# A construction of deformations to general algebras

Dave Bowman, Dora Puljić, Agata Smoktunowicz

###### Abstract

One of the questions investigated in deformation theory is to determine to
which algebras can a given associative algebra be deformed. In this paper we
investigate a different but related question, namely: for a given associative
finite-dimensional $\mathbb{C}$-algebra $A$, find algebras $N$ which can be
deformed to $A$. We develop a simple method which produces associative and
flat deformations to investigate this question. As an application of this
method we answer a question of Michael Wemyss about deformations of
contraction algebras.

## Introduction

Michael Wemyss and Will Donovan developed a method for characterising
commutative rings which appear in algebraic geometry, by using methods from
noncommutative ring theory. In [DW16] they introduced contraction algebras
which provide important insight into resolutions of noncommutative
singularities and invariants of flops. Contraction algebras can be described
in a purely algebraic way, by using generators and relations. They appear in
many questions which can be investigated by noncommutative ring theorists who
are not familiar with advanced methods of algebraic geometry. Some of these
questions are related to nilpotent rings. In 2022, Gavin Brown and Michael
Wemyss described all finite dimensional contraction algebras which are local
[BW22]. Notice that the Jacobson radical of a finite dimensional algebra is
nilpotent, and hence contraction algebras which are local are very close to
being nilpotent. Some other questions involve characterisation of contraction
algebras which are not local.

Deformations of contraction algebras give insight into invariants of flops. In
this context Michael Wemyss asked questions about deformations of contraction
algebras which are local. Using geometric methods he was able to conjecture
which rings can be obtained as deformations of these contraction algebras.

Since contraction algebras are noncommutative, the usual methods using
derivations as in [CGG89] cannot be applied. Another approach can be to use
Hochschild cohomology and the Gerstenhaber bracket, however this often leads
to complicated calculations as seen in [SW13].

In this paper, we develop a method to calculate deformations of noncommutative
local algebras. In particular we answer a question of Michael Wemyss regarding
deformations of contraction algebras. For information about contraction
algebras we refer reader to [Wem23, DW16].

In [HT18, Conjecture 4.3] Hua and Toda conjectured that there exists an
algebraic flat deformation of the contraction algebra. As an application we
confirm this conjecture for one of the contraction algebras constructed in
[BW22]. Note that our method can be applied to other contraction algebras
described in [BW22]. Observe that Theorem $4.2$ of [HT18] gives important
information about deformations of contraction algebras which were obtained by
using geometric methods. More information related to the existence of
geometric deformations is given on pages 7-9 in [Tod15].

In [SW13] deformations of graded Hecke algebras were constructed using novel
methods. It is known that deformations of graded Hecke algebras give important
information about the Hecke algebras. Our method could be also used as another
method of choice for constructing deformations of graded Hecke algebras. Some
other methods of constructing deformations of associative algebras were
described in [Wit19].

##  1 Preliminaries

######  Definition 1.1.

Let $S$ be a subset of a $k$-algebra $A$ for a commutative unital ring $k$.
Then $S$ is a generating set if all elements of $A$ can be written as a
$k$-linear sum of products of elements of $S$ using the operations in $A$.

######  Definition 1.2 ([Wit19]).

A formal deformation $(A_{t},\ast)$ of a $k$-algebra $A$ is an associative
$k$-bilinear multiplication $\ast$ on the $k[t]$-module $A[t]$, such that in
the specification at $t=0$, the multiplication corresponds to that on $A$;
this multiplication is required to be determined by a multiplication on
elements of $A$ and extended to $A[t]$ by the Cauchy product rule.

Any multiplication $\ast$ as in Definition 1.2 is determined by products of
pairs of elements of $A$: for $a,b\in A$, write

$a\ast b=\sum_{i\geq 0}\mu_{i}(a\otimes b)t^{i},$

where $\mu_{0}$ denotes the usual product in $A$ and $\mu_{i}$ for $i\geq 1$
are $k$-linear functions from $A\otimes A$ to $A$.

######  Definition 1.3 ([FO19]).

Let $A$ be a $k$-algebra and for $t\in\mathbb{C}$ let $\\{A_{t}\\}$ be the
family of algebras arising from a deformation $*$ of $A$. Then $*$ is a flat
deformation if each $A_{t}$ has the same dimension.

######  Notation 1.4.

We will denote by $\mathbb{C}[x,y]$ the polynomial ring over $\mathbb{C}$ in
two non-commuting variables, $x$ and $y$.

######  Notation 1.5.

Consider monomials of the ring $\mathbb{C}[x,y]$. We can order the monomials
using shortlex ordering by specification $1<x<y$. We denote the monomials of
$\mathbb{C}[x,y]$ by $p_{1},p_{1},p_{2}\dots$ where $p_{i}<p_{j}$ for $i<j$.

######  Notation 1.6.

We will denote by $\mathbb{C}[x,y][t]$ the polynomial ring in variable $t$
with coefficients from the polynomial ring in two non-commuting variables $x$
and $y$, with coefficients from $\mathbb{C}$.

######  Notation 1.7.

For $a_{i},b_{i}\in A$ we define $f:\mathbb{C}[x,y][t]\rightarrow A[t]$ to be
a homomorphism of $\mathbb{C}$-algebras such that

$f:x\mapsto\sum_{i=1}^{n}a_{i}t^{i},\quad
f:y\mapsto\sum_{i=1}^{n}b_{i}t^{i},\quad f:t\mapsto t.$

We will denote $\ker f$ by $I$.

######  Notation 1.8.

For an element $p_{i}$ in $\mathbb{C}[x,y]$ we will denote by
$\overline{p_{i}}$ the same element $p_{i}$, but with all instances of $x$
replaced by $\sum_{i=1}^{n}a_{i}t^{i}$, and all instances of $y$ replaced by
$\sum_{i=1}^{n}b_{i}t^{i}$. We will denote by $\overline{\overline{p_{i}}}$
the same element as $p_{i}$, but with all instances of $x$ replaced by
$\sum_{i=1}^{n}a_{i}$, and all instances of $y$ replaced by
$\sum_{i=1}^{n}b_{i}$.

######  Notation 1.9.

Let $R$ be a ring and $I$ be an ideal of $R$, then elements of the factor ring
$R/I$ will be denoted as $r+I$, where $r\in R$. Notice that $r+I=s+I$ for
$r,s\in R$ if and only if $r-s\in I$.

##  2 Algorithm

In this section we shall describe the simplest case of our proposed method.
This requires the data of a $\mathbb{C}$-algebra $A$ and a two element
generating set $\\{a,b\\}$. We shall state the steps of this algorithm and
then provide notes regarding important components. Then some examples will be
displayed.

######  Method 1.

We fix a finite dimensional $\mathbb{C}$-algebra $A$ with two generators
$a,b$. Then:

  1. 1.

We consider $A[t]$, the polynomial algebra over $A$. We will use $t$ as the
parameter of our deformation. We define:

$\displaystyle x$ $\displaystyle:=ta,$ $\displaystyle y$ $\displaystyle:=tb.$

  2. 2.

We calculate:

$\displaystyle x^{2}$ $\displaystyle=t^{2}a^{2},$ $\displaystyle xy$
$\displaystyle=t^{2}ab,$ $\displaystyle yx$ $\displaystyle=t^{2}ba,$
$\displaystyle y^{2}$ $\displaystyle=t^{2}b^{2},$

and continue to calculate larger products of $x,y$. We shall then proceed with
a Diamond Lemma111See section 1 of [Ber78] like decomposition of large
products of $a,b$ in terms of smaller products of $a,b$. This will cause all
elements of sufficiently large length to have a power of $t$ as a factor. In
doing so we will obtain relations on $x,y$ and products thereof. We terminate
this process when we have enough relations to decompose any large product into
only multiples of $x,y$ and powers of $t$. Our relations will be given by
polynomials $p_{1},\dots,p_{m}\in\mathbb{C}[x,y][t]$ for some
$m\in\mathbb{N}.$ This terminates finitely because $A$ finite dimensional.

  3. 3.

We present the algebra:

$\mathcal{N}:={\mathbb{C}[x,y][t]}/{\left<p_{1},\dots,p_{m}\right>}.$

We note that sufficiently large products of $x,y$ will obtain a factor of $t$.
Now we evaluate $t$ at various values in $T=[0,\infty)$. We denote by $N_{s}$
the algebra that arises from $\mathcal{N}$ by evaluating $t$ at $s$ and in
particular we write $N=N_{0}$. By step (2) $N$ is local, in Section 5 we
consider a more general method where $N$ is not necessarily local. The algebra
$\mathcal{N}$ and the family of algebras $\\{N_{t}\\}$ are the output of this
method.

By Theorem 3.8 the family $\\{N_{t}\\}$ is a deformation of $N$ and by
Proposition 3.9 it holds that $A\in\\{N_{t}\\}$.

Changing the chosen generators $a,b$ can result in different algebras $N_{0}$
that have $A$ as an associative deformation.

###  2.1 Notes on the method

We give some notes before the examples:

  * •

In step (1) we are trying to capture the behaviour of the elements in $A$, but
in a way that is controlled by $t$. This enables us to make use of a
presentation while still having access to our deformation parameter $t$.

  * •

We note that since $a,b$ generate $A$ some linear combination of products of
$a,b$ will be the identity. In step (2) it is important we find a relation on
$x,y,t$ and $1\in A$.

  * •

In step (3) we discard $A$ entirely. However, because of the relations
$p_{i}$, $x$ and $y$ “remember” that they came from $A$ and that they form a
generating set. This step results in something subtle: the monomials $x,y\in
N$ are no longer dependent on $t$ (and thus do not vanish when we set $t=0$)
but their products maintain their dependence on $t$. We have loosened our grip
on $x,y$ just enough for the deformation to take place.

  * •

We note that our deformation will be associative since the multiplication of
$N_{t}$ will inherit from the associative multiplication $*$ of the algebra
$\mathcal{N}$.

  * •

This algorithm is easily generalised to algebras that require a larger
generating set. Suppose a finite dimensional $\mathbb{C}$-algebra $A$ has $n$
generators $\\{a_{1},\dots,a_{n}\\}$. Then in step (1) we would define
$x_{i}:=ta_{i}$ for $1\leq i\leq n$ and proceed as described in Method 1. The
resulting family of algebras will also be a flat and associative deformation
of $A$.

###  2.2 Examples

Here we shall work through some examples. We shall start with a simple example
where $A=\mathbb{C}\oplus\mathbb{C}$ and then give a more complicated one
where $A=M_{2}(\mathbb{C})$, the algebra of $2\times 2$ matrices over
$\mathbb{C}$. Practically it can be useful to fix a basis for the algebra $A$
as this helps find the decompositions in step (2).

######  Example 2.1.

We consider $A=\mathbb{C}\oplus\mathbb{C}$ and fix $a=(i,0),\;b=(0,1)$.
Clearly $\mathbb{C}$-linear combinations of $a$ and $b$ span $A$ so our basis
is just $\\{a,b\\}$. We write $\mathbbm{1}:=(1,1)$. Now we consider $A[t]$ and
define:

$\displaystyle x:=ta=(ti,0),$ $\displaystyle y:=tb=(0,t).$

We calculate:

$\displaystyle x^{2}=t^{2}a^{2}=(-t^{2},0)=tix,$ $\displaystyle
xy=t^{2}ab=(0,0)=0,$ $\displaystyle yx=t^{2}ab=(0,0)=0,$ $\displaystyle
y^{2}=t^{2}b^{2}=(0,t^{2})=ty,$ $\displaystyle t\mathbbm{1}=-ix+y.$

Thus we have relations:

$\displaystyle p_{1}$ $\displaystyle=x^{2}-tix=0,$ $\displaystyle p_{2}$
$\displaystyle=xy=0,$ $\displaystyle p_{3}$ $\displaystyle=yx=0,$
$\displaystyle p_{4}$ $\displaystyle=y^{2}-ty=0,$ $\displaystyle p_{5}$
$\displaystyle=t\mathbbm{1}-(-ix+y).$

Since $xy=yx=0$ and squares of $x,y$ can be reduced, we have enough relations
to decompose any large product into only multiples of $x,y$ and powers of $t$.

Thus we present:

$N:={\mathbb{C}[x,y][t]}/{\left<p_{1},p_{2},p_{3},p_{4},p_{5}\right>}.$

We observe:

$N_{0}={\mathbb{C}[x,y]}/{\left<x^{2},xy,yx,y^{2},-ix+y\right>},$

and

$N_{1}={\mathbb{C}[x,y]}/{\left<x^{2}-ix,xy,yx,y^{2}-y,1+ix-y\right>.}$

We see that $x,y$ behave exactly as $a,b\in A$ and by Proposition 3.9. It is
easily checked that the isomorphism is given by:

$\displaystyle\phi:N_{1}$ $\displaystyle\to A$ $\displaystyle 1$
$\displaystyle\mapsto\mathbbm{1}$ $\displaystyle x$
$\displaystyle\mapsto(i,0)$ $\displaystyle y$ $\displaystyle\mapsto(0,1).$

Thus we have produced an algebra $N_{0}$ that has
$A=\mathbb{C}\oplus\mathbb{C}$ as a deformation.

######  Example 2.2.

We consider $A=M_{2}(\mathbb{C})$ and fix:

$a=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\quad b=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$

We denote by $\mathbbm{1}$ the multiplicative unit in $A$. Next we compute:

$\displaystyle\begin{pmatrix}1&0\\\
0&0\end{pmatrix}=\frac{1}{2}(a+b^{2})=e_{1},$
$\displaystyle\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}=\frac{1}{2}(ab+b)=e_{2},$
$\displaystyle\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}=\frac{1}{2}(b-ab)=e_{3},$
$\displaystyle\begin{pmatrix}0&0\\\
0&1\end{pmatrix}=\frac{1}{2}(b^{2}-a)=e_{4}$

and thus $a,b$ generate $A$ as an algebra. Now we consider $A[t]$ and define:

$\displaystyle x:=ta,$ $\displaystyle y:=tb.$

We compute:

$\displaystyle x^{2}=t^{2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}=t^{2}\mathbbm{1},$ $\displaystyle
xy=t^{2}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}=t^{2}(e_{2}-e_{3}),$
$\displaystyle yx=t^{2}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}=t^{2}(e_{3}-e_{2}),$ $\displaystyle
y^{2}=t^{2}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}=t^{2}\mathbbm{1}.$

We have obtained the relations:

$\displaystyle p_{1}=x^{2}-t^{2}\mathbbm{1},$ $\displaystyle
p_{2}=x^{2}-y^{2}=0,$ $\displaystyle p_{3}=xy+yx=0,$ $\displaystyle
p_{4}=y^{2}-t^{2}\mathbbm{1}.$

Now we consider larger products to produce more relations:

$\displaystyle x^{3}=t^{3}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}=t^{2}x,$
$\displaystyle y^{3}=t^{3}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}=t^{2}y.$

Hence we have also obtained:

$\displaystyle p_{5}=x^{3}-t^{2}x=0,$ $\displaystyle p_{6}=y^{3}-t^{2}y=0.$

These allow us to derive the following:

$\displaystyle x^{2}y=y^{3}=yx^{2}=y^{3}=t^{2}y,$ $\displaystyle
y^{2}x=x^{3}=xy^{2}=x^{3}=t^{2}x,$

which are enough to reduce an arbitrary product of $x,y$. Thus we present:

$N:={\mathbb{C}[x,y][t]}/{\left<p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}\right>}.$

We observe:

$N_{0}={\mathbb{C}[x,y]}/{\left<x^{2},y^{2},xy+yx\right>},$

and

$N_{1}={\mathbb{C}[x,y]}/{\left<x^{2}-1,y^{2}-1,xy+yx,x^{3}-x,y^{3}-y\right>}.$

By Proposition 3.9 we have $N_{1}\cong A$. It is easily checked that the
isomorphism is given by:

$\displaystyle\phi:N_{1}$ $\displaystyle\to A$ $\displaystyle 1$
$\displaystyle\mapsto\mathbbm{1}$ $\displaystyle x$
$\displaystyle\mapsto\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ $\displaystyle
y$ $\displaystyle\mapsto\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

Thus we have produced an algebra $N_{0}$ that has $A=M_{2}(\mathbb{C})$ as a
deformation.

##  3 Deformations to $A$

We now consider the general case in which $A$ is an associative finite
dimensional $\mathbb{C}$-algebra generated by some elements
$\sum_{i=1}^{n}a_{i},\sum_{i=1}^{n}b_{i}$ for some $a_{i},b_{i}\in A$.

###  3.1 Specification at $t=0$

In this section we define a multiplication on $\mathbb{C}[x,y][t]$ which gives
the algebra $N$ at specification $t=0$.

######  Proposition 3.1.

Let $\mathbb{C}[t]$ denote the polynomial ring in variable $t$. Further,
assume that for every $r\in A$ there exists $i=i(r)$ such that
$rt^{i}\in\text{im}f$. There exist elements
$q_{1},q_{2},\dots,q_{n}\in{\mathbb{C}}[x,y][t]$ such that for every $k$ we
have

$f(p_{k})\in\sum_{i=1}^{n}\mathbb{C}[t]f(q_{i}).$ (1)

Moreover $n=\dim A.$

###### Proof.

We let $\\{e_{1},\dots,e_{n}\\}$ be a basis of $A$. Let $k_{1}$ be the
smallest possible integer such that for some integer $j_{1}$ we have

$e_{j_{1}}t^{k_{1}}+\sum_{\begin{subarray}{c}i=1\\\ i\neq
j_{1}\end{subarray}}^{n}\alpha_{i}(t)e_{i}\in\text{im}f$

for some $\alpha_{i}(t)\in\mathbb{C}[t]$. Let $k_{2}$ be the smallest possible
integer such that for some integer $j_{2}\neq j_{1}$ we have

$e_{j_{2}}t^{k_{2}}+\sum_{\begin{subarray}{c}i=1\\\ i\neq
j_{1},j_{2}\end{subarray}}^{n}\alpha^{\prime}_{i}(t)e_{i}\in\text{im}f$

for some $\alpha^{\prime}_{i}(t)\in\mathbb{C}[t]$. Continuing in this way, we
define a basis

$\\{q_{1},\dots,q_{n}\\}$ such that

$f(q_{i})=e_{j_{m}}t^{k_{m}}+\sum_{\begin{subarray}{c}i=1\\\ i\neq
j_{1},\dots,j_{m}\end{subarray}}^{n}\alpha_{i}(t)e_{i}.$

Now notice that $f(q_{1}),f(q_{2}),\dots,f(q_{n})$ are linearly independent
over $\mathbb{C}$ as $e_{1},e_{2},\dots,e_{n}$ are linearly independent over
$\mathbb{C}$.

We now prove that $f(q_{1})+t\text{im}f,\dots,f(q_{n})+t\text{im}f$ span the
$\mathbb{C}$-algebra $\frac{\text{im}f}{t\text{im}f}$ over $\mathbb{C}$, and
hence they are a basis for $\frac{\text{im}f}{t\text{im}f}$. Let
$z_{1}\in\text{im}f$ and write

$z_{1}=\sum_{i=1}^{n}r_{i}(t)e_{j_{i}}t^{s_{i}}$

for some $r_{i}\in\mathbb{C}[t]-t\mathbb{C}[t]$ and some
$j_{i},s_{i}\in\mathbb{N}$.

Notice that for the smallest $i$ such that $r_{i}(t)\neq 0$, $s_{i}\geq k_{i}$
for $k_{i}$ as above. Let

$z_{2}=z_{1}-f(q_{1})t^{s_{1}-k_{1}}r_{1}(t).$

Note that we have

$z_{2}\in\sum_{i=2}^{n}\mathbb{C}[t]e_{j_{i}}$

so we can write

$z_{2}=\sum_{i=2}^{n}r^{2}_{i}(t)e_{j_{i}}t^{s^{2}_{i}}$

for some $r^{2}_{i}(t)\in\mathbb{C}[t]-t\mathbb{C}[t]$ and
$s^{2}_{i}\in\mathbb{N}$. Notice that $q_{2}\in\text{im}f$. Now let

$z_{3}=z_{2}-f(q_{2})t^{s^{2}_{2}-k_{2}}r^{2}_{2}(t).$

Continuing in this way we eventually arrive at $q_{n+1}=0.$ Hence by

summing the following equations

$\displaystyle 0$
$\displaystyle=z_{n}-f(q_{n})t^{s^{n}_{n}-k_{n}}r^{n}_{n}(t),$
$\displaystyle\vdots$ $\displaystyle z_{3}$
$\displaystyle=z_{2}-f(q_{2})t^{s^{2}_{2}-k_{2}}r^{2}_{2}(t)$

and

$z_{2}=z_{1}-f(q_{1})t^{s_{1}-k_{1}}r_{1}(t)$

we arrive at

$z_{1}=\sum_{l=1}^{n}f(q_{l})t^{s^{\prime}_{l}-k_{l}}r^{l}_{l}(t)\in\sum_{l=1}^{n}\mathbb{C}[t]f(q_{l})$

for some $s^{\prime}_{l}\in\mathbb{N}$ as required. ∎

Remark. Notice that for $k\in\mathbb{N}$ and $\alpha_{j}\in\mathbb{C}[t]$

$f(p_{k})-\sum_{j=1}^{n}\alpha_{j}f(q{{}_{j}})=0$

if and only if

$p_{k}-\sum_{j=1}^{n}\alpha_{j}q_{j}\in I.$

######  Proposition 3.2.

For any $k,m\in\\{1,2,\dots,n\\}$ there exist $\zeta_{i,k}\in\mathbb{C}$,
$\xi_{i,k}(t)\in{\mathbb{C}}[t]$ such that

$p_{k}-\sum_{i=1}^{n}(\zeta_{i,k}q_{i}+t\xi_{i,k}(t)q_{i})\in I.$

In particular, there exist $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ such that

$q_{k}q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+t\xi_{i,k,m}(t)q_{i})\in I.$

###### Proof.

For each element $p_{k}\in{\mathbb{C}}[x,y][t]$ we have

$p_{k}\in\sum_{i=1}^{n}{\mathbb{C}}[t]q_{i}+I.$

This follows by taking $f^{-1}$ of equation (1). Hence, for each
$k\in\mathbb{N}$ there exist $\sigma_{i,k}\in\mathbb{C}[t]$ such that

$p_{k}-\sum_{i=1}^{n}\sigma_{i,k}q_{i}\in I,$

where $I=\ker f$. Notice that we can write
$\sigma_{i,k}(t)=\zeta_{i,k}+t\xi_{i,k}(t)$ for some
$\zeta_{i,k}\in\mathbb{C}$, $\xi_{i,k}(t)\in{\mathbb{C}}[t]$, as to separate
the terms dependent on $t$. The above then becomes

$p_{k}-\sum_{i=1}^{n}(\zeta_{i,k}q_{i}+t\xi_{i,k}(t)q_{i})\in I.$

∎

Reasoning similarly as in the proof of Proposition 3.1 we get the following
corollary.

######  Corollary 3.3.

Suppose $\sum_{i=1}^{n}\alpha_{i}q_{i}\in I$ for some
$\alpha_{i}\in{\mathbb{C}}[t]$. Then $f(\sum_{i}^{n}\alpha_{i}q_{i})=0$ and
consequently $\alpha_{i}=0$ for every $i\in\\{1,2,\dots,n\\}$.

Moreover, all elements of $I$ are $\mathbb{C}[t]$-linear combinations of
elements

$p_{k}-\sum_{i=1}^{n}(\zeta_{i,k}q_{i}+t\xi_{i,k}(t)q_{i})$

for some $k\in\mathbb{N},\zeta_{i,k}\in\mathbb{C}$,
$\xi_{i,k}(t)\in{\mathbb{C}}[t]$.

######  Notation 3.4.

We denote by $J$ be the set consisting of $\mathbb{C}$-linear combinations of
elements

$p_{k}-\sum_{i=1}^{n}\zeta_{i,k}q_{i}\in{\mathbb{C}}[x,y],$ (2)

where $\zeta_{i,k}$ are as in Proposition 3.2. Let $\langle J\rangle$ be the
ideal of ${\mathbb{C}}[x,y]$ generated by elements from $J$. Further, let $N$
be the quotient algebra $\mathbb{C}[x,y]/\langle J\rangle$.

######  Corollary 3.5.

We have $e\in J$ if and only if $e+e^{\prime}\in I$ for some $e^{\prime}\in
t{\mathbb{C}}[x,y][t]$. In particular $J\subseteq I+t{\mathbb{C}}[x,y][t]$,
and hence $\langle J\rangle\subseteq I+t{\mathbb{C}}[x,y][t]$.

######  Proposition 3.6.

The dimension of $N$ equals $n$. Moreover, elements $q_{k}+J\in N$ are a basis
of $N$ as a $\mathbb{C}$-vector space.

###### Proof.

Notice that expression (2) implies that

${\mathbb{C}}[x,y]\subseteq\sum_{k=1}^{n}\mathbb{C}q_{k}+\langle J\rangle,$

and hence $\\{q_{k}+\langle J\rangle\mid k\in\\{1,2,\ldots,n\\}\\}$ span $N$
as a $\mathbb{C}$-vector space. Therefore, the dimension of $N$ does not
exceed $n$.

We now show that the elements $q_{k}+\langle J\rangle\in N$ for
$k\in\\{1,2,\ldots,n\\}$ are linearly independent over $\mathbb{C}$. Suppose
on the contrary that we have

$\sum_{i=1}^{n}\xi_{i}q_{i}\in\langle J\rangle$

for some $\xi_{i}\in\mathbb{C}$. By Corollary 3.5 and Proposition 3.1 we have

$\langle J\rangle\subseteq I+t{\mathbb{C}}[x,y][t]\subseteq
I+t\sum_{i=1}^{n}{\mathbb{C}}[t]q_{i}.$

It follows that there is $e^{\prime}\in t\sum_{i=1}^{n}{\mathbb{C}}[t]q_{i}$
such that $\sum_{i=1}^{n}\xi_{i}q_{i}-e^{\prime}\in I$, contradicting
Corollary 3.3. ∎

Observe now that the following holds:

######  Proposition 3.7.

We can present $N$ as an $\mathbb{C}$-algebra with generators
$d_{1},\dots,d_{n}$, which span $N$ as a $\mathbb{C}$-vector space, subject to
relations

$d_{k}d_{m}=\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}$

where $\zeta_{i,k,m}$ are as in Proposition 3.2.

###### Proof.

We let ${\mathbb{C}}[q_{1},\ldots,q_{n}]$ be the subalgebra of
${\mathbb{C}}[x,y]$ generated by elements
$q_{1},\ldots,q_{n}\in{\mathbb{C}}[x,y]$. Similarly, let
${\mathbb{C}}[d_{1},\ldots,d_{n}]$ be the free $\mathbb{C}$-algebra generated
by elements $d_{1},\ldots,d_{n}$. Let
$\zeta:{\mathbb{C}}[d_{1},\ldots,d_{n}]\rightarrow{\mathbb{C}}[x,y]/\langle
J\rangle$ be defined as $\zeta(d_{i})=q_{i}+\langle J\rangle$.

We now show that $x+\langle J\rangle,y+\langle J\rangle\in\text{im}\zeta$.
Notice that since $f(x)\in\text{im}f$ we have

$f(x)\in\sum_{i=0}^{\infty}{\mathbb{C}}[t]f(q_{i}).$

Hence we have $f(x)=f(x^{\prime})$ for some
$x^{\prime}\in\sum_{i=0}^{\infty}{\mathbb{C}}[t]f(q_{i}).$ Then
$x-x^{\prime}\in I$, and so $x-x^{\prime\prime}\in J$ for some
$x^{\prime\prime}\in\sum_{i=0}^{\infty}{\mathbb{C}}q_{i}.$ Then
$x^{\prime\prime}+\langle J\rangle\in\text{im}\zeta$, and so $x+\langle
J\rangle\in\text{im}\zeta$. An analogous argument shows $y+\langle
J\rangle\in\text{im}\zeta$.

Let

$J^{\prime}=\ker(h)\subseteq{\mathbb{C}}[d_{1},\ldots,d_{n}].$

Then by the First Isomorphism Theorem for rings
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime}$ is isomorphic to
${\mathbb{C}}[x,y]/\langle J\rangle=N$.

Let $J^{\prime\prime}$ be the ideal of ${\mathbb{C}}[d_{1},\ldots,d_{n}]$
generated by elements

$d_{k}d_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}$

where $\zeta_{i,k,m}$ are as in Proposition 3.2. Observe that

$\zeta\left(d_{k}d_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}\right)=q_{k}q_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}q_{i}+\langle
J\rangle=0+\langle J\rangle.$

Therefore $J^{\prime\prime}\subseteq J^{\prime}$. It follows that the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime}$ does not exceed the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime\prime}$.

We now show that $J^{\prime}=J^{\prime\prime}$. Notice that
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime\prime}$ is at most
$n$-dimensional, since every element can be presented as a linear combination
of elements $d_{i}+J^{\prime\prime}$ for $i=1,2,\ldots,n$. On the other hand
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime}$ is isomorphic to
${\mathbb{C}}[x,y]/\langle J\rangle$, and hence is $n$-dimensional (by
Proposition 3.6). Hence by comparing dimensions we get that the dimension of
${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime\prime}$ does not exceed the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime}$. Previously we
showed that ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime}$ does not exceed the
dimension of ${\mathbb{C}}[d_{1},\dots,d_{n}]/J^{\prime\prime}$. It follows
that $J^{\prime}=J^{\prime\prime}$, as required. ∎

We now define a multiplication on $N$ which gives a formal deformation to the
algebra $A$.

######  Theorem 3.8.

Let $d_{1},\dots,d_{n}$ be free generators of the $\mathbb{C}$-algebra
$\mathbb{C}[d_{1},\dots,d_{n}]$ and suppose $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ are as in Proposition 3.2. Then the
multiplication rule

$d_{k}\ast_{t}d_{m}=\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}(t)d_{i}).$

gives a formal deformation such that $\ast_{0}$ gives the multiplication on an
algebra isomorphic to $N$.

###### Proof.

Recall relations from Proposition 3.2 that for any $k,m\in\\{1,2,\dots,n\\}$
there exist $\zeta_{i,k,m}\in\mathbb{C}$, $\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$
such that

$q_{k}q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+t\xi_{i,k,m}(t)q_{i})\in I.$ (3)

We introduce notation $[q_{i}]:=q_{i}+I$ for elements of
${\mathbb{C}}[x,y]/I$. Hence we obtain the following relations corresponding
to (3) which give the multiplicative table on ${\mathbb{C}}[x,y]/I$. We have

$[q_{k}][q_{m}]-\sum_{i=1}^{n}(\zeta_{i,k,m}[q_{i}]+t\xi_{i,k,m}(t)[q_{i}]).$

Notice that these relations give a multiplication

$d_{k}\ast_{t}d_{m}=\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}(t)d_{i})$
(4)

for $d_{i}\in\mathbb{C}[x,y].$ Now recall that $N$ is isomorphic to
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime\prime}$ where $J^{\prime\prime}$
is the ideal of ${\mathbb{C}}[d_{1},\ldots,d_{n}]$ generated by relations

$d_{k}d_{m}-\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}.$

Therefore by setting $t=0$ in (4) we obtain the multiplication rule for
${\mathbb{C}}[d_{1},\ldots,d_{n}]/J^{\prime\prime}$. ∎

###  3.2 Specification at $t=1$

In this section we define a multiplication on $\mathbb{C}[x,y][t]$ which gives
a formal deformation of $N$, such that at specification $t=1$ we get an
algebra isomorphic to $A$.

######  Proposition 3.9.

Let $\xi:\mathbb{C}[x,y][t]\rightarrow A$ be a homomorphism of
$\mathbb{C}$-algebras such that

$\xi(x)=\sum_{i=0}^{n}a_{i},\quad\xi(y)=\sum_{i=0}^{n}b_{i},\quad\xi(t)=1.$

Further, assume that for every $r\in A$ there exists $i=i(r)$ such that
$rt^{i}\in\text{im}f$. Then $\ker\xi=\langle I,t-1\rangle.$

###### Proof.

We let $e=\sum_{i}\alpha_{i}p_{i}t^{\beta{i}}\in\ker\xi$ where $p_{i}$ are
monomials in $\mathbb{C}[x,y]$, $\alpha_{i}\in\mathbb{C}$ and
$\beta_{i}\in\mathbb{N}$. We will show there exists $\gamma_{i}\in\mathbb{N}$
such that $\hat{e}:=\sum_{i}\alpha_{i}p_{i}t^{\gamma_{i}}\in I$, so that $e\in
I+\langle t-1\rangle$ as $e-\hat{e}\in\langle t-1\rangle.$ This follows since

$e-\hat{e}=\sum_{i}\alpha_{i}p_{i}t^{\beta_{i}}-\sum_{i}\alpha_{i}p_{i}t^{\gamma_{i}}=\sum_{i}\alpha_{i}p_{i}(t^{\beta_{i}}-t^{\gamma_{i}})=\sum_{i}\alpha_{i}p_{i}t^{m}(t^{n}-1)\in\langle
t-1\rangle$

for some $m,n\in\mathbb{N}$.

Recall Notation 1.8 and note that
$\xi(e)=\sum_{i}\alpha_{i}\overline{\overline{p_{i}}}=0$. Notice that by
assumption there exist $j_{i}\in\mathbb{N}$ such that
$\overline{\overline{p_{i}}}t^{j_{i}}\in\text{im}f$. We let
$f(c_{i})=\overline{\overline{p_{i}}}t^{j_{i}}$ for some
$c_{i}\in\mathbb{C}[x,y][t]$. Hence for large enough $k\in\mathbb{N}$ we have

$f\left(\sum_{i}\alpha_{i}c_{i}t^{k-j_{i}}\right)=\sum_{i}\alpha_{i}\overline{\overline{p_{i}}}t^{k}=0.$

∎

######  Proposition 3.10.

Let notation be as in Proposition 3.9, and suppose that $\sum_{i=0}^{n}a_{i}$
and $\sum_{i=0}^{n}b_{i}$ generate $A$ as a $\mathbb{C}$-algebra. We have

$\frac{{\mathbb{C}}[x,y][t]/I}{\langle t-1+I\rangle}\cong A.$

###### Proof.

Note that the ideal $\langle t-1+I\rangle$ of ${\mathbb{C}}[x,y][t]/I$ equals
the set

$\frac{{\mathbb{C}}[x,y][t](t-1)+I}{I}=\frac{\langle
I,t-1\rangle}{I}=\\{g(t-1)+I\mid g\in{\mathbb{C}}[x,y][t]\\}.$

By the Third Isomorphism Theorem we have

$\frac{{\mathbb{C}}[x,y][t]/I}{\langle
I,t-1\rangle/I}\cong\frac{{\mathbb{C}}[x,y][t]}{\langle I,t-1\rangle}.$

By Proposition 3.9 we have $\langle I,t-1\rangle=\ker\xi$ and as $\xi$ is
onto,

$\frac{{\mathbb{C}}[x,y][t]}{\ker\xi}\cong A$

by the First Isomorphism Theorem. ∎

We now define a multiplication on ${\mathbb{C}}[x,y][t]/\langle t-1,I\rangle$
which gives a deformation to $A$ at $t=1$.

######  Theorem 3.11.

Let notation be as in Proposition 3.9, and suppose that $\sum_{i=0}^{n}a_{i}$
and $\sum_{i=0}^{n}b_{i}$ generate $A$ as a $\mathbb{C}$-algebra. Let
$d_{1},\dots,d_{n}$ be free generators of the $\mathbb{C}$-algebra
$\mathbb{C}[d_{1},\dots,d_{n}]$ and suppose $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ are as in Proposition 3.2. Then the
multiplication rule

$d_{k}\ast_{t}d_{m}=\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}(t)d_{i}).$

gives a formal deformation such that $\ast_{1}$ gives the multiplication on an
algebra generated by $d_{1},\dots,d_{n}$ isomorphic to
${\mathbb{C}}[x,y][t]/\langle I,t-1\rangle$.

###### Proof.

We show that the algebra ${\mathbb{C}}[x,y][t]/\langle I,t-1\rangle$ is
isomorphic to the algebra ${\mathbb{C}}[d_{1},d_{2},\ldots d_{n}]/I^{\prime}$
where $I^{\prime}$ is the ideal of ${\mathbb{C}}[d_{1},d_{2},\ldots d_{n}]$
generated by elements

$d_{k}\ast_{1}d_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+\xi_{i,k,m}(1)d_{i})$

for $\zeta_{i,k,m}\in\mathbb{C}$, $\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ as in
Proposition 3.2.

We let
$\delta:{\mathbb{C}}[d_{1},\ldots,d_{n}]\rightarrow{\mathbb{C}}[x,y][t]/\langle
I,t-1\rangle$ be such that

$\delta(d_{i})=q_{i}+\langle I,t-1\rangle,$

for $i\in\\{1,2,\ldots,n\\}$. Observe that $I^{\prime}\subseteq\ker(\delta)$
since

$\displaystyle\delta\left(d_{k}*d_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}d_{i}+\xi_{i,k,m}(1)d_{i})\right)$
$\displaystyle=q_{k}*q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+\xi_{i,k,m}(1)q_{i})$
$\displaystyle\quad+\langle I,t-1\rangle$ $\displaystyle=\langle I,t-1\rangle$

since $q_{k}\ast
q_{m}-\sum_{i=1}^{n}(\zeta_{i,k,m}q_{i}+\xi_{i,k,m}(1)q_{i})\in I+\langle
t-1\rangle$.

Therefore the dimension of ${\mathbb{C}}[d_{1},\ldots,d_{n}]/\ker(\delta)$
does not exceed the dimension of
${\mathbb{C}}[d_{1},\ldots,d_{n}]/I^{\prime}$. Notice that
${\mathbb{C}}[d_{1},\ldots,d_{n}]/I^{\prime}$ is spanned by elements $d_{i}+I$
as a vector space, and hence has dimension at most $n$. On the other hand, by
the First Isomorphism Theorem for rings:

${\mathbb{C}}[d_{1},\ldots,d_{n}]/ker(\delta)\cong\text{im}(\delta)={\mathbb{C}}[x,y][t]/\langle
I,t-1\rangle,$ (5)

which in turn is isomorphic to $A$ by Proposition 3.9, and hence has dimension
$n$. It follows that $I^{\prime}=\ker(\delta)$ and hence
${\mathbb{C}}[d_{1},\ldots,d_{n}]/ker(\delta)$ is isomorphic to
${\mathbb{C}}[d_{1},\ldots,d_{n}]/I^{\prime}$. Where the equality in Equation
5 is true by the proof of Proposition 3.7.

∎

######  Remark 3.12.

Deformations at $t\neq 0$ The results from this section, hold with analogous
proofs for a specialisation at other $t\neq 0$. Let $z\in\mathbb{C}$ and
$z\neq 0$, the the deformation at $t=z$ is also isomorphic to $A$, provided
that $\sum_{i=0}^{n}a_{i}z^{i}$ and $\sum_{i=0}^{n}b_{i}z^{i}$ generate $A$ as
a $\mathbb{C}$-algebra.

######  Remark 3.13.

Since the dimension of all the algebras in $\\{N_{t}\\}_{t\in[0,1]}$ remains
$n=\dim(A)$ the deformation arising from $*$ is flat.

##  4 Application

In this section we prove a conjecture of M. Wemyss using the method in section
3. The statement is that of the following theorem.

######  Theorem 4.1.

Let

$N=\frac{\mathbb{C}[x,y]}{\langle xy+yx,x^{3}+y^{2},y^{3}\rangle}.$

Then $N$ deforms into $A=M_{2}(\mathbb{C})\oplus\mathbb{C}^{\oplus 5}$.

We prove this theorem through a series of lemmas. Firstly, we fix notation in
addition to the notation of Theorem 4.1.

######  Notation 4.2.

Let $i_{1},i_{2}$ be roots of the polynomial $x^{2}-x+1$.

######  Notation 4.3.

Let $e\in\mathbb{C}$ be such that for $c=\frac{i_{1}-i_{2}}{\sqrt{1+e^{2}}}$
the polynomial

$g(z)=(2z-1)^{2}z^{3}+c^{2}$

has $5$ distinct roots $\alpha_{1},\dots,\alpha_{5}\in\mathbb{C}$. Further, we
assume that $e\neq 0$, $c^{2}\neq 3$ and $\alpha_{i}\neq\frac{1}{2}.$ Notice
that $g(z)$ and $z^{3}+1$ have no common roots, so $\alpha_{i}\neq-1$ and
$\alpha_{j}\notin\\{i_{1},i_{2}\\}$.

######  Notation 4.4.

Let $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5})$ and
$\beta=(\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5})$ for
$\beta_{i}\in\mathbb{C}$ such that

$(\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5})(2\alpha_{1}-1,2\alpha_{2}-1,2\alpha_{3}-1,2\alpha_{4}-1,2\alpha_{5}-1)=c(1,1,1,1,1).$

Notice that such $\beta_{i}$ exist since $\alpha_{i}\neq\frac{1}{2}$.

######  Lemma 4.5.

Let $\xi:\mathbb{C}[x,y][t]\rightarrow A$ be a homomorphism of
$\mathbb{C}$-algebras such that

$\displaystyle\xi(x)$ $\displaystyle=\begin{pmatrix}i_{1}&0\\\
0&i_{2}\end{pmatrix}+\alpha,\quad\xi(y)=\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+\beta,\quad\xi(t)=1.$

Then $\xi(x),\xi(y)$ and $1$ generate $A$ as an algebra.

###### Proof.

Observe that $\xi(x^{j}(x^{2}-x+1))=\alpha^{j}(\alpha^{2}-\alpha+1)$ (where
$\alpha^{j}(\alpha^{2}-\alpha+1)=(\alpha_{1}^{j}(\alpha_{1}^{2}-\alpha_{1}+1),\dots,\alpha_{5}^{j}(\alpha_{5}^{2}-\alpha_{5}+1))$).
Now let $W$ be the matrix with entries
$W_{ij}=\alpha_{i}^{j}(\alpha_{i}^{2}-\alpha_{i}+1)$. Let $V$ be the
Vandermonde matrix with entries $V_{ij}=\alpha_{i}^{j}$. Let $D$ be a diagonal
matrix with entries $D_{ij}=\delta_{ij}(\alpha_{i}^{2}-\alpha_{i}+1)$. It
follows that $W=DV.$ Now since $V$ is nonsingular, $W$ is nonsingular. Now
this implies that any $5$-dimensional vector in $A$ is in $\text{im}\xi$. Then
$\xi(x)-\alpha\in\text{im}\xi$ and hence

$\begin{pmatrix}i_{1}&0\\\ 0&i_{2}\end{pmatrix}\in\text{im}\xi.$

Similarly, $\xi(y)-\beta\in\text{im}\xi$ and hence

$\begin{pmatrix}1&e\\\ e&-1\end{pmatrix}\in\text{im}\xi.$

Now recall that $i_{1},i_{2}$ are roots of the polynomial $z^{3}+1$, so we
have

$\begin{pmatrix}i_{1}&0\\\ 0&i_{2}\end{pmatrix}^{3}=\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix}.$

Notice that every diagonal matrix is a linear combination of
$\begin{pmatrix}i_{1}&0\\\ 0&i_{2}\end{pmatrix}$ and $\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix}$. Now, it can be checked that these matrices are sufficient
to generate all matrices in $M_{2}(\mathbb{C})$.

∎

######  Notation 4.6.

Let $f:\mathbb{C}[x,y][t]\rightarrow A[t]$ be a homomorphism of
$\mathbb{C}$-algebras such that

$\displaystyle f(x)$ $\displaystyle=t^{2}\begin{pmatrix}i_{1}&0\\\
0&i_{2}\end{pmatrix}+t^{2}\alpha,\quad
f(y)=\frac{1}{\sqrt{1+e^{2}}}t^{3}\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+t^{3}\beta,\quad f(t)=t\mathbbm{1},$

where $\mathbbm{1}$ is the identity element in $A$. We write
$\tilde{1}:=(1,1,1,1,1)$. As usual we denote by $1$ the identity element in
$\mathbb{C}$ and thus in $\mathbb{C}[x,y][t]$.

We now prove Theorem 4.1.

###### Proof.

Referring to Notation 4.4 and 4.6 we calculate:

$\displaystyle f(x^{2})$
$\displaystyle=t^{4}\Big{(}\begin{pmatrix}i_{1}^{2}&0\\\
0&i_{2}^{2}\end{pmatrix}+\alpha^{2}\Big{)},$ $\displaystyle f(y^{2})$
$\displaystyle=t^{6}\Big{(}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\beta^{2}\Big{)},$ $\displaystyle f(xy)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\Big{(}\begin{pmatrix}i_{1}&i_{1}e\\\
i_{2}e&-i_{2}\end{pmatrix}+\alpha\beta\Big{)},$ $\displaystyle f(yx)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\Big{(}\begin{pmatrix}i_{1}&i_{2}e\\\
i_{1}e&-i_{2}\end{pmatrix}+\alpha\beta\Big{)}.$

Note that $f(x^{2}-t^{2}x+t^{4})=t^{4}(\alpha^{2}-\alpha+\tilde{1})$ and hence

$\displaystyle f(x(x^{2}-t^{2}x+t^{4}))$
$\displaystyle=t^{6}\alpha(\alpha^{2}-\alpha+\tilde{1}),$ $\displaystyle
f(x^{2}(x^{2}-t^{2}x+t^{4}))$
$\displaystyle=t^{8}\alpha^{2}(\alpha^{2}-\alpha+\tilde{1}),$ $\displaystyle
f(y(x^{2}-t^{2}x+t^{4}))$
$\displaystyle=t^{7}\beta(\alpha^{2}-\alpha+\tilde{1}).$

Observe that

$\displaystyle f(xy+yx)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\begin{pmatrix}2i_{1}&e\\\
e&-2i_{2}\end{pmatrix}+t^{5}2\alpha\beta,$ $\displaystyle f(t^{2}y)$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+t^{5}\beta,$ $\displaystyle
f(t^{5}\frac{i_{2}-i_{1}}{\sqrt{1+e^{2}}})$
$\displaystyle=\frac{t^{5}}{\sqrt{1+e^{2}}}\begin{pmatrix}i_{2}-i_{1}&0\\\
0&i_{2}-i_{1}\end{pmatrix}+\frac{t^{5}(i_{2}-i_{1})}{\sqrt{1+e^{2}}}\tilde{1}.$

Note that $2i_{1}-1+(i_{2}-i_{1})=0$ and $-2i_{2}+1+(i_{2}-i_{1})=0$, so

$f(xy+yx-t^{2}y+\frac{t^{5}(i_{2}-i_{1})}{\sqrt{1+e^{2}}})=2\alpha\beta-\beta+\frac{i_{2}-i_{1}}{\sqrt{1+e^{2}}}\tilde{1}.$

Now notice that
$2\alpha\beta-\beta+\frac{i_{2}-i_{1}}{\sqrt{1+e^{2}}}\tilde{1}=0$ by
assumptions of Notation 4.4. It follows that
$xy+yx-t^{2}y+\frac{t^{5}(i_{2}-i_{1})}{\sqrt{1+e^{2}}}\in I$ and so
$xy+yx\in\langle J\rangle$ is a relation in $N$ as defined in Notation 3.4.

Notice that by Notation 4.4 we have $\beta^{2}(2\alpha-1)^{2}=c^{2}\tilde{1}$.
Now it follows by Notation 4.3 that

$\alpha^{3}+\beta^{2}=0,$ (6)

and thus $x^{3}+y^{2}\in\langle J\rangle$ is a relation in $N$.

We now proceed to show that $y^{3}\in\langle J\rangle$. Notice that:

$f(y^{3})=\frac{t^{9}}{\sqrt{1+e^{2}}}\begin{pmatrix}1&e\\\
e&-1\end{pmatrix}+t^{9}\beta^{3}.$

Hence $y^{3}-t^{6}y=t^{9}(\beta^{3}-\beta).$ Therefore it is sufficient to
show that $y^{3}-t^{6}y+q\in I$ for some $q\in t\mathbb{C}[x,y][t]$. We
denote:

$\displaystyle r_{1}$ $\displaystyle:=t^{5}(x^{2}-t^{2}x+t^{4})$
$\displaystyle r_{2}$ $\displaystyle:=t^{3}x(x^{2}-t^{2}x+t^{4})$
$\displaystyle r_{3}$ $\displaystyle:=tx^{2}(x^{2}-t^{2}x+t^{4})$
$\displaystyle r_{4}$ $\displaystyle:=t^{2}y(x^{2}-t^{2}x+t^{4})$

Notice that the elements

$\displaystyle f(r_{1})$
$\displaystyle=t^{9}(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{1},$ $\displaystyle
f(r_{2})$
$\displaystyle=t^{9}\alpha(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{2},$
$\displaystyle f(r_{3})$
$\displaystyle=t^{9}\alpha^{2}(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{3},$
$\displaystyle f(r_{4})$
$\displaystyle=t^{9}\beta(\alpha^{2}-\alpha+\tilde{1})=:t^{9}e_{4}$

are in $t\mathbb{C}[x,y][t]$. Note that it suffices to show that
$y^{3}-t^{6}y+q\in I$ for some
$q\in\mathbb{C}r_{1}+\mathbb{C}r_{2}+\mathbb{C}r_{3}+\mathbb{C}r_{4}$.
Therefore it suffices to show that

$\beta^{3}-\beta\in\mathbb{C}e_{1}+\mathbb{C}e_{2}+\mathbb{C}e_{3}+\mathbb{C}e_{4}.$
(7)

Now relation (7) is equivalent to

$(2\alpha-1)\beta(\beta^{2}-1)\in\mathbb{C}(2\alpha-1)e_{1}+\mathbb{C}(2\alpha-1)e_{2}+\mathbb{C}(2\alpha-1)e_{3}+\mathbb{C}(2\alpha-1)e_{4},$

which in turn is equivalent to

$-c(\alpha^{3}+1)\in\mathbb{C}(2\alpha-1)e_{1}+\mathbb{C}(2\alpha-1)e_{2}+\mathbb{C}(2\alpha-1)e_{3}+\mathbb{C}(2\alpha-1)e_{4}.$
(8)

Now, notice that $\alpha^{3}+1=(\alpha+1)(\alpha^{2}+\alpha+1)$ and that
$\alpha+1$ has non-zero entries, so multiplying relation (8) by $\alpha+1$ we
get an equivalent relation

$\displaystyle(\alpha^{3}+1)(\alpha+1)\in\mathbb{C}$
$\displaystyle(2\alpha-1)(\alpha^{3}+1)+\mathbb{C}(2\alpha-1)(\alpha^{3}+1)\alpha$
$\displaystyle+\mathbb{C}(2\alpha-1)(\alpha^{3}+1)\alpha^{2}+\mathbb{C}(\alpha^{3}+1).$

Notice that the right hand side can be rewritten as
$\mathbb{C}(\alpha^{3}+1)+\mathbb{C}(\alpha^{3}+1)\alpha+\mathbb{C}(\alpha^{3}+1)\alpha^{2}+\mathbb{C}(\alpha^{3}+1)\alpha^{3}$.

∎

##  5 A more general method

In this section we consider a more general method that results in algebras $N$
that are not necessarily local and have more generators. Furthermore, we
consider $A$ to have finitely many generators.

Let $A$ be a finitely generated $\mathbb{C}$-algebra.

  1. 1.

We consider a free $\mathbb{C}$-algebra $F$ with $j$ generators
$x_{1},\dots,x_{j}$ for some $j\in\mathbb{N}$. We define:

$\displaystyle f:F[t]$ $\displaystyle\to A[t]$ $\displaystyle x_{i}$
$\displaystyle\mapsto\sum\limits_{l=0}^{m}a_{i,l}t^{l}$

for some $m\in\mathbb{N}$ and some $a_{i,l}\in A$. We also assume that for
each $z\in\mathbb{C}$, $z\neq 0$ elements $\sum\limits_{l=0}^{m}a_{i,l}z^{l}$
for $i=1,2,\ldots j$ generate $A$. In other words, images of the generators
generate A when $t$ is set to non-zero numbers.

  2. 2.

We define

$\mathcal{N}:=\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{I},$

where $I:=\ker(f)$.

  3. 3.

We assume that for every $r\in A$ there exists $i=i(r)$ such that
$rt^{i}\in\text{im}f$.

  4. 4.

Our algebra $N=N_{0}$ is isomorphic to

$\frac{\mathcal{N}}{\langle t+I\rangle}$

and deforms flatly and associatively to $A$. Moreover $N$ is isomorphic to
$\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{\langle I,t\rangle},$ where $\langle
I,t\rangle$ is the ideal of $\mathbb{C}[x_{1},\dots,x_{j}][t]$ generated by
elements of $I$ and by $t$.

If $j=2$ then this method corresponds to the method described in section 3
provided that $a_{i,0}=0$ for all $i$.

Proof. When reasoning as in the previous chapters we will use at all places
$\mathbb{C}[x_{1},\ldots,x_{j}]$ instead of $\mathbb{C}[x,y]$, and we will use
$\mathbb{C}[x_{1},\ldots,x_{j}][t]$ instead of $\mathbb{C}[x,y][t]$. Let $A$
have dimension $n$ over $\mathbb{C}$. Recall that we

$\mathcal{N}:=\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{I},$

where $I:=\ker(f)$.

  * •

Reasoning analogously as in Proposition 3.1 we obtain that there are
$q_{1},\ldots,q_{n}\in F$ such that

$f(F[t])\subseteq\sum_{i=1}^{n}{\mathbb{C}}f(q_{i}),$

and hence, since $I=\ker f$, we get

$F(t)\subseteq\sum_{i=1}^{n}{\mathbb{C}}q_{i}+I.$

In particular, there are $\zeta_{i,k,m}\in\mathbb{C}$,
$\xi_{i,k,m}(t)\in{\mathbb{C}}[t]$ such that

$q_{k}\cdot q_{m}-(\sum_{i=1}^{n}\zeta_{i,k,m}q_{i}+t\xi_{i,k,m}q_{i})\in I.$

  * •

Reasoning similarly as in Theorem 3.8 we obtain that if $d_{1},\ldots,d_{n}$
are free generators then

$d_{k}*d_{m}-(\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}d_{i}),$

gives a formal deformation such that $*_{0}$ gives a multiplication on $N$
where $N$ is the $\mathbb{C}$-algebra with generators $d_{1},\ldots,d_{n}$
subject to relations

$d_{k}*_{0}d_{m}=\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}.$

Observe that $*$ defines a multiplication on the algebra
$\mathbb{C}[d_{1},\ldots,d_{n}][t]/I^{\prime}$, where $I^{\prime}$ is the
ideal of $\mathbb{C}[d_{1},\ldots,d_{n}][t]$ generated by elements

$d_{k}*d_{m}-(\sum_{i=1}^{n}\zeta_{i,k,m}d_{i}+t\xi_{i,k,m}d_{i}).$

Notice that $d_{1},\ldots,d_{n}$ are free generators of the free algebra
$\mathbb{C}[d_{1},\ldots,d_{n}]$. Notice that the algebra
$\mathbb{C}[d_{1},\ldots,d_{n}][t]/I^{\prime}$ is isomorphic to algebra
$\mathcal{N}$. To see this consider the map
$\sigma\mathbb{C}[d_{1},\ldots,d_{n}][t]\rightarrow{\mathcal{N}}$ given by

$d_{i}\rightarrow q_{i}+I.$

Notice that $I^{\prime}\subseteq\ker\sigma$ since

$(q_{k}+I)*(q_{m}+I)=\sum_{i=1}^{n}\zeta_{i,k,m}(q_{i}+I)+t\xi_{i,k,m}(q_{i}+I).$

Recall that $\ker\sigma=I^{\prime}$ because if $a\in ker\sigma$ then by using
relations from $I^{\prime}$ we can present $a$ as $a^{\prime}+i$ where
$i^{\prime}\in I$ and where $a^{\prime}=\sum_{i=1}^{n}\alpha_{i}(t)d_{i}$ for
some $\alpha_{i}(t)\in{\mathbb{C}}[t]$. Notice that then
$\sigma(i^{\prime})=0$ and hence

$\sigma(a)=\sigma(a^{\prime})=\sigma(\sum_{i=1}^{n}\alpha_{i}(t)d_{i})=\sum_{i=1}^{n}\alpha_{i}(t)q_{i}.$

Reasoning as in Corollary $3.3$ we get that this implies that all
$\alpha_{i}=0$, hence $a^{\prime}=0$, hence $a=i^{\prime}$. Therefore,
$\ker\sigma\subseteq I$ and consequently $\ker\sigma=I^{\prime}$.

Therefore, by the First Isomorphism Theorem for algebras
${\mathbb{C}}[d_{1},\ldots,d_{n}][t]/I^{\prime}$ is isomorphic to $Im(\sigma)$
which is equal to ${\mathcal{N}}$. To see that $\sigma$ is surjective, notice
that $q_{1}+I,\ldots,q_{n}+I$ span $\mathcal{N}$ as a linear space (as
mentioned before this can be proved similarly as Proposition 3.1).

  * •

Reasoning as in Proposition 3.6 we obtain that $N$ is $n$ dimensional
$\mathbb{C}$-algebra.

  * •

Reasoning similarly as in Proposition 3.7 we obtain that the algebra $N$ is
isomorphic to the algebra $\frac{\mathbb{C}[x_{1},\dots,x_{j}][t]}{\langle
I,t\rangle}$ where ${\langle I,t\rangle}$ is the ideal of
$\mathbb{C}[x_{1},\dots,x_{j}][t]$ generated by elements from $I$ and by $t$.
By the Third Isomorphism Theorem $N$ is isomorphic to

$\frac{\mathcal{N}}{\langle t+I\rangle}.$

This can be seen by reasoning similarly as in Proposition 3.10 but taking $t$
instead of $t-1$ in the proof.

  * •

We will now consider deformations $*_{1}$ at $t=1$. Consider first the
homomorphism of $\mathbb{C}$-algebras $\xi:F[t]\rightarrow A$ given by
$\xi(x_{i})=\sum_{l=0}^{m}a_{i,l}$ for $i=1,2,\ldots,j$.

Reasoning similarly as in Proposition 3.9 we obtain that $\ker\xi=\langle
I,t-1\rangle$. Reasoning analogously as in the last two lines of Proposition
3.10 we obtain that $F[t]/\ker\xi$ is isomorphic to $A$.

  * •

Reasoning as in Theorem 3.11 we see that the algebra
${\mathbb{C}}[d_{1},\ldots,d_{n}][t]/I$ deforms at $t=1$ to the algebra
$F[t]/{\langle I,t-1\rangle}=F[t]/{\ker\xi}$ which is isomorphic to $A$.

  * •

We will now consider deformations $*_{z}$ at $t=z$ where $z\in\mathbb{C},z\neq
0,$ $z\neq 1$. Consider first the homomorphism of $\mathbb{C}$-algebras
$\xi:F[t]\rightarrow A$ given by $\xi(x_{i})=\sum_{l=0}^{m}a_{i,l}z^{l}$ for
$i=1,2,\ldots,j$.

Reasoning similarly as in Proposition 3.9 we obtain that $\ker\xi=\langle
I,t-z\rangle$ (this can be done by using $t^{\prime}=t/z$ at the place of $t$
in the proof of Proposition 3.9, and in the first line declaring that
$p_{i}\in F$ instead of $p_{i}\in{\mathbb{C}}[x,y]$).

Next, reasoning analogously as in the last two lines of Proposition 3.10 we
obtain that $F[t]/\ker\xi$ is isomorphic to $A$ (this follows since elements
$\xi(x_{i})$ generate $A$).

Next, reasoning as in Theorem 3.11 we see that the algebra
${\mathbb{C}}[d_{1},\ldots,d_{n}][t]/I$ deforms at $t=z$ to the algebra
$F[t]/{\langle I,t-z\rangle}=F[t]/{\ker\xi}$ which is is isomorphic to $A$. ∎

##  6 Future Work

In this section we pose some questions about Method 1 and suggest some
generalisations.

######  Question 6.1.

Let $A$ be a $\mathbb{C}$-algebra and suppose $G_{1}=\\{a,b\\}$ and
$G_{2}=\\{a^{\prime},b^{\prime}\\}$ are two distinct generating sets of $A$.
Under what conditions does Method 1 using $A,G_{1}$ and $A,G_{2}$ result in
the same algebra $N_{0}$ that has $A$ as a deformation?

######  Question 6.2.

Let $A$ be a $\mathbb{C}$-algebra. Under what conditions does Method 1 only
produce one algebra $N$ that has $A$ as a deformation.

######  Question 6.3.

Let $A$ be a $\mathbb{C}$-algebra. Does their exist a finitely terminating
algorithm that will produce all the algebras $N$ arising from Method 1 that
have $A$ as a deformation?

######  Question 6.4.

Let $A$ be a $\mathbb{C}$-algebra. Do ring theoretic properties of the
generators $a,b$ (for example being idempotent, being irreducible idempotent,
prime) determine any properties of the algebra $N$ arising from Method 1 using
$A,\\{a,b\\}$?

######  Question 6.5.

Let $A$ be a $\mathbb{C}$-algebra. Do ring theoretic properties of $A$ (for
example being semi-simple, local) determine any properties of the algebra $N$
arising from Method 1 using $A$?

######  Question 6.6.

Can Method 1 be generalised to be used for algebras over a general commutative
ring?

#### Acknowledgements

We are grateful to Michael Wemyss for suggesting his interesting questions
which inspired this paper, and for useful comments about contraction algebras
and their applications in geometry. The third author acknowledges support from
the EPSRC programme grant EP/R034826/1 and from the EPSRC research grant
EP/V008129/1.

## References

  * [Ber78] George M Bergman.  The diamond lemma for ring theory.  Advances in Mathematics, 29(2):178–218, February 1978.
  * [BW22] Gavin Brown and Michael Wemyss.  Local normal forms of noncommutative functions, 2022.
  * [CGG89] Vincent Coll, Murray Gerstenhaber, and Anthony Giaquinto.  An explicit deformation formula with noncommuting derivations.  Israel Math. Conf. Proc., 1:396–403, 01 1989.
  * [DW16] Will Donovan and Michael Wemyss.  Noncommutative deformations and flops.  Duke Mathematical Journal, 165(8), June 2016.  arXiv:1309.0698 [math].
  * [FO19] Boris Feigin and Alexandre Odesski.  Flat deformations of algebras and functional equations.  Journal of Combinatorial Algebra, 3(3):215–236, September 2019\.
  * [HT18] Zheng Hua and Yukinobu Toda.  Contraction algebra and invariants of singularities.  Int. Math. Res. Not. IMRN, (10):3173–3198, 2018.
  * [SW13] Anne V. Shepler and Sarah Witherspoon.  PBW deformations of skew group algebras in positive characteristic, December 2013.  arXiv:1312.3616 [math].
  * [Tod15] Yukinobu Toda.  Non-commutative width and Gopakumar-Vafa invariants.  Manuscripta Math., 148(3-4):521–533, 2015.
  * [Wem23] Michael Wemyss.  A lockdown survey on cdv singularities, 2023.
  * [Wit19] Sarah J. Witherspoon.  Hochschild cohomology for algebras, volume 204 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, [2019] ©2019.


#  Extracting non-local inter-polaron interactions from collisional dynamics

Junichi Takahashi  Department of Electronic and Physical Systems, Waseda
University, Tokyo 169-8555, Japan  Hiroyuki Tajima  Department of Mathematics
and Physics, Kochi University, Kochi 780-8520, Japan  Quantum Hadron Physics
Laboratory, RIKEN Nishina Center, Wako, Saitama 351-0198, Japan  Eiji Nakano
Department of Mathematics and Physics, Kochi University, Kochi 780-8520, Japan
Kei Iida  Department of Mathematics and Physics, Kochi University, Kochi
780-8520, Japan

###### Abstract

This study develops a novel experimental method of deducing the profile of
interaction induced between impurities in a trapped gas of ultracold
Fermi/Bose atoms, which are often referred to as Fermi/Bose polarons. In this
method, we consider a two-body Fermi/Bose polaron collision experiment in
which impurities and atoms interact only weakly. Numerical simulations of the
quantum dynamics reveal the possibility to obtain information regarding the
non-local induced interaction between two polarons from a measured profile of
the polaron wave packet at several snapshots. This is because the potential of
the induced interaction is well balanced by the quantum potential whenever the
WKB approximation for the relevant Schrödinger equation is applicable.

Introduction— To describe interparticle interactions is indispensable in
various fields of physics. At extremely high energy, fundamental interactions
are mediated by gauge bosons Book_Weinberg1995 , while in atomic nuclei, the
nuclear force at a distance occurs through exchange of mesons Taketani1951 .
In conventional superconductors, attractive electron–electron interactions are
induced by phonons Bardeen1957 . In high-$T_{\rm c}$ superconductors,
furthermore, background spin and/or charge fluctuations are expected to play a
role Scalapino1995 .

Ultracold atomic systems can be used as a platform for studying medium-induced
interactions. Particularly, such systems can exhibit high tunability and
pureness, thereby serving as excellent quantum simulators for studying quantum
many-body theories Bloch2008 ; Giorgini2008 ; Dalfovo1999 . For example, cold
atomic Fermi systems with a long scattering length close to the unitary limit
are good simulators with regard to low density neutron matter Horikoshi2019 .
They also possess the unique feature of having the real-time dynamics and
momentum distribution measured Pethick-Smith . Moreover, atomic Fermi/Bose
polarons, which are quasiparticles that consist of impurities (minority atoms)
immersed in a degenerate Fermi gas Schirotzek2009 ; Frohlich2011 ;
Kohstall2012  or a Bose-Einstein condensate Jorgensen2016 ; Hu2016 ;
Rentrop2016 , have been realized. Since the impurities have the mass and the
interaction with each other modified from those in vacuum by the impurity-
medium interaction, a great deal of research has been conducted both from
theoretical and experimental sides. Theoretically, the Ruderman-Kittel-Kasuya-
Yoshida (RKKY) type Ruderman1 ; Kasuya1 ; Yoshida1  and Yukawa/Efimov type
interactions have been proposed as typical examples of non-local induced
interactions between heavy Fermi and Bose polarons Nishida2009 ; Suchet2017 ;
Heiselberg2000 ; Yu2012 ; Enss2020 ; Naidon2018 , respectively. In connection
with such induced interactions, moreover, bi-polaron formation for Bose
polarons Naidon2018 ; Camacho-Guardian2018 ; Dehkharghani2018 ; Volosniev2015
; Mistakidis2019  and the energy shift and broadening of Fermi polarons
Giraud2012 ; Mistakidis2019c ; Tajima2018  have been actively discussed.
Recently, the presence of non-local interactions between Fermi polarons has
been experimentally confirmed DeSalvo2019 ; Edri2020 . Here, the formation of
solitons through the corruption of a Bose gas was found to be different from
the behavior predicted when only local interactions occur in a boson-fermion
mixture, which is an evidence for the existence of a non-local interaction
mediated by fermions. The non-local nature of such interactions, however,
remains to be investigated experimentally. Given that similar induced
interactions occur in various fields of physics, therefore, real cold atom
experiments in this direction would serve as a cornerstone.

We now address the question “What is enough information to find induced
interactions between particles in a medium?” For concreteness, we consider the
case in which two polarons are approximately described by a two-body time-
dependent Schrödinger equation (TDSE) with a mediated interaction instead of
dealing with the medium explicitly. In this case, the simplest answer is a
two-body polaron wavefunction. That is, the interaction can be obtained from
the wavefunction by transforming the TDSE as
$U(r)=\Psi^{-1}\left[i\hbar\frac{\partial}{\partial t}-h_{1}({\bf
x}_{1})-h_{2}({\bf x}_{2})\right]\Psi$, where $\Psi$ is the wavefunction,
$h_{i}$ denotes the one-particle Hamiltonian, $r=|{\bf x}_{1}-{\bf x}_{2}|$,
and $U$ denotes the interaction between two polarons. This means that one can
in principle constitute the inter-polaron interaction $U$ as function of the
relative coordinate if the $\Psi$ keeps nonvanishing in the range of
$U(r)\not=0$ throughout such experiments as the collisional ones Sommer2011 ;
Joseph2011 ; Valtolina2017 ; Reynold2020 . In reality, however, it is
difficult to simultaneously obtain the phase and amplitude of the wavefunction
by experiments. In fact, at a given snapshot, only the amplitude of the
wavefunction can be measured via the square root of the probability density of
the wavefunction. For trapped cold atomic systems, one can use, e.g., a wave
packet of a non-interacting Bose gas as an impurity and then identify the
number density of the Bose gas with the probability density of the impurity.
In this study, we propose a method of deducing the non-local interaction
between two such impurities in a medium from the density of the Bose gas at
several snapshots. It is noteworthy that the idea of deriving bulk
thermodynamic quantities from the density of a single-species cold atomic gas
has been already proposed Ho2010 . Despite similarity in strategy, there is a
crucial difference from our proposal. Authors in Ref. Ho2010  use the density
distribution to obtain the thermodynamic properties of the corresponding
homogeneous system within local density approximation (LDA), while information
about the non-local interaction between atoms in a trap, which is of interest
here, is out of the reach of LDA.

Formulation— For future possible experimental realizations, we focus on a two-
polaron collision in a harmonic trap filled with a Fermi/Bose atomic gas. Each
impurity, which is a Bose condensate wave packet, is initially localized by a
separate confining harmonic potential, released at $t=0$, and allowed to
collide with another impurity. Note that experimental setups that involve a
scenario highly similar to that considered in this study have been proposed
Mistakidis2019b ; Magierski2019 ; Kwasniok2020 ; Tajima2020 , while a part of
them have been realized in experiment by using optical tweezers Reynold2020 .

We assume that at $t=0$, each impurity has already been immersed in the
majority gas long enough to form a polaron and that at $t\geq 0$ a medium-
induced interaction occurs between the polarons. Furthermore, we consider the
polarons to be distinguishable particles that do not interact directly with
each other and to remain robust during the time evolution owing to coupling
between the impurity and the majority gas that is too weak for the entire
system to be relaxed. In a regime of strong coupling, on the other hand, a
hydrodynamic description looks more relevant. In fact, a hydrodynamic analysis
of collisions of two polaronic clouds in a similar setup has been performed
Tajima2020 ; in this case, induced inter-polaron interactions have been
incorporated even inside a cloud but only within the LDA.

In the weak coupling regime of interest here, the dynamics of the two polarons
essentially obeys the TDSE: $i\hbar\partial_{t}\Psi=[h({\bf x}_{1})+h({\bf
x}_{2})+U_{\rm med}(r)]\Psi$, where $h({\bf x}_{i})=-\hbar^{2}\nabla^{2}_{{\bf
x}_{i}}/2m+V_{\rm HO}({\bf x}_{i})$ with the trap potential $V_{\rm HO}({\bf
x}_{i})=m\omega^{2}{\bf x}_{i}^{2}/2$ and $U_{\rm med}(r)$ denotes an
interaction mediated by the majority gas. Here, for simplicity, we assume that
the trap frequency $\omega$ and the bare mass $m$ of an impurity atom are the
same between the polarons and that the dynamics is 1D. We next rewrite the
TDSE in the center-of-mass frame by using the relative coordinate
$x=x_{1}-x_{2}$ as $i\hbar\frac{\partial\varphi}{\partial
t}=\left[-\frac{\hbar^{2}}{2m_{\rm r}}\frac{\partial^{2}}{\partial
x^{2}}+V_{\rm r,HO}(x)+U_{\rm med}(|x|)\right]\varphi$ where $m_{\rm r}=m/2$
is the reduced mass of the impurities, $V_{\rm r,HO}(x)=\frac{1}{2}m_{\rm
r}\omega^{2}x^{2}$, and $\varphi(x,t)$ is the relative wavefunction of the
polarons. We then substitute the wavefunction
$\psi=\sqrt{\rho(x,t)}\exp[iS(x,t)/\hbar]$ into the TDSE, thereby obtaining
the coupled equations that are the equation of continuity and the quantum
Hamilton–Jacobi equation Book_Tannor2007

$\displaystyle C(S;x,t)+Q(\rho;x,t)+U_{\rm med}(|x|)=0,$ (1)

with $C(S;x,t)=\frac{\partial S}{\partial t}+\frac{1}{2m_{\rm
r}}\left(\frac{\partial S}{\partial x}\right)^{2}+V_{\rm r,HO}$ and
$Q(\rho:x,t)=-\frac{\hbar^{2}}{2m_{\rm
r}\sqrt{\rho}}\frac{\partial^{2}\sqrt{\rho}}{\partial x^{2}}$. Here, $C$ is
the potential given by the phase fluctuations, while $Q$ is the potential
given by the density fluctuaions, which is often referred to as the quantum
potential. Equation (1) shows that the interaction $U_{\rm med}$ is determined
by the balance of those potentials.

Analysis and Results — We proceed to solve the TDSE for a particular
interaction and obtain density profiles of a polaron to calculate the quantum
potential. Since each polaron is initially trapped in the respective confining
harmonic potential, we set the initial two-polaron wavefunction as a pair of
the Gaussian wave packets via
$\Psi(x_{1},x_{2},t=0)=\phi_{+}(x_{1})\phi_{-}(x_{2})$, with
$\phi_{\pm}(x)=\frac{1}{(2\pi\eta^{2})^{\frac{1}{4}}}\exp\left[-\frac{(x\pm
x_{0})^{2}}{4\eta^{2}}\right],$ where the two parameters $x_{0}$ and $\eta$
controls the initial position and the width of the respective wave packet. For
such nodeless wave packets, the WKB approximation is expected to be good. We
solve the TDSE under the corresponding initial condition
$\varphi(x,t=0)=\frac{1}{(4\pi\eta^{2})^{\frac{1}{4}}}\exp\left[-\frac{(x-2x_{0})^{2}}{8\eta^{2}}\right],$
by using the second-order split-step Fourier method. We adopt (i) RKKY, (ii)
Yukawa, and (iii) Efimov type interactions as $U_{\rm med}$. We note that the
interactions (i)–(iii), which are derived for a 3D homogeneous medium, are
used for simplicity and that the concrete form of the interactions is not
important for the present purpose of finding a way of capturing the mediated
interactions through experiments unless the range of the interactions is
significantly long.

(i) The RKKY type interaction is as follows:

$\displaystyle U_{\rm med}(|x|)=\frac{\hbar^{2}Ma^{2}}{2\pi M_{\rm
r}^{2}}\frac{2k_{{\rm F}}|x|\cos(2k_{\rm F}|x|)-\sin(2k_{\rm
F}|x|)}{|x|^{4}},$ (2)

where $a$ is the $s$-wave scattering length between a majority atom of mass
$M$ and an impurity, taken to be independent of which impurity the majority
atom interacts with, $k_{\rm F}$ is the Fermi wavenumber of the majority gas,
and $M_{\rm r}=mM/(m+M)$. Figure 1 presents the probability densities for the
relative two-polaron wave packet before and after the collision of a pair of
the polaron wave packets that evolve from various initial conditions under the
influence of $U_{\rm med}$. We set the number of atoms in the majority gas as
a typical one, $N=10^{5}$, while assuming $(k_{\rm F}a)^{2}=2$ and $M=m$. It
is interesting to note that before the collision, the width of the two-polaron
relative wave packet increases (decreases) with $t$ in the case of Fig. 1(d)
(Fig. 1(a)), while remaining almost unchanged with $t$ in the cases of Figs.
1(b) and 1(c). This behavior stems from the fact that the Gaussian wave packet
for a free particle with mass $m_{\rm r}$ has the width increased with time
like $\sqrt{2}\eta\sqrt{1+(\hbar t/4m_{\rm r}\eta^{2})^{2}}$, while, as two
polarons approach each other, the confining potential acts to reduce the width
of the wave packet like $\hbar/\sqrt{2m_{\rm r}(E-V_{\rm r,HO})}$ with the
relative energy $E$. Incidentally, the collision time is of order $t_{c}$
(quarter of the dipole oscillation period) in the cases of Figs. 1(a)–(c),
while being well before $t_{c}$ in the case of Fig. 1(d).

Figure 1:  Probability density of the two-polaron relative wave packet (time
vs. relative position) for (a) $x_{0}/R_{\rm F}=0.5,\,\eta/R_{\rm F}=0.1$, (b)
$x_{0}/R_{\rm F}=0.5,\,\eta/R_{\rm F}=0.05$, (c) $x_{0}/R_{\rm
F}=0.25,\,\eta/R_{\rm F}=0.05$, and (d) $x_{0}/R_{\rm F}=0.25,\,\eta/R_{\rm
F}=0.01$. Here, $R_{\rm F}$ and $E_{\rm F}$ denote the Thomas–Fermi radius and
Fermi energy, respectively. The Fermi energy is given by $E_{\rm
F}=\frac{1}{2}m\omega^{2}R_{\rm F}^{2}=(6N)^{\frac{1}{3}}\hbar\omega$. The
time $t_{c}$ at which the center of the relative wave packet reaches $x=0$ can
be estimated from $t_{c}\omega_{\rm
F}\simeq\frac{1}{4}\frac{2\pi}{\omega}\omega_{\rm F}\simeq 125,$ where
$\omega_{\rm F}=E_{\rm F}/\hbar$.  Figure 2:  Relative density profile
$\rho(x,t)$ at three values of $t\omega_{\rm F}$ of order or greater than the
collision time. The distinction between (a)–(d) corresponds to the difference
in initial parameters as in Fig. 1.  Figure 3:  Same as Fig. 2 for minus the
quantum potential. For comparison, the RKKY interaction employed to solve the
TDSE is also plotted in dotted line. For detailed movies, see Animation .

Figure 2 depicts the relative density $\rho(x,t)$ at three different times,
most of which are close to the collision time extracted from Fig. 1. Numerical
data in the regions of $x<0$ and $x>0$ represent the behavior of the
transmitted component and of the incident and reflected components of the
relative wave packet, respectively. Once this kind of profiles are obtaind by
experiments, one can use them as input data in deriving the quantum potential.

We proceed to exhibit, in Fig. 3, minus the quantum potential $-Q(\rho;x,t)$
that can be derived from pseudo data for $\rho(x,t)$ shown in Fig. 2. One can
observe from Fig. 3 that the quantum potential in the transmitted wave regime
($x<0$) well reproduces the oscillating pattern of the RKKY interaction for
any initial condition, while the reproducibility of the RKKY interaction
itself depends on the initial condition. Comparing Figs. 3 (a)–(c), in which
cases the width of the transmitted relative wave packet just after the
collision is different, we find it advantageous for the transmitted wave
packet to be sufficiently spread. Remarkably, Fig. 3 (d) shows an even better
reproducibility of the interaction for the initial condition that allows the
two respective wave packets to spread and merge well before the centers of
these packets come together at the origin at a timescale $t_{c}$ of the dipole
oscillation. This supports the tendency that the wider the transmitted wave
packet, the higher the reproducibility of the interaction from the quantum
potential. This tendency in turn ensures $C(S;x,t)\simeq 0$ and $U_{\rm
med}\simeq-Q(\rho;x,t)$ in the region of $x<0$ where the WKB approximation is
a good approximation because of the sufficiently spread wave packet and gentle
spatial dependence of the confining potential.

In the region of $x>0$ in each panel of Figs. 1 and 2, on the other hand, a
density oscillation pattern emerges after the collision. This pattern stems
naturally from interference between the incident and reflected components of
the relative wave packet, while the presence of the reflected component arises
mainly from a repulsive part of $U_{\rm med}$. As can be seen from Fig. 3,
however, the periodicity of this density oscillation is totally different from
that of $U_{\rm med}$. In other words, such quantum interference leads
inevitably to $C\not\simeq 0$. Then, the WKB approximation is no longer valid,
and hence the mediated interaction cannot be reproduced by $-Q$.

Let us turn to the (ii) Yukawa and (iii) Efimov type interactions:

$\displaystyle U_{\rm
med}(|x|)=\begin{cases}U_{0}\frac{\exp(-\kappa_{0}|x|)}{\kappa_{0}|x|}&({\rm
Yukawa}\,{\rm type}),\\\ U_{1}\frac{\kappa_{1}^{-2}}{|x|^{2}}&({\rm
Efimov}\,{\rm type}).\end{cases}$ (3)

We again confine ourselves to sufficiently spread relative wave packets for
the WKB approximation to hold at least in the region of $x<0$. Figure 4 shows
minus the relative quantum potentials obtained by numerically simulating
collision dynamics of two Bose polarons that interact with (ii) or (iii). We
can observe that the quantum potential tends to well reproduce the respective
mediated interaction in the regime ($x<0$), just like the RKKY case shown in
Fig. 3. Comparing Figs. 4 (a) and (b), in which cases the initial position of
the relative wave packet is different under the same Yukawa interaction, one
can see that $U_{\rm med}$ is more reproducible in the former than in the
latter. On the other hand, comparing Figs. 4 (c) and (d), which are the Efimov
counterparts to Figs. 4 (a) and (b), one can realize that $U_{\rm med}$ is
more reproducible in the latter than in the former. These results imply that
tuning of the initial energy would be desirable for better reproduction of
$U_{\rm med}$ in a manner that depends on the range of the mediated
interaction. We remark in passing that in contrast to the RKKY case, almost no
density oscillation emerges in the region of $x>0$ for the Yukawa and Efimov
type interactions, whose purely attractive nature produces almost no reflected
wave packet. Then, the WKB approximation is expected to be valid even for
$x>0$. A significant deviation of $-Q$ from $U_{\rm med}$ that can be observed
in this region for the Efimov type interaction, therefore, suggests that the
long-range nature distorts the incident component of the relative wave packet
even at a semiclassical level.

Figure 4:  Minus the quantum potential at three different times as plotted for
the Yukawa (upper panels) and Efimov (lower panels) type interactions with
$U_{0}/\hbar\omega=U_{1}/\hbar\omega=-0.1$ and $\kappa_{0}R_{\rm
HO}=\kappa_{1}R_{\rm HO}=20$, where $R_{\rm HO}=\sqrt{\hbar/m_{r}\omega}$. In
panels (a) and (c) [(b) and (d)], the initial parameters are set to
$x_{0}/R_{\rm HO}=0.25$ and $\eta/R_{\rm HO}=0.05$ ($x_{0}/R_{\rm HO}=1$ and
$\eta/R_{\rm HO}=0.05$). The dotted line in each panel denotes the mediated
interaction involved. For detailed movies, see Animation .

Summary— This study has developed a novel practical method of deducing the
induced interactions between two impurities from a measured profile of the
polaron wave packet at several snapshots of a two-polaron collisional dynamics
in the case in which impurities and medium atoms interact only weakly. The key
to success in this method is the validity of the WKB approximation, which is
satisfied for sufficiently spread polaron wave packets. We have successfully
demonstrated by solving the TDSE the possibility of reproducing such
interactions as the RKKY, Yukawa, and Efimov type from the measured profile of
the wave packets via the quantum potential.

In real possible experiments, one would be prepared to confine a degenerate
Fermi gas or Bose condensate with a sufficiently large trap and treat two non-
interacting Bose gases with different internal states as impurities by putting
each of them in a sufficiently narrow trap that is located at a symmetrical
position with respect to the large trap of a majority gas and then by
releasing the two Bose gases from the trap at the same time. We expect that
observation of transmitted wave packets after the collision would be possible
owing to the different internal states, which would help to distinguish
between two impurities. We emphasize that our strategy to deduce the mediated
interaction from the probability density of the two-polaron wavefunction is
different from the method of estimating the potential from a phase shift as
used in an inverse scattering method InverseScattering .

In general, the full profile of the medium-induced non-local interaction that
occurs between particles is theoretically unknown. In this regard, we believe
that the method proposed here could help to easily and quantitatively obtain
information regarding mediated interactions from experiments. It is
nevertheless foreseeable that in actual experiments the interaction cannot be
described by the type of mediated interactions used in this study due to
various effects such as the finite volume one. If the experiment is carried
out under appropriate initial conditions, however, the resultant deviation
between the empirically deduced interactions and the full interaction could be
duly reduced. Also, our proposed method might open an opportunity to study
nuclear interactions that are in principle microscopically known from QCD
Ishii2007 ; Ishii2012 ; Iritani2019 , from a different viewpoint that utilizes
a cold atomic system as a quantum simulator. As a next step, we will attempt
to test our method for 3D collision systems and investigate the applicability
to general interactions.

Acknowledgments— We would like to thank K. Nishimura, T. Hata, K. Ochi, and Y.
Yamanaka for useful discussions. This work was supported in part by Grants-in-
Aid for Scientific Research from JSPS (Nos. 17K05445, 18K03501, 18H05406,
18H01211, and 19K14619).

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Supplemental online material for: “Extracting non-local inter-polaron
interactions from collisional dynamics”

In the supplimental material, we show the list of movies.

List of movies:

  * 1.

File: `RKKY_x0_050_eta_010.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(a) and 3(a).
Youtube: `https://youtu.be/By_ZALuI6A8`

  * 2.

File: `RKKY_x0_050_eta_005.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(b) and 3(b).
Youtube: `https://youtu.be/dTKHjs05SfQ`

  * 3.

File: `RKKY_x0_025_eta_005.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(c) and 3(c).
Youtube: `https://youtu.be/9iXiWWM9H6o`

  * 4.

File: `RKKY_x0_025_eta_001.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Figs. 2(d) and 3(d).
Youtube: `https://youtu.be/Q74kZCrYWrY`

  * 5.

File: `Yukawa_x0_025_eta_005.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(a).
Youtube: `https://youtu.be/nTRTT_8gGME`

  * 6.

File: `Yukawa_x0_100_eta_005.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(b).
Youtube: `https://youtu.be/tJkNVh91SrY`

  * 7.

File: `Efimov_x0_025_eta_005.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(c).
Youtube: `https://youtu.be/fK-G-StErAM`

  * 8.

File: `Efimov_x0_100_eta_005.mp4`

Dynamics of the relative density and minus the quantum potential that
corresponds to Fig. 4(d).
Youtube: `https://youtu.be/2RqUnCu-Shg`