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\title{A family of finite dimensional Hopf algebras\texorpdfstring{\\}{ }
over \texorpdfstring{\(\C\)}{C} with varying algebra structure}
\author{ChatGPT 6 Pro}
\date{November 8, 2026}

\begin{document}
\maketitle

\begin{abstract}
We construct an algebraic one-parameter family of complex Hopf algebras
of dimension \(1024\), with fixed coalgebra, whose underlying associative
algebras vary in a continuous family of isomorphism classes. Four
multiplication lines in the second radical layer of a distinguished
block recover the \(j\)-invariant of the elliptic curve
\(y^2=x(x-1)(x-t^2)\). This answers negatively the finiteness question
of Etingof and Gelaki concerning algebra types of Hopf algebras of
fixed dimension.
\end{abstract}

\section*{AI disclaimer}
\addcontentsline{toc}{section}{AI disclaimer}
\noindent\textbf{This text has been produced by ChatGPT at a request by Pavel Etingof.
It was repeatedly checked by AI agents and revised in response to their comments.
Pavel's substantive prompts are reproduced in chronological order at the end.
Feel free to use at your discretion. If you rewrite it or build on it, you may say
that it was inspired by this note.}

\tableofcontents

\section{Introduction}

Etingof and Gelaki ask in \cite[p.~2 of the arXiv version]{EG} whether
there are only finitely many associative algebra isomorphism classes
underlying complex Hopf algebras of a given dimension. We construct
a continuous family with infinitely many algebra types.

\begin{theorem}\label{thm:main}
There exists a Hopf algebra \(\mathcal H\), free of rank \(1024\) over
\(\C[t]\), with constant underlying coalgebra, whose fibers \(H_t\)
have the following property. For \(t,t'\in T:=\C\setminus\{0,1,-1\}\),
\[
H_t\cong H_{t'}\ \text{as associative algebras}
\quad\Longrightarrow\quad
\JJ(t)=\JJ(t'),
\]
where
\begin{equation}\label{eq:invariant}
\JJ(t)=256\frac{(t^4-t^2+1)^3}{t^4(t^2-1)^2}.
\end{equation}
Each associative algebra isomorphism class occurs at most twelve times
on \(T\). On some open complex disc \(U\subset T\), the fibers \(H_t\)
are pairwise nonisomorphic as associative algebras.
\end{theorem}

\section{The Hopf family}\label{sec:family}

We use the convention \([x,y]=xy-yx\). The algebra \(H_t\) has commuting
generators \(g,r,s\) satisfying \(g^4=r^2=s^2=1\), and four further
generators \(x_1,y_1,x_2,y_2\). Put
\[
a=gr,\qquad b=gs,\qquad c=g^2,\qquad d=ab=c\,rs.
\]
Thus \(c\) is central, \(a^2=b^2=c\), and \(d^2=1\). The group action is
specified by the table below, where an entry \(\epsilon\) means
\(hz=\epsilon zh\):
\begin{equation}\label{eq:action}
\begin{array}{c|rrrr}
 &x_1&y_1&x_2&y_2\\ \hline
g&-1&1&-1&1\\
r&1&1&1&-1\\
s&1&-1&1&1
\end{array}.
\end{equation}
The remaining relations are
\begin{equation}\label{eq:relations}
\begin{aligned}
x_1^2=x_2^2&=\frac{1-c}{2},&
y_1^2=y_2^2&=0,\\
x_1x_2+x_2x_1&=t(1-d),&
y_1y_2&=y_2y_1,\\
x_1y_2+y_2x_1&=0,&
x_2y_1+y_1x_2&=0,\\
[x_1,y_1]^2&=0,&
[x_2,y_2]^2&=0.
\end{aligned}
\end{equation}
The elements \(g,r,s\) are group-like, and
\begin{equation}\label{eq:coproduct}
\begin{aligned}
\Delta(x_1)&=x_1\otimes1+a\otimes x_1,&
\Delta(y_1)&=y_1\otimes1+b\otimes y_1,\\
\Delta(x_2)&=x_2\otimes1+b\otimes x_2,&
\Delta(y_2)&=y_2\otimes1+a\otimes y_2.
\end{aligned}
\end{equation}
The counit is zero on \(x_i,y_i\). For a generator \(z\) with
\(\Delta(z)=z\otimes1+h_z\otimes z\), the antipode is
\(S(z)=-h_z^{-1}z\).

\begin{proposition}\label{prop:family}
These formulas define Hopf algebras \(H_t\) of dimension \(1024\).
Writing \(z_i=[x_i,y_i]\), a basis is
\begin{equation}\label{eq:pbw}
x_1^{a_1}z_1^{b_1}y_1^{c_1}
x_2^{a_2}z_2^{b_2}y_2^{c_2}g^i r^j s^k,
\qquad
\begin{gathered}
a_\ell,b_\ell,c_\ell,j,k\in\{0,1\},\\
0\leq i<4.
\end{gathered}
\end{equation}
They are the fibers of a free Hopf algebra of rank \(1024\) over
\(\C[t]\), with constant coalgebra.
\end{proposition}

\begin{proof}
Let
\(\Gamma=C_4(h)\times C_2(v)\times C_2(w)\), and put \(u=h^2\).
Define an auxiliary Hopf algebra \(K\) by adjoining
\(X_1,Y_1,X_2,Y_2\) to \(\C\Gamma\). The group action and the group
degrees \(q_Z\) are
\begin{equation}\label{eq:auxiliary}
\begin{array}{c|rrrr}
 &X_1&Y_1&X_2&Y_2\\ \hline
h&\ii&\ii&\ii&\ii\\
v&1&-1&-1&1\\
w&-1&1&1&-1\\ \hline
q_Z&u&v&u&w
\end{array},
\qquad
\Delta(Z)=Z\otimes1+q_Z\otimes Z.
\end{equation}
Its defining relations, besides those of the group and its action, are
\begin{gather*}
X_i^2=Y_i^2=(X_iY_i+Y_iX_i)^2=0\quad(i=1,2),\\
X_1X_2+X_2X_1=X_1Y_2+Y_2X_1=X_2Y_1+Y_1X_2=0,
\qquad Y_1Y_2=Y_2Y_1.
\end{gather*}
For either pair \(X,Y\), the algebra
\(\C\langle X,Y\rangle/(X^2,Y^2,(XY+YX)^2)\) has basis
\begin{equation}\label{eq:eight}
1,\ X,\ Y,\ XY,\ YX,\ XYX,\ YXY,\ XYXY.
\end{equation}
Indeed, only alternating words survive, \(YXYX=-XYXY\), every word
of length five vanishes, and there are no further relations in degrees
at most four. The cross relations are scalar commutation relations
between the two factors, so their product has dimension \(64\).
Adjoining \(\Gamma\) gives \(\dim K=1024\).

The coproduct respects the relations. For the only quartic relation
in the first pair, put \(Z_1=X_1Y_1+Y_1X_1\). Then
\[
\Delta(Z_1)=Z_1\otimes1+uv\otimes Z_1+2vX_1\otimes Y_1.
\]
The quadratic relations give \([Z_1,X_1]=[Z_1,Y_1]=0\), while
\[
Z_1(uv)=-(uv)Z_1,\qquad
Z_1(vX_1)=-(vX_1)Z_1,\qquad
(uv)(vX_1)=-(vX_1)(uv).
\]
Together with \((uv)^2=1\) and \(Y_1^2=0\), these identities give
\[
\Delta(Z_1^2)=Z_1^2\otimes1+1\otimes Z_1^2,
\]
so the quartic relation is preserved. The second pair
is identical with \(v\) replaced by \(w\), and the quadratic relations
are checked directly from \eqref{eq:auxiliary}. The usual antipode on
the skew-primitive generators makes \(K\) a Hopf algebra.

The elements \(u,X_1,X_2\) generate the Hopf subalgebra
\(\bigwedge\langle X_1,X_2\rangle\rtimes\C\langle u\rangle\). Set
\begin{equation}\label{eq:twist}
(b_{ij})=-\frac12
\begin{pmatrix}1&t\\t&1\end{pmatrix},
\qquad
T_t=\sum_{i,j=1}^2 b_{ij}X_i u\otimes X_j,
\qquad
F_t=e^{T_t}=1+T_t+\tfrac12T_t^2.
\end{equation}
Here \(T_t^3=0\). To check the twist equation, for
\(L=\sum b_{ij}X_i u\otimes X_j\) consider
\[
L_{12}=L\otimes1,\qquad L_{23}=1\otimes L,\qquad
L_{13}^{u}=\sum b_{ij}X_i u\otimes u\otimes X_j.
\]
Each summand of one tensor commutes with each summand of the other
two: the signs from \(uX_i=-X_i u\) and \(X_iX_j=-X_jX_i\) cancel.
Thus these tensors commute pairwise. Moreover,
\[
(\Delta\otimes\mathrm{id})L=L_{13}^{u}+L_{23},
\qquad
(\mathrm{id}\otimes\Delta)L=L_{12}+L_{13}^{u}.
\]
Exponentiation proves the twist equation; the counit normalization
is immediate. Thus \(K\otimes\C[t]\), with coproduct
\(F_t^{-1}\Delta(-)F_t\), is a Hopf algebra. Its \(\C[t]\)-linear dual
\(\mathcal H\) is free of rank \(1024\), with coalgebra
\(K^*\otimes\C[t]\). Write \(\widetilde H_t\) for its fibers.

We identify these fibers with the stated presentation.
Take \(g,r,s\in\widehat\Gamma\) to be the characters
\[
\begin{array}{c|rrr}
 &h&v&w\\ \hline
g&\ii&1&1\\
r&1&1&-1\\
s&1&-1&1
\end{array},
\]
extended by zero in positive degree. Define degree-one functionals by
\[
x_i(X_j\gamma)=\delta_{ij},\qquad
y_i(Y_j\gamma)=\delta_{ij}\qquad(\gamma\in\Gamma),
\]
and zero on the other homogeneous basis elements.
Their coproducts and group action are \eqref{eq:coproduct} and
\eqref{eq:action}.
Together with \(g,r,s\), they generate the graded algebra \(K^*\):
on either factor, pairing
the words \(x,y;\ xy,yx;\ xyx,yxy;\ xyxy\) with the corresponding
words in \(X,Y\) gives the invertible matrices
\[
I_2,\qquad
\begin{pmatrix}1&1\\-1&1\end{pmatrix},\qquad
2I_2,\qquad (2),
\]
and the assertion follows for \(K^*\) by the scalar cross relations
and Fourier duality on \(\Gamma\). The twisted dual multiplication
is filtered, with this graded multiplication as its leading term:
the positive-degree terms of \(F_t\) contribute only lower-degree
terms. Hence the same elements generate \(\widetilde H_t\).

Quadratic products acquire only degree-zero corrections, since
\(F_t-1\) starts in total degree two.
For \(\gamma\in\Gamma\),
\begin{align*}
F_t^{-1}(\gamma\otimes\gamma)F_t
&=\gamma\otimes\gamma+(\gamma\otimes\gamma)T_t
-T_t(\gamma\otimes\gamma)\\
&\quad+\text{terms of total degree at least four}.
\end{align*}
Applying \(x_i\otimes x_j\), and writing
\(\chi_1=a,\ \chi_2=b\) for the characters of \(X_1,X_2\), gives
\[
(x_i x_j)(\gamma)
=b_{ij}\bigl(\chi_i(\gamma)\chi_j(\gamma)-1\bigr).
\]
Together with the unchanged degree-two terms, this gives the square
and linking relations in \eqref{eq:relations}. The other quadratic
relations acquire no correction, since the twist involves only
\(X_1,X_2\).

For the quartic relations, put \(z_1=[x_1,y_1]\). Centrality of
\(x_1^2\) and the relation \(y_1^2=0\) give
\(z_1x_1=-x_1z_1\) and \(z_1y_1=-y_1z_1\). Moreover,
\[
\Delta(z_1)=z_1\otimes1+d\otimes z_1-2b x_1\otimes y_1.
\]
Using \(d^2=1\) and \eqref{eq:action}, its square is
\(z_1^2\otimes1+1\otimes z_1^2\).
A finite-dimensional Hopf algebra in characteristic zero has no
nonzero primitive elements, as follows by applying the coproduct
to a minimal polynomial. Thus \(z_1^2=0\); the same argument gives
\(z_2^2=0\).

There is therefore a surjection from the presented algebra onto
\(\widetilde H_t\). To obtain a spanning set, move group elements
to the right and move the second pair past the first using the
cross relations. The linking correction lowers the number of
non-group generators. Within each pair, the square relations and
\[
y_i x_i y_i x_i=-x_i y_i x_i y_i
\]
reduce every word to the span of eight monomials, equivalently
\(x_i^{a_i}z_i^{b_i}y_i^{c_i}\) with exponents in \(\{0,1\}\).
Thus \eqref{eq:pbw} spans. It has \(1024\) elements, so the surjection
is an isomorphism and \eqref{eq:pbw} is a basis.
\end{proof}

\section{A distinguished block and its radical layers}\label{sec:radical}

Fix \(t\in T\). The central involution \(c\) gives
\[
H_t=H_t^+\oplus H_t^-,
\qquad
H_t^\pm=\frac{1\pm c}{2}H_t,
\qquad
\dim H_t^\pm=512.
\]
Regard \(A_t=H_t^-\) as a unital algebra with identity \((1-c)/2\).
In this algebra,
\[
g^2=-1,\qquad x_1^2=x_2^2=1,\qquad
x_1x_2+x_2x_1=t(1+rs).
\]
Give \(y_1,y_2\) degree one and \(g,r,s,x_1,x_2\) degree zero.
The basis \eqref{eq:pbw} gives
\[
\operatorname{Hilb}(A_t;q)=32(1+q)^4.
\]

\subsection{The semisimple quotient}

Let \(D_t=(A_t)_0\). Its simple modules are indexed by eight triples
\begin{equation}\label{eq:vertices}
v=(r_v,s_v,\alpha_v),\qquad
r_v,s_v\in\{1,-1\},\qquad
\alpha_v+\alpha_v^{-1}=t(1+r_vs_v),
\end{equation}
and are given by
\begin{equation}\label{eq:matrices}
g=\begin{pmatrix}\ii&0\\0&-\ii\end{pmatrix},\quad
x_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad
x_2=\begin{pmatrix}0&\alpha_v\\\alpha_v^{-1}&0\end{pmatrix},
\quad r=r_vI_2,\quad s=s_vI_2.
\end{equation}
For \(r_vs_v=-1\), the roots are \(\pm\ii\); for \(r_vs_v=1\), they
are \(t\pm\sqrt{t^2-1}\). The eight representations are simple and
inequivalent. Since \(\dim D_t=32\), it follows that
\(D_t\cong\Mat_2(\C)^{\oplus8}\).
The positive-degree ideal is nilpotent with semisimple quotient, so
it is \(\rad A_t\).

The elements \(g,x_1\) generate a copy of \(\Mat_2(\C)\) in degree
zero. The corner
\[
B_t=pA_t p,\qquad p=\frac{1-\ii g}{2},
\]
is basic, and
\begin{equation}\label{eq:basic}
A_t\cong\Mat_2(B_t),\qquad
\operatorname{Hilb}(B_t;q)=8(1+q)^4.
\end{equation}
Inserting the degree-zero matrix units between factors shows that
\(B_t\), like \(A_t\), is generated in degrees zero and one.
Consequently, for \(J=\rad B_t\),
\begin{equation}\label{eq:radical}
J^k=\bigoplus_{j\geq k}(B_t)_j,\qquad
\gr_J B_t\cong B_t.
\end{equation}
In particular, \((B_t)_0=\C^8\), \(\dim(B_t)_1=32\), and
\(\dim(B_t)_2=48\).

\subsection{Arrows}

Let \(e_v\) be the primitive idempotents of \((B_t)_0\), indexed
by \eqref{eq:vertices}. Under \eqref{eq:basic}, the component of
\(y_i\) from \(v\) to \(w\) is a \(2\times2\) matrix with entries
in \(e_w(B_t)_1e_v\). The group relations force \(y_1\) to change
\((r_v,s_v)\) to \((r_v,-s_v)\), and \(y_2\) to change it to
\((-r_v,s_v)\). The relations with \(g,x_2\), respectively \(g,x_1\),
force each component to be a single element of \(e_w(B_t)_1e_v\)
times, respectively, the matrix
\begin{equation}\label{eq:arrows}
Y_{wv}=\begin{pmatrix}1&0\\0&-\alpha_v/\alpha_w\end{pmatrix},
\qquad
Q_{wv}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\end{equation}
Since \((A_t)_1=D_ty_1D_t+D_ty_2D_t\), each of the \(8\cdot4=32\)
possible arrow spaces has dimension at most one.
As \(\dim(B_t)_1=32\), every one has dimension one.
We take the nonzero upper-left entries as arrow generators.
The resulting quiver has underlying graph \(K_{4,4}\), with parts
distinguished by \(r_vs_v\).

The graph is connected, so \(A_t\) is a block. It is the unique block
of \(H_t\) of dimension \(512\) all of whose simple modules have
dimension two. Indeed, the complementary summand \(H_t^+\) also has
dimension \(512\) and contains the one-dimensional counit module.
Any other block with the stated dimension would exhaust that summand.

\subsection{Products of two arrows}

For vertices \(v,w\), put
\[
W_{wv}=e_w(J^2/J^3)e_v=e_w(B_t)_2e_v.
\]
There are four length-two paths from \(v\) to each \(w\) in the
same bipartite part, and none to the opposite part.

First suppose \((r_w,s_w)=(r_v,s_v)\). There are two paths of type
\(y_1y_1\) and two of type \(y_2y_2\), with intermediate scalars
\(\beta,\beta^{-1}\). For either type \(i\), write their products as
\(P_\beta,P_{\beta^{-1}}\). The relations \(y_i^2=z_i^2=0\), using
\eqref{eq:matrices} and \eqref{eq:arrows}, give
\[
P_\beta+P_{\beta^{-1}}=0,\qquad
f_i(\beta)P_\beta+f_i(\beta^{-1})P_{\beta^{-1}}=0,
\]
where
\begin{equation}\label{eq:weights}
f_1(\beta)=
(1+\beta/\alpha_w)(1+\alpha_v/\beta),
\qquad
f_2(\beta)=\beta+\alpha_v+\alpha_w+
\frac{\alpha_v\alpha_w}{\beta}.
\end{equation}
For the second weight, the coefficient matrix of \(z_2=[x_2,y_2]\)
on a \(y_2\)-arrow with source scalar \(\gamma\) and target scalar
\(\delta\) is
\[
R(\delta,\gamma)=
\begin{pmatrix}
0&-(\delta+\gamma)\\
\delta^{-1}+\gamma^{-1}&0
\end{pmatrix}.
\]
The upper-left entry of the two-step product through \(\beta\) is
\[
\bigl(R(\alpha_w,\beta)R(\beta,\alpha_v)\bigr)_{11}
=-(\alpha_w+\beta)(\beta^{-1}+\alpha_v^{-1})
=-\frac{f_2(\beta)}{\alpha_v},
\]
which gives the stated relation.

If \(w=v\), the differences of the second coefficients are
\[
(\beta-\beta^{-1})(\alpha_v^{-1}-\alpha_v),
\qquad
(\beta-\beta^{-1})(1-\alpha_v^2),
\]
respectively. Both are nonzero, so all return-path products vanish.
If \(\alpha_w=\alpha_v^{-1}\), the coefficients agree; each type
then spans at most one line. Thus \(W_{vv}=0\), and the other target
with the same \(r,s\) has dimension at most two.

Now let \((r_w,s_w)=(-r_v,-s_v)\). Write \(U_\beta\) for the product
through \((-r_v,s_v,\beta)\), and \(V_\beta\) for the product through
\((r_v,-s_v,\beta)\). The two matrix entries of \(y_1y_2=y_2y_1\)
give
\begin{equation}\label{eq:mixed}
\begin{aligned}
U_\beta+U_{\beta^{-1}}
&=V_\beta+V_{\beta^{-1}},\\
\beta U_\beta+\beta^{-1}U_{\beta^{-1}}
&=\alpha_v\alpha_w
\bigl(\beta^{-1}V_\beta+\beta V_{\beta^{-1}}\bigr).
\end{aligned}
\end{equation}
These relations are independent because \(\beta\ne\beta^{-1}\),
so each of the two mixed targets has dimension at most two.
Since \(B_t\) is generated in degrees zero and one, \((B_t)_2\) is
spanned by length-two paths. Thus only the four targets considered
above occur for each source, and the exhaustive upper bounds sum to
\[
8(0+2+2+2)=48=\dim(B_t)_2.
\]
Every bound is therefore attained. For each mixed target, the map
from the four-dimensional path space onto \(W_{wv}\) has a
two-dimensional kernel. The two independent relations
\eqref{eq:mixed} form a basis of this kernel.

\section{The elliptic invariant}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Proposition~\ref{prop:family} constructs the required algebraic family.
We recover \(\JJ(t)\) from the associative algebra.

The radical multiplication maps
\[
e_w(J/J^2)e_k\otimes e_k(J/J^2)e_v
\longrightarrow W_{wv}
\]
attach a product line to each intermediate vertex \(k\), whenever
the product is nonzero. Here the \(e_v\) may be viewed in \(B_t/J\):
each radical layer \(J^m/J^{m+1}\) is canonically a \(B_t/J\)-bimodule.
Indeed, changing a lift by an element of \(J\) changes its action
only in \(J^{m+1}\). The construction is therefore independent of
the chosen idempotent lifts.
If \(W_{wv}\) is two-dimensional and the
four product lines are distinct, their unordered configuration in
\(\mathbb P(W_{wv})\) is intrinsic. Indeed, an algebra isomorphism
preserves the radical filtration and permutes the vertices and arrow
spaces of its associated graded algebra. Passing to a basic algebra
does not affect the construction, since basic algebras in a given
Morita class are isomorphic.

We determine all configurations of four distinct lines.
The return target has zero product space. For the other target
with the same \(r,s\), the relation \(y_i^2=0\) makes the two paths
of each type span the same line, so there are at most two distinct
lines. For the mixed target with
\(\alpha_w=\alpha_v^{-1}\), equation \eqref{eq:mixed} gives
\[
V_\beta=U_{\beta^{-1}},\qquad
V_{\beta^{-1}}=U_\beta,
\]
again leaving only two lines.

It remains to take \(w=(-r_v,-s_v,\alpha_v)\).
Put \(\alpha=\alpha_v=\alpha_w\). Equation \eqref{eq:mixed} expresses
the two \(V\)'s in terms of the two \(U\)'s, and conversely; each pair
is therefore a basis of the two-dimensional space \(W_{wv}\).
Solving these equations gives the cross-ratio of their four spans,
up to ordering:
\begin{equation}\label{eq:crossratio}
\begin{aligned}
\lambda
&=-\frac{(\alpha^2\beta-\beta^{-1})
(\beta-\alpha^2\beta^{-1})}{(\alpha^2-1)^2}\\
&=\frac{(\alpha+\alpha^{-1})^2-(\beta+\beta^{-1})^2}
{(\alpha-\alpha^{-1})^2}.
\end{aligned}
\end{equation}
By \eqref{eq:vertices}, the two sums in the numerator are
\((0,2t)\) or \((2t,0)\). Hence
\[
\lambda=t^2\qquad\text{or}\qquad
\lambda=\frac{t^2}{t^2-1}.
\]
These values are related by an anharmonic transformation, and neither
is \(0,1,\infty\). Thus the four lines are distinct, and every such
configuration is projectively equivalent to
\(\{0,1,\infty,t^2\}\).

For an unordered quadruple, the expression
\[
j(\lambda)=256\frac{(1-\lambda+\lambda^2)^3}
{\lambda^2(1-\lambda)^2}
\]
is independent of ordering: it is unchanged by
\(\lambda\mapsto1-\lambda\) and \(\lambda\mapsto1/\lambda\), which
generate the six changes of cross-ratio. Every configuration just
found has the same value \(j(t^2)=\JJ(t)\).
Thus \(\JJ(t)\) is intrinsic to \(B_t\), hence to \(A_t\), and hence
to \(H_t\), because \(A_t\) is the unique block identified in
Section~\ref{sec:radical}. This proves the asserted implication.

For fixed \(j_0\), the equation \(\JJ(t)=j_0\) is a polynomial equation
of degree twelve after clearing the denominator. It therefore has
at most twelve solutions in \(T\). Since \(\JJ\) is nonconstant,
choose \(t_0\in T\) with \(\JJ'(t_0)\ne0\). The inverse function theorem
gives a small complex disc about \(t_0\) on which \(\JJ\) is injective.
The corresponding
fibers are pairwise nonisomorphic as associative algebras.
\end{proof}

\begin{remark}\label{rem:elliptic}
The invariant \(\JJ\) is the classical elliptic \(j\)-invariant.
The smooth projective double cover of the projective line branched
over the recovered quadruple has affine equation
\[
E_t:\qquad y^2=x(x-1)(x-t^2).
\]
Choosing a ramification point as origin gives an elliptic curve, with
isomorphism class independent of this choice.
Its invariants are
\[
c_4=16(1-t^2+t^4),\qquad
\Delta=16t^4(1-t^2)^2,\qquad
j(E_t)=c_4^3/\Delta=\JJ(t).
\]
Thus \([H_t]\mapsto[E_t]\) defines a map from the associative algebra
isomorphism classes occurring in this family to the moduli of complex
elliptic curves.
\end{remark}

\Needspace{15\baselineskip}
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}

This paper was written in response to the following substantive prompts
by Pavel Etingof, reproduced in chronological order:
\begin{enumerate}
\item
In the paper \url{https://arxiv.org/pdf/math/0110043} (top of p.~2)
it is asked if there are finitely many algebra structures for Hopf
algebras over \(\C\) of a given dimension \(n\). Can you find a
counterexample? Work hard!

\item
It's a sequence, but is there a continuous family?

\item
We don't need the sequence, the continuous family is good. So the
moduli of these Hopf algebras are moduli of elliptic curves, and
\(J\) is the \(J\)-invariant? If so, you may make a remark to that
extent. Write a high-quality paper with abstract and introduction.
State the main result in the introduction. In the body of the paper
give a proof. Be concise, absolutely nothing not needed for the
proof. Make the presentation clear and easily human-readable!
\end{enumerate}

\Needspace{7\baselineskip}
\phantomsection
\addcontentsline{toc}{section}{References}
\begin{thebibliography}{9}
\bibitem{EG}
P.~Etingof and S.~Gelaki,
\emph{On families of triangular Hopf algebras},
International Mathematics Research Notices \textbf{2002}, no.~14,
757--768.
\href{https://arxiv.org/abs/math/0110043}{arXiv:math/0110043}.
\end{thebibliography}

\end{document}