
\draftcut \section{Generating Functions for Maps Between Polynomial Rings}

\subsection{The Abstract Principle}

This section may seem like an awful way to start a topology paper --- it's
all about formula-based technicalities. Here are its redeeming features
(beyond its usefulness for the later parts of the paper): \begin{itemize}
\item Did you know that quadratic forms (aka ``Gaussians'') form a
category in a natural way?  (Theorem~\ref{thm:GDO}).  \item Did you know
that Feynman diagrams arise in pure algebra in a completely natural way?
\end{itemize}

\begin{motivation} \label{mot:PBW}
The ``PBW Principle'' says that many algebras $U$ are isomorphic, as vector spaces, to
polynomial rings (hence as algebras they are ``polynomial rings with funny multiplications''). Many times
one needs to understand maps between algebras. Primarily, the algebra's own structure: the multiplication
map $m\colon U\otimes U\to U$, perhaps a co-multiplication $\Delta\colon U\to U\otimes U$, and more. Sometimes
one may care about specific special elements in $U$ or some tensor power thereof; say, $R\in U\otimes
U=\Hom(U^{\otimes\emptyset}\to U^{\otimes 2})$. So we need to understand the category of maps between
algebras and their tensor powers, and hence, by PBW, the category of maps between polynomial rings. This
category is way too big --- one can encode an infinite amount of information into a map between polynomial
rings (no matter the base fields) --- and so no finite computer can fully store a general such map. Hence we
develop a theory of ``maps between polynomial rings that can be described using finite formulas (of a
certain kind)'' and we are lucky that the maps we care about later in this paper can indeed be described by
formulas of that kind. Those maps/formulas are ``{\bf Do}cile {\bf Pe}rturbed
{\bf G}aussian differential operators'', and they make a category, \dpg, which is the main
object of study for this section.
\end{motivation}

\begin{convention} Throughout this paper we will use lower case Latin letters such as $z$, $y$, $b$, $a$,
$x$, and $t$ to denote the generators of polynomial rings. Each such generator comes with a dual (whose
purpose will be explained shortly), and the dual will always be denoted by the corresponding Greek letter:
$z^\ast=\zeta$, $y^\ast=\eta$, $b^\ast=\beta$, $a^\ast=\alpha$, $x^\ast=\xi$, and $t^\ast=\tau$. If $C$ is a
finite set, we will denote by $z_C=\{z_c\}_{c\in C}$ the set of variables denoted by the letter $z$ with an
index $c\in C$; likewise there's $y_C$, $x_C$, etc. We will regard $z_C$ sometimes as a set and sometimes as
a column vector, as appropriate. We extend duality to indexed variables:
$z^\ast_C=\zeta_C=\{z^\ast_c=\zeta_c\}_{c\in C}$. We will sometimes treat $\zeta_C$ (or $\eta_C$, etc) as a
row vector.
\end{convention}

Next, we establish a bijection
\begin{equation} \label{eq:calG}
  \calG\colon\Hom(\bbQ[z_A]\to\bbQ[z_B])\to\bbQ[z_B]\llbracket\zeta_a\rrbracket
\end{equation}
between linear maps from polynomials in variables $z_A$ to polynomials
in variables $z_B$ ($A$ and $B$ are finite sets) and a certain class of
power series in the output variables and the duals of the input variables
(more precisely, power series in the Greek variables corresponding to
the inputs, with coefficients that are polynomials in the Latin variables
corresponding to the outputs).

\begin{definition} Let $A$ and $B$ be finite sets and let
$L\colon\bbQ[z_A]\to\bbQ[z_B]$ be linear. Let $\calL=\calG(L)$, the
generating function of $L$, be defined as follows:
\begin{equation} \label{eq:calG1}
  \calL = \calG(L)
  \coloneqq \sum_{n\in\bbN^A}\frac{\zeta_A^n}{n!}L(z_A^n)
  \in \bbQ[z_B]\llbracket\zeta_a\rrbracket.
\end{equation}
Here $\bbN$ denotes the non-negative integers,
$n=(n_a)_{a\in A}$ is a multi-index, $\zeta_A^n\coloneqq\prod_{a\in A}\zeta_a^{n_a}$ and likewise
$z_A^n\coloneqq\prod_{a\in A}z_a^{n_a}$, and $n!\coloneqq\prod_{a\in A}n_a!$. Extending $L$ without changing
its name to an operator $L\colon\bbQ[z_A]\llbracket\zeta_a\rrbracket\to\bbQ[z_B]\llbracket\zeta_a\rrbracket$
by treating the $\zeta_A$'s as scalars, and recalling the definition of the exponential function, we find
that~\eqref{eq:calG1} can also be written as
\[ \calL = \calG(L) = L\left(\bbe^{\zeta_A\cdot z_A}\right), \]
where $\zeta_A\cdot z_A\coloneqq\sum_{a\in A}\zeta_az_a$.
\end{definition}

\begin{proposition} $\calG\colon\Hom(\bbQ[z_A]\to\bbQ[z_B])\to\bbQ[z_B]\llbracket\zeta_a\rrbracket$ is a
bijection. If $\calL\in\bbQ[z_B]\llbracket\zeta_a\rrbracket$ and $p\in\bbQ[z_A]$ then 
\[ \calG^{-1}(\calL)(p)
  = \left.p(\partial_{\zeta_a})\calL(\zeta_a,z_b)\right|_{\zeta_a=0}
  = \left.\calL(\partial_{z_a},z_b)p(z_a)\right|_{z_a=0}
\]
\qed
\end{proposition}

\begin{example} \label{exa:GenFun} Consider $L_i\colon\bbQ[z]\to\bbQ[z]$ for $i=1,2,3,4$,
where $L_1(p)=p$ is the identity, $L_2(p)=p(z+1)$ is the shift,
$L_3(p)=p'$ is differentiation, and $L_4(p)=\int_0^zp$ is definite
integration. Then
\[ \calG(L_1)=\bbe^{\zeta z},
  \qquad \calG(L_2)=\bbe^{\zeta(z+1)},
  \qquad \calG(L_3)=\zeta\bbe^{\zeta z},
  \qquad\text{and}\qquad \calG(L_4)=(\bbe^{\zeta z}-1)/\zeta.
\]
A few further examples of generating functions, closer in spirit to the ones we care for the most in this paper,
are in Section~\ref{ssec:realistic}, right below.
\end{example}

Linear maps between polynomial rings can be composed, and it is useful to know how their corresponding
generating functions compose\footnotemark:

\footnotetext{Below and throughout we use ``$\act$'' for left-to-right composition: $L\act M=M\circ L$.}

\begin{proposition} \label{prop:LMcomposition} Let $A$, $B$, and $C$ be finite
sets, and let $L\in\Hom(\bbQ[z_A]\to\bbQ[z_B])$ and
$M\in\Hom(\bbQ[z_B]\to\bbQ[z_C])$. Then, with $b$ standing for all
elements of $B$,
\begin{equation} \label{eq:LMcomposition}
  \calG(L\act M)
  = \left(\calG(L)|_{z_b\to\partial_{\zeta_b}}\calG(M)\right)_{\zeta_b=0}
  = \left(\calG(M)|_{\zeta_b\to\partial_{z_b}}\calG(L)\right)_{z_b=0}.
\end{equation}
\qed
\end{proposition}

Said differently, $\calG$ is an isomorphism of categories from the
category of polynomial rings in finitely many generators to the category
$\frakG$ whose objects are finite sets with morphisms $\mor_\frakG(A\to
B)=\bbQ[z_B]\llbracket\zeta_A\rrbracket$ and compositions
\begin{equation} \label{eq:frakGComposition}
  \calL\act\calM = \left(\calL|_{z_b\to\partial_{\zeta_b}}\calM\right)_{\zeta_b=0}
  = \left(\calM|_{\zeta_b\to\partial_{z_b}}\calL\right)_{z_b=0},
\end{equation}
where $\calL\in\mor_\frakG(A\to B)$ and $\calM\in\mor_\frakG(B\to C)$.

\begin{comment} \label{com:contraction}
We call the operation in~\eqref{eq:frakGComposition} ``contraction of the variable pairs
$(\zeta_b,z_b)$ for $b\in B$''.
\end{comment}

\begin{comment} \label{com:DoubleContraction}
There is an easily-provable third version for the composition
formula~\eqref{eq:frakGComposition}, which treats $\calL$ and $\calM$
and Greek and Latin letters more symmetrically:
\begin{equation} \label{eq:DoubleContraction}
  \calL\act\calM =
    \left.\bbe^{\sum\partial_{z_b}\partial_{\zeta_b}}(\calL\cdot\calM)\right|_{z_b=\zeta_b=0},
\end{equation}
where the indices $b$ run through the set $B$. Here
$\calL\cdot\calM$ stands for the ordinary product
of power series $\bbQ[z_B]\llbracket\zeta_A\rrbracket
\otimes \bbQ[z_C]\llbracket\zeta_B\rrbracket \to \bbQ[z_{A\cup
B}]\llbracket\zeta_{B\cup C}\rrbracket$.\footnote{Strictly speaking this
is valid only if there are no name clashes, namely if $A\cap B=B\cap
C=\emptyset$. That's a non-issue --- if needed the labels in $B$ can be
temporarily renamed before the formula is applied.}
\end{comment}

\begin{comment} \label{com:integration}
If you are familiar with formal Gaussian integration, especially as
it is used in physics and especially in perturbation theory where one
allows themselve to pretend that integrals always converge
(e.g.~\cite{Polyak:Feynman}), then there is another easily verified form
for the composition formula (see
also~\cite{Abdesselam:FeynmanDiagramsInAlgebraicCombinatorics}):
\begin{equation} \label{eq:integration}
  \calL\act\calM
  = \left.\bbe^{\sum\partial_{z_b}\partial_{\zeta_b}}(\calL\cdot\calM)\right|_{z_b=\zeta_b=0}
  \propto \int\bbe^{-\sum_bz_b\zeta_b}(\calL\cdot\calM)\prod_{b\in B}dz_bd\zeta_b.
\end{equation}

Much of this paper can be re-written in terms of the above formula
and Gaussian integration, yet we prefer to use this fact only
for inspiration\footnotemark.  There is simply nothing to gain:
everything one can do with integration we can also do directly
with~\eqref{eq:DoubleContraction}, a bit more simply. Yet there
is a lesson to learn from~\eqref{eq:integration}: compositions
may have simple formulas (and indeed they do) if $\calL$ and
$\calM$ are themselves Gaussians or perturbed Gaussians, for
then the integral in~\eqref{eq:integration} would be Gaussian
or perturbed Gaussian, and these are known to be computable.
\endpar{\ref{com:integration}}
\end{comment}

\footnotetext{The constant of proportionality in Equation~\eqref{eq:integration}
has some $2\pi$ factors in it.  We don't really want dreadful transcendental
numbers in an algebra paper.}

\begin{discussion} \label{disc:Qh}
Later in this paper we will also want to consider power series in
the mold of $\bbe^{z}\in\bbQ\llbracket z\rrbracket$ or $(1-z)^{-1}$.
The generating function formalism does not extend to power series
in the most naive way: the space $\Hom\left(\bbQ\llbracket
z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket\right)$ is {\em not}
isomorphic to some space of ``generating functions'' such as
$\bbQ\llbracket\zeta_A,z_B\rrbracket$. Indeed, $\bbQ\llbracket
z_A\rrbracket$ is of uncountable dimension over $\bbQ$, and
$\Hom\left(\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket\right)$
is quite wild\footnotemark. One standard way to get around this is to introduce a
``small'' parameter $\hbar$ and insist that it be present in power series,
as in $\bbe^{\hbar z}$ and $(1-\hbar z)^{-1}$. But first, a discussion and a
convention.

\footnotetext{\label{foot:ContHom}\
One may be tempted to restrict attention in $\Hom\left(\bbQ\llbracket
z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket\right)$ to {\em
continuous} homomorphisms (relative to the power series topology;
see e.g.~\cite[Chapter XVI]{Kassel:QuantumGroups}). That's wrong in
our context --- many of the homomorphisms we care about are simply not
continuous relative to the power series topology. See an example in
Footnote~\ref{foot:ncont}.}

In analysis the identity $(1-\hbar z)^{-1} = \sum \hbar^nz^n$ holds true even if $|z|$ isn't small, provided
$\hbar$ is small enough\footnote{How small? $|h|$ must be smaller than $|z|^{-1}$, so $\hbar$ must be
determined {\em after} $z$.}. In algebra, if we want to enrich $\bbQ[z]$ so as to allow such
identities\footnote{And yet without making $z$ small, that is, without switching to $Q\llbracket z\rrbracket$,
which our formalism can't handle.} we need to do two things:
\begin{itemize}
\item Tensor multiply $\bbQ[z]$ with $\bbQ[\hbar]$ to get $\bbQ[z,\hbar]$, so as to allow coefficient
depending on~$\hbar$.
\item Complete relative to the $\hbar$-adic topology so as to get $\bbQ[z]\llbracket\hbar\rrbracket$, where
series like $\sum \hbar^nz^n$ make sense.
\end{itemize}
{\red MORE. ??? Add somewhere a comment that exponentials make sense in both
$\bbQ\llbracket z\rrbracket$ and $\bbQ[z]\llbracket\hbar\rrbracket$,
yet $\Hom(\bbQ\llbracket z\rrbracket\to\bbQ\llbracket
z\rrbracket)$ is of uncountable dimension while
$\Hom_{\bbQ\llbracket\hbar\rrbracket}(\bbQ[z]\llbracket\hbar\rrbracket\to\bbQ[z]\llbracket\hbar\rrbracket)$
is countable.}

{\red MORE. This whole discussion is still murky. Does God really care about $\hbar$?}
\endpar{\ref{disc:Qh}}
\end{discussion}

\begin{convention}[and subtle point] \label{conv:Qh}
We slightly abuse notation and use $\Qh$ as a symbol for both steps:
\[ \Qh[x,y,z] \coloneqq \bbQ[x,y,z]\llbracket\hbar\rrbracket. \]
Note that $\Qh$ is not a ring but a name for an operator: tensor with $\bbQ[\hbar]$ and complete relative to
the $\hbar$-adic topology. In particular, $\Qh$ isn't $\bbQ\llbracket\hbar\rrbracket$ and $\Qh[z]$ isn't
$\bbQ\llbracket\hbar\rrbracket[z]$. Indeed, $\bbe^{\hbar z}$ and $(1-\hbar z)^{-1}$ are both members of
$\Qh[z]$ but not of $\bbQ\llbracket\hbar\rrbracket[z]$.

Yet we further abuse notation, and when $\Qh$ is on its own, we will regard it as the ring
$\bbQ\llbracket\hbar\rrbracket$. So ``$\omega\in\Qh$'' means that $\omega$ is a power series in $\hbar$
with rational coefficients.

With all this said, in much of this paper one can read $\Qh$ to simply
mean ``$\bbQ$, also with a small parameter $\hbar$'', with only a minor
disloyalty to precision. \endpar{\ref{conv:Qh}}
\end{convention}

Everything said so far work over $\Qh$ as well as over $\bbQ$. The same bijection as in~\eqref{eq:calG},
\[ \calG\colon\Hom_\Qh(\Qh[z_A]\to\Qh[z_B])\to\Qh[z_B]\llbracket\zeta_a\rrbracket, \]
with the same definition~\eqref{eq:calG1} and the same composition law~\eqref{eq:LMcomposition}.

\begin{convention} \label{conv:GreekCompletion}
We also automatically complete spaces relative to the Greek letters
$\alpha$, $\beta$, $\pi$, $\tau$, $\eta$, $\xi$, and $\zeta$, and also when
they come with subscripts. So if $a$ and $b$ are elements of some algebra
$U$ then $\bbe^{\alpha_1a+\beta_2b}$ always makes sense, and should be
regarded as an element of $U\llbracket\alpha_1,\beta_2\rrbracket$.
\end{convention}

\draftcut \subsection{Some Real Life Examples} \label{ssec:realistic}
Let us briefly meet a few generating functions of the type that we will
care about the most in this paper. But first,

\begin{convention} \label{conv:id} Throughout this paper we often put labels on tensor
factors in a tensor product instead of ordering them; hence we
often write $U^{\otimes A}$, where $U$ is a vector space and $A$
is a finite set, instead of $U^{\otimes n}$, where $n$ is a natural
number\footnotemark. If $U$ has a prescribed unit $1\in U$ and if $z\in
U$ and $i\in A$, we write $z_i$ for ``$z$ placed in tensor factor $i$
(with $1$ in all other tensor factors)''. Thus for example, we often write $z_1+z_2$ for $z\otimes 1+1\otimes z$.
If $\psi\colon U^{\otimes
A}\to U^{\otimes B}$ is a map, we often emphasize its domain and range
by writing ``$\psi^A_B$''.
\end{convention}

\footnotetext{These conventions only make sense in strict monoidal categories. They are consistent with the
``identity'' world view as opposed to the ``geography'' view; see~\cite{Talk:Toronto-1912}.}

We start with some examples from the realm of commutative polynomials. Here $U=\bbQ[z]$ denotes the ring of
commutative polynomials in a variable $z$.

\begin{example} \label{exa:m}
Let $m\colon U\otimes U\to U$ be the
multiplication of polynomials. With the language of Convention~\ref{conv:id}, we choose labels $i,j,k$ and write
$m^{ij}_k\colon\bbQ[z_i,z_j]\simeq U_i\otimes U_j\to
U_k\simeq\bbQ[z_k]$. But now $m^{ij}_k$ is given by $z_i,z_j\mapsto z_k$,
and so \[ \calG(m^{ij}_k) = m^{ij}_k(\bbe^{\zeta_iz_i+\zeta_jz_j}) =
\bbe^{(\zeta_i+\zeta_j)z_k}. \]
Note that $\calG(m^{ij}_k)$ is a Gaussian --- the exponential of a quadratic expression.
\end{example}

\begin{example} Similarly, there is a coproduct $\Delta\colon U\to U\otimes U$, better written
as $\Delta^i_{jk}$, given by $z_i\mapsto z_j+z_k$. We have
\[ \calG(\Delta^i_{jk}) = \Delta(\bbe^{\zeta_i z_i}) = \bbe^{\zeta_i(z_j+z_k)}. \]
Again, this is a Gaussian expression.
\end{example}

\begin{example} \label{exa:sigma}
A bit silly but nevertheless useful is the relabelling map $\sigma^i_j\colon U_i\to U_j$, which is
merely the identity map $U\to U$, albeit with a change-of-label for the unique tensor factor that appears. We have
\[ \calG(\sigma^i_j) = \sigma^i_j(\bbe^{\zeta_iz_i}) = \bbe^{\zeta_iz_j}. \qquad\text{(A Gaussian!)} \]
\end{example}

\begin{example} There is an inner product $P\colon U\otimes U\to\bbQ$ given by $\langle z^n,z^m\rangle=\delta_{nm}n!$,
and by a quick computation we have
\[ \calG(P^{ij}) = \bbe^{\zeta_i\zeta_j}. \qquad\text{(A Gaussian!)} \]
Note that there are no Latin letters in the above expression, because it is the generating function of a morphism
whose target space is a polynomial ring on $0$ variables.
\end{example}

\begin{example} On a finite dimensional vector space $V$ an inner product
$P$ would have an inverse $R\in V\otimes V$ such that $R_{ij}\act
P^{jk}=\sigma^k_i$ (with $\sigma$ like in Example~\ref{exa:sigma}). We
cannot have that here because $U$ is infinite-dimensional. We come close
with $R_{ij}=\bbe^{\hbar z_iz_j}\in\Qh U^{\otimes\{i,j\}}$, which satisfies
$R_{ij}\act P^{jk} = \hbar^{\deg}\sigma^k_i$, where $\hbar^{\deg}$ is the operator defined by
$\hbar^{\deg}(z^n)=\hbar^nz^n$. We then have
\[ \calG(R_{ij}) = \bbe^{\hbar z_iz_j}. \qquad\text{(A Gaussian!)} \]
Note that there are no Greek letters in the above expression, because it is the generating function of a morphism
whose domain space is a polynomial ring on $0$ variables.
\end{example}

Our next few examples are minimally-non-commutative having to do first with the Heisenberg algebra $\bbH$ and then with
the unique non-commutative 2D Lie algebra $\fraka$. But first,

\begin{convention} In support of the PBW principle
(Motivation~\ref{mot:PBW}) we will often consider both commutative and
non-commutative algebras generated by the same set of generators. In such
cases we will use ordinary {\it italics} for the generators regarded
within commutative algebras, yet {\bf boldface} letters for the same
generators regarded within non-commutative algebras. The following
definition is an example.
\end{convention}

\begin{definition} Let $\bbH$ denote the Heisenberg algebra, the free
associative algebra with generators $\bp$ and $\bx$ modulo the ``canonical commutation relation''
$[\bp,\bx]=1$. The ``$\bp$ before $\bx$'' PBW ordering map (or ``normal ordering'',
as physicists would call it) $\bbO\colon\bbQ[p,x]\to\bbH$ defined by
$p^mx^n\mapsto\bp^m\bx^n$ is a vector space isomorphism of a (commutative)
polynomial algebra with the (non-commutative) algebra $\bbH$.
\end{definition}

\begin{example} \label{exa:Heis}
Let $hm$ be the multiplication map of $\bbH$, turned into a map between
polynomial rings by using $\bbO$ to identify $\bbH$ with $\bbQ[p,x]$;
namely, let $hm^{ij}_k$ be the composition
\[ \begin{CD} \bbQ[p_i,x_i,p_j,x_j] @>\bbO_i\otimes\bbO_j>> \bbH_i\otimes\bbH_j
  @>m^{ij}_k>> \bbH_k @>\bbO_k^{-1}>> \bbQ[p_k,x_k], \end{CD}
\]
where $m\colon\bbH\otimes\bbH\to\bbH$ is the (non-commutative) multiplication map of $\bbH$.
Then
\begin{equation} \label{eq:Ghm}
  \calG(hm^{ij}_k) = \bbe^{-\xi_i\pi_j+(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}, \qquad \text{(a Gaussian!),}
\end{equation}
for indeed, using the Weyl form of the canonical commutation relation,
\begin{equation} \label{eq:Weyl}
  \bbe^{\xi\bx}\bbe^{\pi\bp} = \bbe^{-\xi\pi}\bbe^{\pi\bp}\bbe^{\xi\bx}
  \qquad\text{(in $\bbH\llbracket\pi,\xi\rrbracket$; see Convention~\ref{conv:GreekCompletion})},
\end{equation}
we have
\begin{multline*}
  \calG(hm^{ij}_k)
  = \bbe^{\pi_ip_i+\xi_ix_i+\pi_jp_j+\xi_jx_j} \act \bbO_i\otimes\bbO_j \act m^{ij}_k \act \bbO_k^{-1} 
  = \bbe^{\pi_i\bp_i}\bbe^{\xi_i\bx_i}\bbe^{\pi_j\bp_j}\bbe^{\xi_j\bx_j} \act m^{ij}_k \act \bbO_k^{-1} \\
  = \bbe^{\pi_i\bp_k}\bbe^{\xi_i\bx_k}\bbe^{\pi_j\bp_k}\bbe^{\xi_j\bx_k} \act \bbO_k^{-1}
  = \bbe^{-\xi_i\pi_j}\bbe^{(\pi_i+\pi_j)\bp_k}\bbe^{(\xi_i+\xi_j)\bx_k} \act \bbO_k^{-1}
  = \bbe^{-\xi_i\pi_j+(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}.
\end{multline*}
Note that as in Example~\ref{exa:m}, $\frakg(m^{ij}_k)=\bbe^{(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}$, so the only
``contribution'' of the non-commutativity of $hm$ is the term $-\xi_i\pi_j$ in~\eqref{eq:Ghm}.
\end{example}

Our last example for this section is split between a definition, a proposition, two proofs, and a discussion.

\begin{definition} \label{def:am}
Let $\eps$ be a parameter, let $\fraka_\eps$ be the 2D Lie algebra with generators $\ba$ and
$\bx$ and relation $[\ba,\bx]=\eps\bx$, and let $\bbA_\eps=\calU(\fraka_\eps)$ be the universal enveloping
algebra of $\fraka_\eps$. Let $\bbO\colon\bbQ[a,x]\to\bbA_\eps$ be the ``$\ba$ before $\bx$'' ordering map given
by $a^mx^n\mapsto\ba^m\bx^n$ (by PBW, it is a vector space isomorphism). Let $am^{ij}_{\eps;k}$ be the
composition
\[ \begin{CD} \bbQ[a_i,x_i,a_j,x_j] @>\bbO_i\otimes\bbO_j>> \bbA_{\eps;i}\otimes\bbA_{\eps;j}
  @>m^{ij}_{\eps;k}>> \bbA_{\eps;k} @>\bbO_k^{-1}>> \bbQ[a_k,x_k], \end{CD}
\]
where $m^{ij}_{\eps;k}$ is the multiplication map of $\bbA_\eps$.
\end{definition}

\begin{proposition} \label{prop:am}
\[
  \calG(am^{ij}_{\eps;k}) = \exp\left(
    (\alpha_i+\alpha_j)a_k+(\bbe^{-\eps\alpha_j}\xi_i+\xi_j)x_k
  \right)
  \qquad
  \left(\parbox{1.35in}{\rm\centering nearly Gaussian, see Discussion~\ref{disc:gam}}\right)
\]
\end{proposition}

\begin{proof}[Proof 1] We first need a Weyl-style exponentiated
relation (cf.\ \eqref{eq:Weyl}). Start with $\bx\ba = (\ba-\eps)\bx$,\footnotemark\
iterate to get $\bx\ba^n = (\ba-\eps)^n\bx$, sum over $n$ with coefficients
$\frac{\alpha^n}{n!}$ to get $\bx\bbe^{\alpha\ba} = \bbe^{\alpha(\ba-\eps)}\bx =
\bbe^{\alpha\ba}\bbe^{-\eps\alpha}\bx$, iterate again to get $\bx^n\bbe^{\alpha\ba} =
\bbe^{\alpha\ba}(\bbe^{-\eps\alpha})^n\bx^n$, and sum again with coefficients $\frac{\xi^n}{n!}$ to get the
exponentiated relation $\bbe^{\xi\bx}\bbe^{\alpha\ba} = \bbe^{\alpha\ba}\bbe^{\bbe^{-\eps\alpha}\xi\bx}$.

\footnotetext{\label{foot:ncont}\
In continuation of Footnote~\ref{foot:ContHom}: We have
just shown that $am(x\otimes a^n) = (a-\eps)^nx = (-\eps)^nx+\text{higher
powers}$. But $x\otimes a^n\rightarrow 0$ while $(-\eps)^nx\nrightarrow 0$, so $am$
is not continuous.}

Now proceed as in Example~\ref{exa:Heis}:
\begin{multline} \label{eq:GamMain}
  \calG(am^{ij}_{\eps;k})
  = \bbe^{\alpha_ia_i+\xi_ix_i+\alpha_ja_j+\xi_jx_j} \act \bbO_i\otimes\bbO_j \act m^{ij}_{\eps;k} \act \bbO_k^{-1}
  = \bbe^{\alpha_i\ba_i}\bbe^{\xi_i\bx_i}\bbe^{\alpha_j\ba_j}\bbe^{\xi_j\bx_j} \act m^{ij}_{\eps;k} \act \bbO_k^{-1} \\
  = \bbe^{\alpha_i\ba_k}\bbe^{\xi_i\bx_k}\bbe^{\alpha_j\ba_k}\bbe^{\xi_j\bx_k} \act \bbO_k^{-1}
  \overset{\text{\footnotesize(key)}}{=}
     \bbe^{(\alpha_i+\alpha_j)\ba_k}\bbe^{(\bbe^{-\eps\alpha}\xi_i+\xi_j)\bx_k} \act \bbO_k^{-1}
  = \bbe^{(\alpha_i+\alpha_j)a_k+(\bbe^{-\eps\alpha}\xi_i+\xi_j)x_k}.
\end{multline}
\end{proof}

\noindent{\em Proof 2.} We reprove the
key equality of Equation~\eqref{eq:GamMain},
$\bbe^{\alpha_i\ba}\bbe^{\xi_i\bx}\bbe^{\alpha_j\ba}\bbe^{\xi_j\bx} =
\bbe^{(\alpha_i+\alpha_j)\ba}\bbe^{(\bbe^{-\eps\alpha}\xi_i+\xi_j)\bx}$.
Let $\rho$ be the 2-dimensional representation of $\fraka_\eps$
given by $\rho(\ba)=\begin{pmatrix}\eps&0\\0&0\end{pmatrix}$
and $\rho(\bx)=\begin{pmatrix}0&1\\0&0\end{pmatrix}$. The
represenattion $\rho$ is faithful on $\fraka_\eps$ so it extends
to a faithful representation of group-like elements in (the
Greek-letter completion of) $\bbA_\eps$\footnote{That's an algebraic version of the fact that faithful
representations of a Lie algebra are also faithful on a neighborhood of the identity element of the corresponding
Lie group.}, so it is enough to prove that
$\bbe^{\alpha_i\rho(\ba)}\bbe^{\xi_i\rho(\bx)}\bbe^{\alpha_j\rho(\ba)}\bbe^{\xi_j\rho(\bx)}
= \bbe^{(\alpha_i+\alpha_j)\rho(\ba)}\bbe^{(\bbe^{-\eps\alpha}\xi_i+\xi_j)\rho(\bx)}$.
This we do by brute force matrix exponentiation (see also \cite[Gam.nb]{Self}):

\input{Gam.tex}
\vskip -7mm \ \qed

\begin{discussion} \label{disc:gam}
The exponent of $\calG(am^{ij}_{\eps;k})$ in itself has an exponential
term in it ($\bbe^{-\eps\alpha_j}$) hence $\calG(am^{ij}_{\eps;k})$ is
not a Gaussian in $\{\alpha_i,\alpha_j,\xi_i,\xi_j,a_k,x_k\}$, and hence
some of the techniques that we introduce in later sections, to compose
Gaussian generating functions, appear to break. We have two ways around
this, we need both of them below, and in fact, one reason we included
Proposition~\ref{prop:am} is to forewarn that these two ways are needed:

\begin{enumerate}

\item If $\eps$ is considered as ``small'' we can expand relative to $\eps$ and find
\[ \calG(am^{ij}_{\eps;k}) = \exp\left(
    (\alpha_i+\alpha_j)a_k+(\xi_i+\xi_j)x_k + \sum_{m\geq 1}\frac{(-\eps)^m\alpha_j^m}{m!}\xi_i x_k.
  \right)
\]
This is a prime example of a ``perturbed Gaussian'', and the lesson to
take is that we will need to look beyond Gaussians and at perturbation
theory.

\item Even if $\eps=1$, $\calG(am^{ij}_{k})$ is a Gaussian
in the variables $\{\xi_i,\xi_j,x_k\}$ if the variables
$\{\alpha_i,\alpha_j,a_k\}$ are held fixed, so contractions involving
$\xi$'s and $x$'s create no problems. As for contractions of
$\alpha$'a and $a$'s, $\calG(am^{ij}_{k})$ is {\em nearly} Gaussian
in $\{\alpha_i,\alpha_j,a_k\}$ for fixed $\{\xi_i,\xi_j,x_k\}$:
the offending term is the term $\bbe^{-\alpha_j}\xi_ix_k$. That
term is a manageable perturbation. It is a bit hard to summarize
what ``manageable'' means beyond saying ``whatever is subject to the
techniques of Section~\ref{ssec:PDE}''. Yet in short, the manageability
here stems from the fact that the quadratic term $(\alpha_i+\alpha_j)a_k$
is ``bipartite'', involving only $\alpha a$ terms but no $\alpha\alpha$'s
or $aa$'s, while the perturbation term $\bbe^{-\alpha_j}\xi_ix_k$ involves
only variables from one side of the partition: only the $\alpha$'s.

The lesson to take is that sometimes we will need to use Gaussian
techniques twice, relative to to different sets of variables, while
holding the variables from the other set fixed.

\end{enumerate}
\end{discussion}
