition of equivalent deformations

Two Hopf algebra deformations $(A\left[\left[h\right]\right],m_h,{\iota }_h,{\epsilon }_h,{\Delta }_h,S_h)$ and $(A\left[\left[h\right]\right],m_h^{\prime },{\iota }_h^{\prime },{\Delta }_h^{\prime },S_h^{\prime })$ of a Hopf algebra $(A,m,\iota ,\Delta ,\epsilon ,S)$ are equivalent if there is an isomorphism

$$f_h:(A\left[\left[h\right]\right],m_h,{\iota }_h,{\epsilon }_h,{\Delta }_h,S_h)⟶(A\left[\left[h\right]\right],m_h^{\prime },{\iota }_h^{\prime },{\Delta }_h^{\prime },S_h^{\prime })$$

of $h\text{-adically}$ complete Hopf algebras over $k\left[\left[h\right]\right]$ which can be written in the form

$$f_h={\text{id}}_A+f_{1}h+f_2{h}^2+\dots$$

such that, for each positive integer $i,$ $f_i:A\to A$ is a $k\text{-linear}$ map which is extended $k\left[\left[h\right]\right]\text{-linearly}$ to $k\left[\left[h\right]\right]{\otimes }_{k}A$ and then to the $h\text{-adic}$ completion $A\left[\left[h\right]\right]\text{.}$

Definition of the trivial deformation as a Hopf algebra

Let $(A,m,\iota ,\Delta ,\epsilon ,S)$ be a Hopf algebra. The trivial deformation of $A$ as a Hopf algebra is the Hopf algebra $(A\left[\left[h\right]\right],m_h,{\iota }_h,{\epsilon }_h,{\Delta }_h,S_h)$ over $k\left[\left[h\right]\right]$ such that $m_h=m,$ ${\iota }_h=\iota ,$ ${\Delta }_h=\Delta ,$ ${\epsilon }_h=\epsilon$ and $S_h=S$ (extended to $A\left[\left[h\right]\right]\text{).}$

Deformation as an algebra

Assume that $(A,m,\iota )$ is an algebra over $k\text{.}$ Let $A\left[\left[h\right]\right]⨶A\left[\left[h\right]\right]$ denote the completion of $A\left[\left[h\right]\right]{\otimes }_{k\left[\left[h\right]\right]}A\left[\left[h\right]\right]$ in the $h\text{-adic}$ topology. A deformation of $A$ as an algebra is a tuple $(A\left[\left[h\right]\right],m_h,{\iota }_h)$ where

$$m_h:A\left[\left[h\right]\right]⨶A\left[\left[h\right]\right]⟶A\left[\left[h\right]\right],{\iota }_h:k\left[\left[h\right]\right]⟶A\left[\left[h\right]\right],$$

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

$$\begin{array}{ccc}{m}_h& =& m+m_{1}h+m_2{h}^2+\dots \\ {\iota }_h& =& \iota +{\iota }_{1}h+{\iota }_2{h}^2+\dots \end{array}$$

where, for each positive integer $i,$

$$m_i:A\otimes A⟶A,{\iota }_i:k⟶A,$$

are $k\text{-linear}$ maps which are extended first $k\left[\left[h\right]\right]\text{-linearly}$ and then to the $h\text{-adic}$ completion. We shall abuse language (only slightly) and call $(A\left[\left[h\right]\right],m_h,{\iota }_h)$ an algebra over $k\left[\left[h\right]\right]\text{.}$

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 $f_h$ to be an algebra isomorphism instead of a Hopf algebra isomorphism.

The trivial deformation as an algebra

Let $(A,m,\iota )$ be an algebra. The trivial deformation of $A$ as an algebra is the algebra $(A\left[\left[h\right]\right],m_h,{\iota }_h)$ over $k\left[\left[h\right]\right]$ such that $m_h=m$ and ${\iota }_h=\iota$ (extended to $A\left[\left[h\right]\right]\text{).}$ 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 $𝔤\text{.}$ 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 $𝔘𝔤\text{.}$

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.

page history