PRIMITIVE ELEMENTS IN THE HALL ALGEBRA OF A CYCLIC QUIVER
by

Renda Ma

December 2022

A dissertation submitted to the

Faculty of the Graduate School of

the University at Buffalo, The State University of New York

in partial fulfilment of the requirements for the

degree of

Doctor of Philosophy

Department of Mathematics

Copyright by

Renda Ma

2022

## Acknowledgments

I would like to thank Professor Yiqiang Li for his patience, guidance and
support towards me for my studies at University at Buffalo. I would also like
to thank Professor Joseph Hundley and Professor David Hemmer for their
comments and help on my thesis.

###### Table of Contents

  1. Acknowledgments
  2. 1 Introduction
  3. 2 Ring of symmetric functions
    1. 2.1 Introduction
    2. 2.2 Ring of Symmetric functions
    3. 2.3 Cyclic Symmetric Functions
  4. 3 Formula of $p_{n}$ in ring of symmetric functions
    1. 3.1 Introduction
  5. 4 Hall algebra
    1. 4.1 Introduction
    2. 4.2 Finitary category
    3. 4.3 Hall algebra of a finitary category
      1. 4.3.1 Comultiplication on ${\mathbf{H}_{\mathcal{A}}}$
      2. 4.3.2 ${\mathbf{H}_{\mathcal{A}}}$ is a bialgebra over hereditary finitary category
  6. 5 Categories of representations of quivers
    1. 5.1 Introduction
    2. 5.2 Representations of quivers
      1. 5.2.1 Representations of Jordan quiver
      2. 5.2.2 Representations of Cyclic quiver
    3. 5.3 Hall algebra of quiver representations
      1. 5.3.1 Hall algebra is a Hopf algebra
    4. 5.4 Isomorphism between Hall algebra and quantum Lie group
  7. 6 Primitive elements in the Hall algebra
    1. 6.1 Introduction
    2. 6.2 Primitive element of Jordan Quiver case
    3. 6.3 Primitive elements in Hall algebra for Cyclic quiver
      1. 6.3.1 Isomorphism between $\Lambda[t]$ and central subalgebra of $\mathbf{H}_{\hat{A}_{N}}^{nil}$
      2. 6.3.2 Primitive elements of Hall algebra of cyclic quiver
  8. 7 Examples
    1. 7.1 Examples of primitive elements
    2. 7.2 An identity of Hall number
    3. 7.3 Primitive elements for acyclic quiver
    4. 7.4 Primitive elements for Hall algebra of cyclic quiver for all case
    5. 7.5 Primitive elements for Hall algebra of 2-periodic complexes
    6. 7.6 Examples of $z_{n}$

## Abstract

We provide an explicit formula for primitive elements in the Hall algebras of
nilpotent representations of cyclic quivers.

##  Chapter 1 Introduction

To a finitary category, Ringel attached an associative algebra which is
commonly called the Ringel-Hall algebra or just the Hall algebra. The
multiplication of the Hall algebra is related to the number of short exact
sequences with fixed isomorphism classes. Later, Ringel showed that the
structure constants are specializations of certain polynomials, called the
Hall polynomials when the finitary category is the category of representations
of a Dynkin quiver over a finite field. Further, J.A.Green proved that there
is a topological coproduct for the Ringel-Hall algebra associated with any
finitary category, and that, this coproduct gives the structure of a bialgebra
in the case of the category of representation of a quiver [4]. Xiao discovered
an antipode [7] which provides the Hall algebra with a twisted Hopf algebra
structure. The twisting is related with the coproduct. One interesting
property of the Hall algebra provided by Ringel is that for Dynkin quivers,
there is an isomorphism between the Borel part of the quantum universal
enveloping algebra and the extended Hall algebra of the category of
representations of the corresponding quiver. Further, it is shown that for a
general quiver without loops, the positive part of the quantum universal
enveloping algebra is isomorphic to the composition algebra of the Hall
algebra of the corresponding quiver. Moreover, this isomorphism is a twisted
Hopf algebra isomorphism. By the Milnor-Moore theorem, any Hopf algebra
generated by primitive elements is the universal enveloping algebra of the Lie
algebra structure on these elements. Thus, it is an interesting problem to
describe primitive elements of the Hall algebra.

The Hall algebra of a cyclic quiver has been intensively studied since its
incarnation. When the cyclic quiver is the Jordan quiver, the study of this
Hall algebra goes back to Steinitz, Hall and Macdonald. Their work is the
inspiration of Ringel’s work. The Hall algebra of the Jordan quiver is a
genuine Hopf algebra instead of a twisted one. Steinitz, Hall and Macdonald[6]
showed that there is an isomorphism between the Hall algebra of nilpotent
representations of the Jordan quiver and ring of symmetric functions over the
same field. Based on this result, Macdonald showed that every primitive
element in the Hall algebra of the category of nilpotent representations of a
Jordan quiver has the formula

$\displaystyle\sum_{\lambda\vdash n}(1-q)...(1-q^{l(\lambda)-1})I_{\lambda}$

where $\lambda$ is a partition of $n$, $l(\lambda)$ is the length of
$\lambda$, $I_{n}$ is the $n-dimensional$ indecomposable representation of the
Jordan quiver, and $I_{\lambda}=I_{\lambda_{1}}\oplus...\oplus
I_{\lambda_{n}}$. It was shown by Ringel, Hubery and Schiffmann that the Hall
algebra of the category of nilpotent representations of a cyclic quiver is the
tensor product of the center with its composition algebra. Hubery and
Schiffmann extended Steinitz, Hall and Macdonald result, proving that there is
an isomorphism between the ring of symmetric functions and the center of the
Hall algebra of nilpotent representations of a cyclic quiver over a finite
field [5]. Hubery asked a question about what primitive elements are like in
the Hall algebra of nilpotent representations of a cyclic quiver[5]. In this
paper, we answer this question, showing that primitive elements $p_{n}$ for
case $\hat{A}_{N}$ have the formula as in Theorem 6.3.2

$\displaystyle p_{n}=\frac{n}{1-q^{-Nn}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}z_{\lambda}$ (1.0.1)

In the process of proving the above result, we made use of the following
ingredients. The first one is the isomorphism between the Hall algebra of the
Jordan quiver and the ring of symmetric functions. The second one is the
recursive formula for primitive elements of the ring of symmetric functions in
a base denoted as $c_{\lambda}(x,t)$. We are somewhat surprised to see that
the formula in Prop 3.1.3

$\displaystyle p_{n}=\frac{n}{1-t^{n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}c_{\lambda}(X;t)$

is neglected in the literature. The third ingredient is the isomorphism
between the ring of symmetric functions and center of Hall algebra of cyclic
quiver. We can use this isomorphism to deduce result (3.1.7). This result can
provide us some applications. For example, by comparing the two equations of
primitive elements (3.1.7), (6.2.2) in the case of the Jordan quiver, we get
an identity in proposition 6.1 which is related to the Hall number for a
special case.

In section 2 we will recall some basic information about the ring of symmetric
functions. Special symmetric functions called cyclic symmetric functions
$c_{n}(x)$ are introduced. These are mapped to the generators of the center of
the Hall algebra. In section 3, we present the formula for primitive elements
in the ring of symmetric functions in terms of the basis of cyclic symmetric
functions in Proposition 3.1.3. In section 4, Some basic information about
representations of quiver are introduced. In section 5, our paper will present
the main theorem for the case of a cyclic quiver and in Propostion 6.3 give a
detailed description of the primitive elements of the Hall algebra over
nilpotent representations of a cyclic quiver. We will give some examples of
applications of our formula in section 7.

##  Chapter 2 Ring of symmetric functions

###  2.1 Introduction

Here we give a brief review of the ring of symmetric functions. The main
reference of this chapter is [6].

###  2.2 Ring of Symmetric functions

Let $\mathbb{Q}[X_{1},X_{2},...,X_{n}]$ be the polynomial ring with $n$
independent variables $X_{1},...,X_{n}$. The symmetric group $S_{n}$ acts on
this ring by permutating variables $\sigma(X_{i})=X_{\sigma(i)}$ where
$\sigma$ is in $S_{n},1\leq i\leq n$. The symmetric polynomial ring
$\mathbb{Q}[X_{1},X_{2},...,X_{n}]^{S_{n}}$ is the fixed point locus in
$\mathbb{Q}[X_{1},X_{2},...,X_{n}]$ under this action. The symmetric
polynomials form a subring

$\Lambda_{n}=\mathbb{Q}[X_{1},X_{2},...,X_{n}]^{S_{n}}$

Moreover $\Lambda_{n}$ is a graded ring, we have

$\Lambda_{n}=\bigoplus_{k\geq 0}\Lambda_{n}^{k}$

where $\Lambda_{n}^{k}$ is the homogeneous symmetric polynomials of degree
$k$. For $m\geq n$, consider the homomorphism
$\mathbb{Q}[X_{1},X_{2},...,X_{m}]\to\mathbb{Q}[X_{1},X_{2},...,X_{n}]$ by
sending $X_{n+1},...,X_{m}$ to 0, and each other $X_{i}$ to itself. This
homomorphism is compatible with the actions of $S_{n}$ and $S_{m}$.
Restricting this homomorphism to $\Lambda_{n}$, gives a homomorphism
$\rho_{m,n}:\Lambda_{m}\to\Lambda_{n}$. Further restriction to
$\Lambda_{n}^{k}$, we have a homomorphism
$\rho_{m,n}^{k}:\Lambda_{m}^{k}\to\Lambda_{n}^{k}$. We now form an inverse
limit $\Lambda^{k}=\varprojlim\Lambda_{n}^{k}$ relative to the homomorphisms
$\rho_{m,n}^{k}$. Now let $\Lambda=\bigoplus_{k\geq 0}\Lambda^{k}$. We call
$\Lambda$ the ring of symmetric functions. Let $\\{p_{n}(X)\\}$ represent
power sum symmetric functions, that is $p_{n}(X)=\sum_{i}X_{i}^{n}$. Then
$\Lambda=\mathbb{Q}[p_{1},p_{2},...]$ admits a Hopf algebra structure by
setting

$\Delta(p_{n}(X))=p_{n}(X)\otimes 1+1\otimes
p_{n}(X),\quad\epsilon(p_{n}(X))=0,\quad S(p_{n})=-p_{n}(X).$

By definition, we see that $p_{n}$ are primitive elements of $\Lambda$.
Because $p_{n}(X)$ are also generators of $\Lambda$, so all primitive elements
of $\Lambda$ are $p_{n}(X)$.

###  2.3 Cyclic Symmetric Functions

We now explain an important class of symmetric functions called cyclic
symmetric functions, which are denoted by $c_{n}(X;t)$. They will be used
later. Denote the extension $\Lambda\otimes_{\mathbb{Q}}\mathbb{Q}[t]$ by
$\Lambda[t]$. Along this extension, there is a generalization of symmetric
functions in $\Lambda$. In $\Lambda$, the complete symmetric functions
$h_{n}(X)$ is defined as

$h_{n}(X)=\sum_{i_{1}\leq...\leq i_{n}}X_{i_{1}}...X_{i_{n}}$

The generating function for $h_{n}(X)$ is defined as,

$H(T)=1+\sum_{n\geq 1}h_{n}(X)T^{n}=\prod_{i\geq 1}(1-X_{i}T)^{-1}$

The relation between $H(T)$ and $p_{n}$ is shown in [6]. Let $exp$ denote
exponential function, we have

$H(T)=\exp{\sum_{n\geq 1}\frac{p_{n}}{n}T^{n}}$

By following [5], define a generalization $c_{n}(X;t)\in\Lambda[t]$ of
$h_{n}(X)$ in the following way:

$\displaystyle\begin{split}1+\sum_{n\geq
1}c_{n}(X;t)T^{n}=\frac{H(T)}{H(tT)}=\exp{(\sum_{n\geq
1}\frac{1-t^{n}}{n}p_{n}T^{n})}\end{split}$ (2.3.1)

We take the natural logarithm $\ln$ of the equation (2.3.1) to get the
relation between $c_{n}(X;t)$ and $p_{n}(X)$ in the following way:

$\displaystyle\begin{split}\sum_{n\geq
1}\frac{1-t^{n}}{n}p_{n}T^{n}=\ln(1+\sum_{n\geq 1}c_{n}(X;t)T^{n})\end{split}$
(2.3.2)

By taking derivative with respect to $T$ for the above equation, we have

$\displaystyle\begin{split}\sum_{n\geq
1}(1-t^{n})p_{n}T^{n-1}&=\frac{\sum_{n\geq 1}nc_{n}(X;t)T^{n-1}}{1+\sum_{n\geq
1}c_{n}(X;t)T^{n}}\\\ .\end{split}$

Multiplying the denominator on both sides of the equation, we have

$\displaystyle\begin{split}\sum_{n\geq 1}nc_{n}(X;t)T^{n-1}&=(\sum_{n\geq
1}(1-t^{n})p_{n}T^{n-1})(1+\sum_{n\geq 1}c_{n}(X;t)T^{n})\\\ &=(\sum_{n\geq
1}(1-t^{n})p_{n}T^{n-1})(c_{0}(X;t)T^{0}+\sum_{n\geq 1}c_{n}(X;t)T^{n})\\\
&=(\sum_{a\geq 1}(1-t^{a})p_{a}T^{a-1})(\sum_{b\geq 0}c_{b}(X;t)T^{b})\\\
&=\sum_{b\geq 0}\sum_{a\geq 1}(1-t^{a})p_{a}c_{b}(X;t)T^{a+b-1}\\\
\end{split}$

Let $c_{0}(X;t)=1$, $a+b=n$ and matching the coefficient correspondingly, we
have

$\displaystyle nc_{n}(X;t)$
$\displaystyle=\sum_{a=1}^{n}(1-t^{a})p_{a}c_{n-a}(X;t)$ (2.3.3)

##  Chapter 3 Formula of $p_{n}$ in ring of symmetric functions

###  3.1 Introduction

Here we provide our formula to describe primitive elements in $\Lambda[t]$.
Solving the above equation $(\ref{eq:2.3.3})$ recursively, we have

######  Lemma 3.1.1.

The following formula for primitive elements in $\Lambda[t]$

$\begin{split}p_{n}&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\sum_{\begin{split}i_{1}+...+i_{k}&=n\\\
i_{1},...,i_{k}&\geq
1\end{split}}i_{k}c_{i_{k}}(X;t)c_{i_{k-1}}(X;t)...c_{i_{1}}(X;t)\\\
\end{split}$ (3.1.1)

###### Proof.

We will prove the lemma by induction.

  * 1

When $n=1$, by equation (3.1.1) we have $p_{1}=\frac{1}{1-t}c_{1}(X;t)$ which
satisfies equation (2.3.3).

  * 2

Suppose $p_{j}$ also satisfies equation (2.3.3) for $j\leq{n-1}$.

  * 3

Simplifying equation (3.1.1) we have,

$\displaystyle\begin{split}(1-t^{n})p_{n}&=nc_{n}(X;t)-\sum_{i_{1}+i_{2}=n}i_{2}c_{i_{2}}(X;t)c_{i_{1}}(X;t)+...\\\
&+(-1)^{n+1}\sum_{i_{1}+...+i_{n}=n}i_{n}c_{i_{n}}(X;t)...c_{i_{1}}(X;t)\\\
&=nc_{n}(X;t)-\sum_{i_{1}=1}^{n-1}\Big{(}\sum_{i_{2}=n-i_{1}}i_{2}c_{i_{2}}(X;t)+...\\\
&+(-1)^{n}\sum_{i_{n}+...+i_{2}=n-i_{1}}i_{n}c_{i_{n}}(X;t)..c_{i_{2}}(X;t)\Big{)}c_{i_{1}}(X;t)\\\
&=nc_{n}(X;t)-\sum_{i_{1}=1}^{n-1}\sum_{k=1}^{n-i_{1}}(-1)^{k+1}\sum_{i_{k+1}+...+i_{2}=n-i_{1}}i_{k+1}c_{i_{k+1}}(X;t)...c_{i_{2}}(X;t)c_{i_{1}}(X;t)\\\
\end{split}$ (3.1.2)

Substituting $n-i_{1}=a$ to the right hand of last equation as follows,

$\begin{split}(1-t^{n})p_{n}&=nc_{n}(X;t)-\sum_{a=1}^{n-1}\sum_{k=1}^{a}(-1)^{k+1}\sum_{i_{k+1}+...+i_{2}=a}i_{k+1}c_{i_{k+1}}(X;t)...c_{i_{2}}(X;t)c_{n-a}(X;t)\\\
&=nc_{n}(X;t)-\sum_{a=1}^{n-1}\sum_{k=1}^{a}(-1)^{k+1}\sum_{j_{k}+...+j_{1}=a}j_{k}c_{j_{k}}(X;t)...c_{j_{1}}(X;t)c_{n-a}(X;t)\\\
\end{split}$ (3.1.3)

By assumption, $p_{a}$ satisfies (3.1.1), so

$p_{a}=\frac{1}{1-t^{a}}\sum_{k=1}^{a}(-1)^{k+1}\sum_{j_{1}+...+j_{k}=a}j_{k}c_{j_{k}}(X;t)...c_{j_{1}}(X;t)$

. Substituting $p_{a}$ in (3.1.3), we have,

$\begin{split}(1-t^{n})p_{n}=nc_{n}(X;t)-\sum_{a=1}^{n-1}(1-t^{a})p_{a}c_{n-a}(X;t)\end{split}$
(3.1.4)

The formula (3.1.1) satisfies equation (2.3.3) for the case of $n$. So by
induction, the formula for $p_{n}$ satisfies equation (2.3.3) for all n.
Because $\\{c_{n}(X;t)\\}$ is a basis of $\Lambda[t]$, by equation (2.3.3),
$p_{n}$ are uniquely determined. Since the formula (3.1.1) also satisfies
equation (2.3.3), so the formula (3.1.1) is a correct representation of
$p_{n}$.

∎

We can simplify equation $(\ref{eq:3.1.1})$ further. The goal is to express
the sum

$\sum_{\begin{split}i_{1}+...+i_{k}&=n\\\ i_{1},...,i_{k}&\geq
1\end{split}}i_{k}c_{i_{k}}(X;t)c_{i_{k-1}}(X;t)...c_{i_{1}}(X;t)$ (3.1.5)

as the tautology

$\sum_{\begin{aligned} r_{1}+2r_{2}...+nr_{n}&=n\\\
r_{1}+...+r_{n}&=k\end{aligned}}\sum_{l=1}^{n}lg_{r_{1},...,r_{n}}^{l}c_{1}^{r_{1}}(X;t)...c_{n}^{r_{n}}(X;t)$
(3.1.6)

Here $r_{1},...,r_{n}\geq 0$. $g_{r_{1},...,r_{n}}^{l}$ representing the
cardinality of monomials $c_{i_{k}}(X;t)...c_{i_{1}}(X;t)$ with a turple of
fixed multiplicity $m(c_{1}(X;t))=r_{1},...,m(c_{n}(X;t))=r_{n}$. $l$ here is
an index $1\leq l\leq n$. In other words, if $|\cdot|$ represents cardinality
then,

$\displaystyle g_{r_{1},...,r_{n}}^{l}=|\biggl{\\{}(j_{k-1},...,j_{1})\vdash
n-l:|\\{s\in\\{1,...,k-1\\}:j_{s}=1\\}|$ $\displaystyle=r_{1},...,$
$\displaystyle|\\{s\in\\{1,...,k-1\\}:j_{s}=l\\}|$
$\displaystyle=r_{l}-1,...,$
$\displaystyle|\\{s\in\\{1,...,k-1\\}:j_{s}=n\\}|$
$\displaystyle=r_{n}\biggr{\\}}|$

######  Lemma 3.1.2.

For a fixed $(r_{1},...,r_{n})$, we have
$g_{r_{1},...,r_{n}}^{l}={r_{1}+...+r_{n}-1\choose
r_{1},...,r_{l}-1,...r_{n}}.$

###### Proof.

By definition

$\displaystyle g_{r_{1},...,r_{n}}^{l}=|\biggl{\\{}(j_{k-1},...,j_{1})\vdash
n-l:|\\{s\in\\{1,...,k-1\\}:j_{s}=1\\}|$ $\displaystyle=r_{1},...,$
$\displaystyle|\\{s\in\\{1,...,k-1\\}:j_{s}=l\\}|$
$\displaystyle=r_{l}-1,...,$
$\displaystyle|\\{s\in\\{1,...,k-1\\}:j_{s}=n\\}|$
$\displaystyle=r_{n}\biggr{\\}}|$

Simply we can get that

$\displaystyle g_{r_{1},...,r_{n}}^{l}$
$\displaystyle={r_{1}+...+r_{n}-1\choose r_{1}}{r_{2}+...+r_{n}-1\choose
r_{2}}...{r_{n-1}+r_{n}\choose r_{n-1}}$
$\displaystyle={r_{1}+...+r_{n}-1\choose r_{1},...,r_{l}-1,...,r_{n}}$

∎

Let $r_{1},...,r_{n}\geq 0$, if the formula $(\ref{eq:3.1.1})$

$\displaystyle\begin{split}p_{n}&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\sum_{\begin{aligned}
\hfil\displaystyle\begin{split}i_{1}+...+i_{k}&=n\\\ i_{1},...,i_{k}&\geq
1\end{split}\end{aligned}}i_{k}c_{i_{k}}(X;t)c_{i_{k-1}}(X;t)...c_{i_{1}}(X;t)\\\
\end{split}$

being simplified as the tautology

$\displaystyle\begin{split}&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\sum_{\begin{aligned}
\hfil\displaystyle\begin{split}1r_{1}+...+nr_{n}&=n\\\
r_{1}+...+r_{n}&=k\end{split}\end{aligned}}\sum_{l=1}^{t}lg_{r_{1},...,r_{n}}^{l}c_{1}(X;t)^{r_{1}}...c_{n}(X;t)^{r_{n}}\\\
\end{split}$

Substituting formula of $g_{r_{1},...,r_{n}}^{l}$ into equation above, we have

$\displaystyle\begin{split}p_{n}&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\sum_{\begin{aligned}
\hfil\displaystyle\begin{split}1r_{1}+...+nr_{n}&=n\\\
r_{1}+...+r_{n}&=k\end{split}\end{aligned}}\sum_{l=1}^{t}l{r_{1}+...+r_{n}-1\choose
r_{1},...,r_{l}-1,...,r_{n}}c_{1}(X;t)^{r_{1}}...c_{n}(X;t)^{r_{n}}\\\
&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\sum_{\begin{aligned}
\hfil\displaystyle\begin{split}1r_{1}+...+nr_{n}&=n\\\
r_{1}+...+r_{n}&=k\end{split}\end{aligned}}\sum_{l=1}^{t}lr_{l}{r_{1}+...+r_{n}-1\choose
r_{1},...,r_{l},...r_{n}}c_{1}(X;t)^{r_{1}}...c_{n}(X;t)^{r_{n}}\\\
&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\sum_{\begin{aligned}
\hfil\displaystyle\begin{split}1r_{1}+...+nr_{n}&=n\\\
r_{1}+...+r_{n}&=k\end{split}\end{aligned}}\sum_{l=1}^{t}lr_{l}\frac{1}{k}{r_{1}+...+r_{n}\choose
r_{1},...,r_{n}}c_{1}(X;t)^{r_{1}}...c_{n}(X;t)^{r_{n}}\\\
&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\frac{1}{k}\sum_{\begin{aligned}
\hfil\displaystyle\begin{split}1r_{1}+...+nr_{n}&=n\\\
r_{1}+...+r_{n}&=k\end{split}\end{aligned}}\sum_{l=1}^{t}lr_{l}{r_{1}+...+r_{n}\choose
r_{1},...,r_{t}}c_{1}(X;t)^{r_{1}}...c_{n}(X;t)^{r_{n}}\\\ \end{split}$

After summing over $l$, we have

$\displaystyle\begin{split}p_{n}&=\frac{1}{1-t^{n}}\sum_{k=1}^{n}(-1)^{k+1}\frac{n}{k}\sum_{\begin{aligned}
\hfil\displaystyle\begin{split}1r_{1}+...+nr_{n}&=n\\\
r_{1}+...+r_{n}&=k\end{split}\end{aligned}}{r_{1}+...+r_{n}\choose
r_{1},...,r_{n}}c_{1}(X;t)^{r_{1}}...c_{n}(X;t)^{r_{n}}\\\
&=\frac{n}{1-t^{n}}\sum_{1r_{1}+...+nr_{n}=n}(-1)^{r_{1}+...+r_{n}+1}\frac{1}{r_{1}+...+r_{n}}{r_{1}+...+r_{n}\choose
r_{1},...,r_{n}}c_{1}(X;t)^{r_{1}}...c_{n}(X;t)^{r_{n}}\\\ \end{split}$
(3.1.7)

So we have the following proposition.

######  Proposition 3.1.3.

Let $\lambda$ be a partition of $n$ in the form that
$\lambda=(1^{m_{1}(\lambda)},...,n^{m_{n}(\lambda)})$, $m_{i}(\lambda)$ is the
multiplicity of $i$ in $\lambda$. One explicit formula for primitive elements
in $\Lambda[t]$ is

$\displaystyle\begin{split}p_{n}=\frac{n}{1-t^{n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}c_{\lambda}(X;t)\end{split}$ (3.1.8)

where $l(\lambda)$ is the length of $\lambda$,
$c_{\lambda}(X;t)=c_{1}(X;t)^{m_{1}(\lambda)}...c_{n}(X;t)^{m_{n}(\lambda)}$.

The equation $(\ref{eq:3.1.8})$ is important in our later discussion. One
interesting point is that the right hand of $(\ref{eq:3.1.8})$ is in
$\Lambda[t]$. We will begin introducing the Hall algebra of nilpotent
representations of a cyclic quiver over a finite field.

##  Chapter 4 Hall algebra

###  4.1 Introduction

We will now introduce the Hall algebra of a finitary category, the main
reference of this section is [9]

###  4.2 Finitary category

Shiffmann introduced a category called finitary category in [9], we will
review it here. A category is called a small category if all of its objects
form a set and all of its morphisms form a set. A small abelian category
$\mathcal{A}$ is called finitary if for any two objects $M,N\in\mathcal{A}$,
they satisfy the condition,

$\displaystyle|Hom_{\mathcal{A}}(M,N)|<\infty,\quad|Ext_{\mathcal{A}}^{1}(M,N)|<\infty$

Here $Ext_{\mathcal{A}}^{n}(M,N)$ is the $n-th$ cohomology group. Let
$\mathcal{A}$ be a finitary category, we make additional assumptions that for
arbitrary objects $M,N$ in $\mathcal{A}$, there exists a number $c_{M,N}>0$
such that

$\displaystyle Ext_{\mathcal{A}}^{i}(M,N)=\\{0\\},\text{for }i>c_{M,N}$
$\displaystyle|Ext_{\mathcal{A}}^{i}(M,N)|<\infty\text{ for all }i.$

For any two objects $M,N$ of $\mathcal{A}$, we put

$\langle
M,N\rangle=\left(\prod_{i=0}^{\infty}|Ext_{\mathcal{A}}^{i}(M,N)|^{(-1)^{i}}\right)^{\frac{1}{2}}$

By the assumption, the quantity $\langle M,N\rangle$ is well defined because
for there exists an $c_{M,N}$ such that any $i>c_{M,N}$,
$Ext_{\mathcal{A}}^{i}(M,N)=\\{0\\}$ for any objects $M,N$. $\langle,\rangle$
is called multiplicative Euler form. It is also useful to introduce the
symmetric Euler form $(M,N)=\langle M,N\rangle\cdot\langle N,M\rangle$. Now we
can introduce the definition of the Hall algebra of a finitary category.

###  4.3 Hall algebra of a finitary category

Here we will introduce Hall algebra of a finitary category. The main reference
is in [9].
Let $\mathcal{A}$ be a finitary category and for an object $M$ of
$\mathcal{A}$, let $[M]$ denote the isomorphism class of $M$. With fixed
objects $M,N,R\in\mathcal{A}$, we denote $\mathbf{P}_{M,N}^{R}$ as the
cardinality of all short exact sequenes $0\rightarrow N\rightarrow
R\rightarrow M\rightarrow 0$. Set $a_{M}=|Aut(M)|$. The Hall algebra
${\mathbf{H}_{\mathcal{A}}}$ is defined to be

${\mathbf{H}_{\mathcal{A}}}=\bigoplus_{[M]}\mathbb{C}[M]$

as vector space where $[M]$ runs over the set of isoclasses of objects in
$\mathcal{A}$ with multiplication given by

$[M]\cdot[N]=\langle
M,N\rangle\sum_{R}\frac{1}{a_{M}a_{N}}\mathbf{P}_{M,N}^{R}[R]$

The summation in above definition is well-defined because
$|Ext^{1}(M,N)|<\infty$, so there are only finitely many $R$. It is known that
${\mathbf{H}_{\mathcal{A}}}$ is an associative algebra with unit
$i:\mathbb{C}\rightarrow{\mathbf{H}_{\mathcal{A}}}$ given by $i(c)=c[0]$,
where $0$ is the zero object of $\mathcal{A}$.

####  4.3.1 Comultiplication on ${\mathbf{H}_{\mathcal{A}}}$

The main reference in this section is from [9] and [4]. Paraphrasing what
Schiffmann said in [9], we can say that the multiplication in
${\mathbf{H}_{\mathcal{A}}}$ encodes all the ways of forming short exact
sequences with object $M$ on top of object $N$. It would be interesting to
have an operation which is breaking an object $R$ into two objects. By the
assumption of $\mathcal{A}$, there are only finitely many extensions of two
objects $M,N$. But for a fixed object $R$, there are possible infinitely many
pairs $(M,N)$ such that short exact sequence $0\rightarrow N\rightarrow
R\rightarrow M\rightarrow 0$ exist. So we need to consider certain completion
of ${\mathbf{H}_{\mathcal{A}}}$ and
${\mathbf{H}_{\mathcal{A}}}\otimes{\mathbf{H}_{\mathcal{A}}}$. Let
$K(\mathcal{A})$ denote the Grothendieck group of $\mathcal{A}$. For
$\alpha\in K(\mathcal{A})$,
${\mathbf{H}_{\mathcal{A}}}[\alpha]=\bigoplus_{\bar{M}=\alpha}\mathbb{C}[M]$.
Here $\bar{M}$ stands for the class of $M$ in Grothendieck group. Set

${\mathbf{H}_{\mathcal{A}}}[\alpha]\widehat{\otimes}{\mathbf{H}_{\mathcal{A}}}[\beta]=\prod_{[M]\in\alpha,[N]\in\beta}\mathbb{C}[M]\otimes\mathbb{C}[N]$
${\mathbf{H}_{\mathcal{A}}}\widehat{\otimes}{\mathbf{H}_{\mathcal{A}}}=\prod_{\alpha,\beta}{\mathbf{H}_{\mathcal{A}}}[\alpha]\widehat{\otimes}{\mathbf{H}_{\mathcal{A}}}[\beta]$

In other words,
${\mathbf{H}_{\mathcal{A}}}\widehat{\otimes}{\mathbf{H}_{\mathcal{A}}}$ is the
space of all formal (infinite) linear combinations
$\sum_{M,N}c_{M,N}[M]\otimes[N],c_{M,N}\in\mathbb{C}.$

######  Proposition 4.3.1.

[4] The following defines on ${\mathbf{H}_{\mathcal{A}}}$ the structure of a
topological coassociative coproduct: for any object
$R\in{\mathbf{H}_{\mathcal{A}}}$,

$\Delta([R])=\sum_{M,N}\langle
M,N\rangle\frac{\mathbf{P}_{M,N}^{R}}{a_{R}}[M]\otimes[N]$

with counit $\epsilon:{\mathbf{H}_{\mathcal{A}}}\rightarrow\mathbb{C}$ defined
by $\epsilon([M])=\delta_{M,0}$.

####  4.3.2 ${\mathbf{H}_{\mathcal{A}}}$ is a bialgebra over hereditary
finitary category

By the result of the previous section, ${\mathbf{H}_{\mathcal{A}}}$ is an
algebra and coalgebra. But in general ${\mathbf{H}_{\mathcal{A}}}$ is not a
bialgebra. An algebra $B(\cdot,i,\epsilon,\delta)$ is called a bialgebra, if
it satisfies the following conditions:

  * 1

B is a vector space over a field k.

  * 2

There are k-linear map (multiplication) $\cdot:B\otimes B\mapsto B$ and unit
$i:k\mapsto B$ such that $(B,\times,i)$ is an unital associative algebra.

  * 3

There are $\Delta:B\mapsto B\otimes B$ and counit $\epsilon:B\mapsto k$ such
that $(B,\Delta,\epsilon)$ is a coalgebra.

  * 4

Compatibility conditions expressed by the following commutative diagrams:

$\textstyle{B\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot}$$\scriptstyle{\cdot\otimes\Delta}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{B\otimes
B}$$\textstyle{B\otimes B\otimes B\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mbox{id}\otimes\tau\otimes\mbox{id}}$$\textstyle{B\otimes
B\otimes B\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot\otimes\cdot}$
$\textstyle{B\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot}$$\scriptstyle{\epsilon\otimes\epsilon}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{k\otimes
k\cong k}$ $\textstyle{k\otimes k\cong
k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i\otimes
i}$$\scriptstyle{i}$$\textstyle{B\otimes
B}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$
---
$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{i}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{k}$

Green proved that ${\mathbf{H}_{\mathcal{A}}}$ is a bialgebra when
$\mathcal{A}$ is hereditary.[4] From now on we assume that $\mathcal{A}$ is
hereditary, i.e., that is for any two objects $M,N\in\mathcal{A}$,
$Ext^{i}(M,N)=0$ for $i\geqslant 2.$ We slightly twist the product on
${\mathbf{H}_{\mathcal{A}}}\otimes{\mathbf{H}_{\mathcal{A}}}:$ Because
${\mathbf{H}_{\mathcal{A}}}$ is a graded algebra, so it contains homogeneous
elements in the sense of graded algebra. If $x,y,z,w$ are homogeneous elements
of ${\mathbf{H}_{\mathcal{A}}}$ of respect weight $wt(x),wt(y),wt(z),wt(w)\in
K(\mathcal{A})$, then define

$(x\otimes y)\cdot(z\otimes w)=\langle wt(y),wt(z)\rangle\langle
wt(z),wt(y)\rangle(xz\otimes yw)$

The following is proved by Green [4].

######  Proposition 4.3.2.

(Green) Assume that $\mathcal{A}$ is hereditary. The map
$\Delta:{\mathbf{H}_{\mathcal{A}}}\rightarrow{\mathbf{H}_{\mathcal{A}}}\widehat{\otimes}{\mathbf{H}_{\mathcal{A}}}$
is an algebra homomorphism, i.e. for any two
$x,y\in{\mathbf{H}_{\mathcal{A}}}$, $\Delta(x\cdot
y)=\Delta(x)\cdot\Delta(y)$.

Because the above proposition, we have that ${\mathbf{H}_{\mathcal{A}}}$ is a
bialgebra.

##  Chapter 5 Categories of representations of quivers

###  5.1 Introduction

In the previous section, we showed that ${\mathbf{H}_{\mathcal{A}}}$ is a
bialgebra. Furthermore Xiao in [7] proved that for the category of
representations of a quiver, the extended hall algebra
$\tilde{{\mathbf{H}}}_{\mathcal{A}}$, which we define in section 5.3.1, is a
Hopf algebra. So we introduce representations of quivers in this section. The
main reference in this section is from [9]

###  5.2 Representations of quivers

######  Definition 5.2.1.

A quiver $Q$ is an oriented graph with vertices and arrows between vertices.
(Multiple arrows and loop arrows are allowed in $Q$.) Let $Q_{0}$ denote the
set of vertices and $Q_{1}$ denote the set of arrows. For $a\in Q_{1}$, denote
$h(a)$ the vertex $a$ points to, and $t(a)$ the other endpoint of a.

######  Definition 5.2.2.

For a quiver $Q$, a representation of $Q$ over a field $k$ is an assignment of
a vector space $V_{i}$ over $k$ for each $i\in Q_{0}$ and a linear morphism
$\rho_{a}:V_{i}\rightarrow V_{j}$ for each $a\in Q_{1}$ from $i$ to $j$. So,
we denote a representation of a fixed $Q$ as $(\bigoplus_{i\in
Q_{0}}V_{i},(\rho_{a})_{a\in Q_{1}})$.

For a fixed quiver, a morphism between two representations $(V_{i},\rho_{a})$,
$(V_{i}^{\prime},\rho^{\prime}_{a})$ is $(f_{i}:V_{i}\rightarrow
V^{\prime}_{i})$, $i\in Q_{0}$ such that the following diagram commute for any
$i,j\in Q_{0}$ and any $a\in Q_{1}$ such that $h(a)=j,t(a)=i$.

${V_{i}}$${V_{j}}$${V_{i}^{\prime}}$${V_{j}^{\prime}}$$f_{i}$$\rho_{a}$$\rho_{a}$$f_{j}$

A representation is called a nilpotent representation if there exist an
integer $n$, such that $\rho_{a_{1}}\rho_{a_{2}}...\rho_{a_{n}}=0$ for any
$a_{i}\in Q_{0},i=1,...,n-1$, satisfying $h(a_{i})=t(a_{i+1})$.
A representation is called an indecomposable representation if it cannot be
decomposed as a direct sum of subrepresentations.
A representation is called simple if there is no subrepresentation. We denote
simple representations as $S_{i},i\in Q_{0}$

####  5.2.1 Representations of Jordan quiver

######  Example 1.

A quiver is called a Jordan quiver if $|Q_{0}|=1$ and $|Q_{1}|=1$. Up to
isomorphism, there is only one Jordan quiver. All indecomposable
representations of the Jordan quiver is given by a $n\times n$ matrix,

$\rho=\begin{pmatrix}\mu&1&0&\cdots&0\\\ 0&\mu&1&\cdots&0\\\
0&0&\ddots&\ddots\\\ 0&0&\dots&\mu&1\\\ 0&0&\dots&0&\mu\end{pmatrix}$

where $n=\text{dim }V$. Denote an indecomposable representation of dimension
$n$, diagonal $\mu$ by $I_{[n;\mu]}$. When we consider nilpotent
representations of Jordan quiver, then $\rho^{t}=0$ for some $t\geq 0$, then
the eigenvalue $\mu$ of $\rho$ is $\mu=0$. If we introduce the notation that
$I_{[n]}=I_{[n;0]}$, the nilpotent representation looks like
$(V,\rho)=I_{[1]}^{\oplus r_{1}}\oplus\dots\oplus I_{[n]}^{\oplus r_{n}}$. So
Krull-Schmidt theorem for representations asserts that every representation
$(V,\rho)=I_{[1]}^{\oplus r_{1}}\oplus\dots\oplus I_{[n]}^{\oplus r_{n}}$
where $n=1r_{1}+...+nr_{n},r_{i}\geq 0$. This works for a field no matter that
is algebraically closed or not. So if a partition $\lambda$ of $n$ is
$\lambda=(1^{r_{1}},...,n^{r_{n}})$, and $I_{\lambda}=I_{[1]}^{\oplus
r_{1}}\oplus\dots\oplus I_{[n]}^{\oplus r_{n}}$, then every nilpotent
representation $(V,\rho)=I_{\lambda}$.

####  5.2.2 Representations of Cyclic quiver

######  Example 2.

The quiver is called the cyclic quiver of $n$ vertices, denoted as
$\hat{A}_{n}$, is the unique (up to isomorphism) quiver with the diagram looks
like the following

${0}$${1}$${2}$${...}$${n-1}$

For $i\in\\{0,1,...,n-1\\}$ and $l\geq 1$. Consider
$V^{\prime}=\oplus_{j=i+1-l}^{i}ke_{j}$ and define $x\in End(V^{\prime})$ by
$x(e_{j})=e_{j-1},j\neq i+1-l$ and $x(e_{i+1-l})=0$. Let
$V_{h}=\oplus_{j\equiv h}ke_{j},V=\oplus_{h\in\mathbb{Z}/n\mathbb{Z}}V_{h}$.
The indecomposable nilpotent representations of $\hat{A}_{n}$ are all $V$ with
different $i$, we denote all these representations as $I_{[i:l]}$. Then any
nilpotent representation is a direct sum of $I_{[i;l]}$ with various value for
$i,l$

###  5.3 Hall algebra of quiver representations

Denote category of representations of quiver $Q$ over field $k$ as $Rep_{k}Q$,
$Rep_{k}Q$ is a hereditary finitary category. So by the definition, there is a
Hall algebra $\mathbf{H}_{Rep_{k}Q}$ for an arbitrary quiver $Q$, we shorten
the denotation $\mathbf{H}_{Rep_{k}Q}$ as $\mathbf{H}_{Q}$. Furthermore we
denote the category of nilpotent representation of quiver $Q$ over field $k$
as $Rep_{k}^{nil}Q$, and the Hall algebra over $Rep_{k}^{nil}Q$ as
$\mathbf{H}^{nil}_{Q}$.

####  5.3.1 Hall algebra is a Hopf algebra

Xiao in [7] proves that extended Hall algbra of category of representations of
quivers are Hopf algebras. We will introduce it here.

######  Definition 5.3.1.

Let $\mathbf{K}=\mathbb{C}[K(\mathcal{A})]$. We define an $extended$ Hall
algebra
$\tilde{\mathbf{H}}_{\mathcal{A}}={\mathbf{H}_{\mathcal{A}}}\otimes_{\mathcal{C}}\mathbf{K}$
where by adding the relations

$\mathbf{k}_{\alpha}[M]\mathbf{k}_{\alpha}^{-1}=(\alpha,M)[M]$

The extended Hall algebra $\tilde{\mathbf{H}}_{\mathcal{A}}$ is still graded
where $deg(k_{\alpha})=0$. If we define the multiplication on
$\tilde{\mathbf{H}}_{\mathcal{A}}\otimes\tilde{\mathbf{H}}_{\mathcal{A}}$ as
$(x\otimes y)(z\otimes w)=xz\otimes yw$, then
$\tilde{\mathbf{H}}_{\mathcal{A}}$ is still a bialgebra when $\mathcal{A}$ is
hereditary with comultiplication

$\Delta(\mathbf{k}_{\alpha})=\mathbf{k}_{\alpha}\otimes\mathbf{k}_{\alpha},$
$\Delta([R]\mathbf{k}_{\alpha})=\sum_{M,N}\langle
M,N\rangle\frac{1}{a_{R}}\mathbf{P}_{M,N}^{R}[M]\mathbf{k}_{[N]+\alpha}\otimes[N]\mathbf{k}_{\alpha}.$

We can extend the antipode $S$ in Xiao’s paper [7] to
$S([M]\mathbf{k}_{\alpha})=\mathbf{k}_{\alpha}^{-1}S([M])$, then
$\tilde{\mathbf{H}}_{\mathcal{A}}$ is a Hopf algebra.

###  5.4 Isomorphism between Hall algebra and quantum Lie group

Suppose that $\mathbf{U}_{v}(\mathfrak{b}_{+}^{\prime})$ is the positive half
of the quantum group $\mathbf{U}_{v}(\mathfrak{g^{\prime}})$ corresponding to
$\mathfrak{g}$. The detailed information about quantum group can be found in
[9] appendix A.4. Here we have the following nice property.

######  Proposition 5.4.1.

(Ringel [8],Green [4]) There is an embedding of Hopf algebra

$\Phi:\mathbf{U}_{v}(\mathfrak{b}_{+}^{\prime})\rightarrow\tilde{\mathbf{H}}_{Q}$

by sending $E_{i}\rightarrow[S_{i}],K_{i}\rightarrow\mathbf{k_{S_{i}}}$ for
$i\in I$. The map $\Phi$ is an isomorphism when $Q$ is of finite type.

We can restrict the above map $\Phi$ to
$\mathbf{U}_{v}(\mathfrak{n}_{+}^{\prime})$ which gives an embedding
$\mathbf{U}_{v}(\mathfrak{n}_{+}^{\prime})\rightarrow\tilde{\mathbf{H}}_{Q}$.
The image of this map is called composition algebra, which is the subalgebra
of $\tilde{\mathbf{H}}_{Q}$ generated by simple representations.

##  Chapter 6 Primitive elements in the Hall algebra

###  6.1 Introduction

From this section, we introduce an isomorphism between ring of symmetric
functions and center of Hall algebra over nilpotent representation of cyclic
quiver. In Jordan quiver $\hat{A}_{0}$, there is an formula for primitive
elements in $\mathbf{H}_{\hat{A}_{0}}^{nil}$ calculated by Hall, Steinitz and
Macdonald [6]. By the isomorphism we will introduce, we can give an explicit
formula for primitive elements for all $H_{\hat{A}_{n}}^{nil}$. The main
reference is from [5].

###  6.2 Primitive element of Jordan Quiver case

For $\hat{A}_{0}$ being the Jordan quiver, $\mathbf{H}_{\hat{A}_{0}}$ is
defined over field $k=\mathbb{F}_{q}$. we introduce the isomorphism between
$\mathbf{H}_{\hat{A}_{0}}^{nil}$ and ring of symmetric functions. Suppose
$P_{\lambda}(\boldsymbol{X},t)$ denote Hall-Littlewood functions in
$\Lambda[t]$.

######  Proposition 6.2.1.

(Steinitz, Hall, Macdonald [6]). For $\hat{A}_{0}$ being the Jordan quiver,
the field $k=\mathbb{F}_{q}$. There is an isomorphism $\Phi$ between
$\Lambda[t]\otimes\mathbb{Q}(t^{1/2})$ and $\mathbf{H}_{\hat{A}_{0}}^{nil}$
where

$\displaystyle\begin{split}\Phi:\Lambda[t]\otimes_{\mathbb{Q}}\mathbb{Q}(t^{\frac{1}{2}})&\rightarrow\mathbf{H}_{\hat{A}_{0}}^{nil}\\\
t&\mapsto q^{-1}\\\ t^{n(\lambda)}P_{\lambda}(\boldsymbol{X},t)&\mapsto
I_{\lambda}\end{split}$ (6.2.1)

where $n(\lambda)=\sum_{i}(i-1)\lambda_{i}$.

In $\Lambda[t]$, power sum $p_{n}(\boldsymbol{X})=\sum_{i}X_{i}^{n}$ are the
primitive elements. In [6], Macdonald shows that
$p_{n}(\boldsymbol{X})=\sum\limits_{\lambda\vdash
n}t^{n(\lambda)}\prod\limits_{i=1}^{l(\lambda)-1}(1-t^{-i})P_{\lambda}(\boldsymbol{X};t).$
Under proposition 6.2.1, we have

$\displaystyle\begin{split}p_{n}(\boldsymbol{X})\mapsto\sum_{\lambda\vdash
n}(1-q)...(1-q^{l(\lambda)-1})I_{\lambda}\end{split}$ (6.2.2)

Because $p_{n}(\boldsymbol{X})$ in $\Lambda[t]$ are primitive elements, so
$\sum_{\lambda\vdash n}(1-q)...(1-q^{l(\lambda)-1})I_{\lambda}$ are primitive
elements in $\mathbf{H}_{\hat{A}_{0}}^{nil}$.

###  6.3 Primitive elements in Hall algebra for Cyclic quiver

Now we extend the above map to the case of cyclic quiver $\hat{A}_{N}$,
similarly the field is $k=\mathbb{F}_{q}$.

####  6.3.1 Isomorphism between $\Lambda[t]$ and central subalgebra of
$\mathbf{H}_{\hat{A}_{N}}^{nil}$

######  Proposition 6.3.1.

(Shiffmann[10], Hubery[5])

  * 1

$\mathbf{H}_{\hat{A}_{N}}^{nil}\cong C_{N}\otimes Z_{N}$ as self-dual graded
Hopf algebra, where $Z_{N}$ is the center of $\mathbf{H}_{\hat{A}_{N}}^{nil}$.
$C_{N}$ is the composition algebra.

  * 2

$z_{N}=\mathbb{Q}(q^{1/2})[z_{1},z_{2},....]$ where

$\displaystyle z_{r}=(-q^{-1})^{rN}\sum_{\begin{aligned} &dim[M]=r\delta\\\
&socle(M)\text{ is square-free}\\\ \end{aligned}}(-1)^{\text{dim
End(M)}}|Aut(M)|[M]$ (6.3.1)

Here socle (M) = $\sum_{\text{N is a simple submodule of M}}N$

  * 3

There is a natural isomorphism

$\displaystyle\begin{split}\Lambda[t]\otimes\mathbb{Q}(t^{\frac{1}{2N}})&\rightarrow
Z_{N}\\\ t&\mapsto q^{-N}\\\ c_{r}(X;t)&\mapsto z_{r}\end{split}$ (6.3.2)

####  6.3.2 Primitive elements of Hall algebra of cyclic quiver

By our equation (3.1.8), primitive elements $p_{n}\in\Lambda[t]$ has formula,

$\displaystyle p_{n}=\frac{n}{1-t^{n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}c_{\lambda}(X;t)$ (6.3.3)

Taking $p_{n}$ under the above isomorphism (5.3.1), we have

######  Theorem 6.3.2.

Primitive elements $p_{n}$ of $H_{\hat{A}_{N}}^{nil}$ has formula that

$\displaystyle p_{n}=\frac{n}{1-q^{-Nn}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}z_{\lambda}$ (6.3.4)

where
$z_{\lambda}=z_{\lambda_{1}}^{m_{1}(\lambda)}...z_{\lambda_{n}}^{m_{n}(\lambda)}$

.

##  Chapter 7 Examples

###  7.1 Examples of primitive elements

For the case of $\mathbf{H}_{\hat{A}_{0}}^{nil}$, we show some calculations of
by using our formula.

$\displaystyle\begin{split}z_{r}&=(-q)^{-r}(-1)^{dimEnd(I_{(r)})}|Aut({I_{(r)}})|[I_{(r)}]\\\
&=(-q)^{-r}(-1)^{dimEnd(I_{(r)})}q^{r}(1-q^{-1})[I_{(r)}]\\\
&=(1-q^{-1})[I_{(r)}]\end{split}$

So,

$\displaystyle p_{1}=\frac{1}{1-q^{-1}}z_{1}=I_{(1)}$
$\displaystyle\begin{split}p_{2}&=\frac{2}{1-q^{-2}}[(-1)^{2}{1\choose
0,1}z_{2}+(-1)^{3}\frac{1}{2}{2\choose 2,0}z_{1}^{2}]\\\
&=\frac{2}{1-q^{-2}}[z_{2}-\frac{1}{2}z_{1}^{2}]\\\
&=\frac{2}{1-q^{-2}}[(1-q^{-1})I_{(2)}-\frac{1}{2}(1-q^{-1})^{2}I_{1}^{2}]\\\
&=\frac{2}{1-q^{-2}}[(1-q^{-1})I_{(2)}-\frac{1}{2}(1-q^{-1})^{2}((q+1)I_{1}\oplus
I_{1}+I_{2})]\\\ &=I_{2}+\frac{q^{-1}-1}{q^{-1}}I_{1}\oplus I_{1}\\\
&=I_{2}+(1-q)I_{1}\oplus I_{1}\\\ \end{split}$

$\displaystyle\begin{split}&p_{3}=\frac{3}{1-q^{-3}}\sum_{\lambda\vdash
3}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}z_{1}^{m_{1}(\lambda)}z_{2}^{m_{2}(\lambda)}z_{3}^{m_{3}(\lambda)}\\\
&=\frac{3}{1-q^{-3}}\Bigg{\\{}(-1)^{2}\cdot 1\cdot
1z_{3}+(-1)^{3}\frac{1}{2}\cdot 2z_{1}z_{2}+(-1)^{4}\frac{1}{3}\cdot
3z_{1}^{3}\Bigg{\\}}\\\
&=\frac{3}{1-q^{-3}}\Bigg{\\{}(1-q^{-1})I_{3}-(1-q^{-1})^{2}I_{1}\cdot
I_{2}+(1-q^{-1})^{3}I_{1}^{3}\Bigg{\\}}\\\
&=\frac{3}{1-q^{-3}}\Bigg{\\{}(1-q^{-1})I_{3}-(1-q^{-1})^{2}(qI_{2}\oplus
I_{1}+I_{3})+(1-q^{-1})^{3}\big{\\{}{2\choose 1}_{+}{3\choose
1}_{+}I_{1}^{\oplus 3}\\\ &+({2\choose 1}_{+}+q)I_{2}\oplus
I_{1}+I_{3}\big{\\}}\Bigg{\\}}\\\
&=\frac{3}{q^{3}-1}\Bigg{\\{}I_{3}+(q-1)^{2}(q^{2}-q-1)I_{2}\oplus
I_{1}+(q-1)^{3}(q+1)(q^{2}+q+1)I_{1}^{\oplus 3}\Bigg{\\}}\\\ \end{split}$

###  7.2 An identity of Hall number

In $\mathbf{H}_{\hat{A}_{0}}^{nil}$, the two formulas

$p_{n}=\sum_{\lambda\vdash n}(1-q)...(1-q^{l(\lambda)-1})I_{\lambda}$

and

$p_{n}=\frac{n}{1-q^{-n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}(1-q^{-1})^{l(\lambda)}I_{(1)}^{m_{1}(\lambda)}...I_{(n)}^{m_{n}(\lambda)}$

represent same elements. Suppose that

$\displaystyle
I_{(1)}^{m_{1}(\lambda)}...I_{(n)}^{m_{n}(\lambda)}=\sum_{\mu\vdash
n}f_{(1)^{m_{1}(\lambda)},...,(n)^{m_{n}(\lambda)}}^{\mu}I_{\mu}$

where $f_{(1)^{m_{1}(\lambda)},...,(n)^{m_{n}(\lambda)}}^{\mu}$ is the
corresponding Hall number, then

$\displaystyle\begin{split}&\sum_{\lambda\vdash
n}(1-q)...(1-q^{l(\lambda)-1})I_{\lambda}\\\
&=\frac{n}{1-q^{-n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}(1-q^{-1})^{l(\lambda)}I_{(1)}^{m_{1}(\lambda)}...I_{(n)}^{m_{n}(\lambda)}\\\
&=\frac{n}{1-q^{-n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}(1-q^{-1})^{l(\lambda)}\sum_{\mu\vdash
n}f_{(1)^{m_{1}(\lambda)},...,(n)^{m_{n}(\lambda)}}^{\mu}I_{\mu}\\\
&=\frac{n}{1-q^{-n}}\sum_{\lambda\vdash n}\sum_{\mu\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}(1-q^{-1})^{l(\lambda)}f_{(1)^{m_{1}(\lambda)},...,(n)^{m_{n}(\lambda)}}^{\mu}I_{\mu}\end{split}$

If we switch $\lambda,\mu$, we get that

$\displaystyle\begin{split}&=\frac{n}{1-q^{-n}}\sum_{\mu\vdash
n}\sum_{\lambda\vdash n}(-1)^{l(\mu)+1}\frac{1}{l(\mu)}{l(\mu)\choose
m_{1}(\mu),...,m_{n}(\mu)}(1-q^{-1})^{l(\mu)}f_{(1)^{m_{1}(\mu)},...,(n)^{m_{n}(\mu)}}^{\lambda}I_{\lambda}\\\
&=\frac{n}{1-q^{-n}}\sum_{\lambda\vdash n}\sum_{\mu\vdash
n}(-1)^{l(\mu)+1}\frac{1}{l(\mu)}{l(\mu)\choose
m_{1}(\mu),...,m_{n}(\mu)}(1-q^{-1})^{l(\mu)}f_{(1)^{m_{1}(\mu)},...,(n)^{m_{n}(\mu)}}^{\lambda}I_{\lambda}\end{split}$

So we get the following theorem.

######  Theorem 7.2.1.

For a fixed $\lambda$, we have the identity for
$f_{(1)^{m_{1}(\mu)},...,(n)^{m_{n}(\mu)}}^{\lambda}$ that

$(1-q)...(1-q^{l(\lambda)-1})=\frac{n}{1-q^{-n}}\sum_{\mu\vdash
n}(-1)^{l(\mu)+1}\frac{1}{l(\mu)}{l(\mu)\choose
m_{1}(\mu),...,m_{n}(\mu)}(1-q^{-1})^{l(\mu)}f_{(1)^{m_{1}(\mu)},...,(n)^{m_{n}(\mu)}}^{\lambda}$

###  7.3 Primitive elements for acyclic quiver

For a tame acyclic quiver $Q$, tubes of homogeneous regular representation are
parametrized by the set $\mathbf{Proj}k[x,y]$. So for
$\rho\in\mathbf{Proj}k[x,y]$, the corresponding tubes denoted as
$\tau_{\rho}$. Because prime ideals in $k[x,y]$ are $(0)$, $(f(y))$ where
$f(y)$ is an irreducible polynomial in $(k[x])[y]$, and $(p,f(y))$ where $p\in
k[x]$ is prime, and $f(y)$ is irreducible in $(k[x]/(p))[y]$. For
$\mathbf{Proj}k[x,y]$ are homogeneous prime ideals in $k[x,y]$, not containing
$k[x,y]_{+}=(x,y).$ So only elements in $\mathbf{Proj}k[x,y]$ are $(0)$ and
homogeneous cases for $(f(y))$. Then denote deg$\rho$ as the degree of the
generator of prime ideals. $\tau_{\rho}$ is equivalent to category of
nilpotent representations of Jordan quiver. Consider the Hall algebra
$H_{\tau_{\rho}}$ which is isomorphic to the classical Hall algebra, so
primitive elements in $H_{\tau_{\rho}}$ are,

$P_{n}(\rho)=\sum_{\lambda\vdash
n}\left(\prod_{j=1}^{l(\lambda)-1}(1-q^{j\text{deg}{\rho}})\right)I_{\lambda}$

For a fixed $\tau_{\rho}$, $P_{n}(\rho)=p_{n}$ in the Hall algebra of Jordan
quiver. But the above elements are not necessarily primitive in Hall algebra
$\mathbf{H}_{Q}$ because an object in $\tau_{\rho}$ can have preprojective
subobjects and preinjective quotients. In paper by [1], the authors make
conjecture 3.3 that the elements

$\displaystyle\begin{split}P_{n}(\rho)-\frac{1}{N(deg\rho)}\sum_{\rho^{\prime}\in\mathbb{KP}^{1}:deg{\rho^{\prime}}=deg{\rho}}P_{n}(\rho^{\prime})\
\end{split}$ (7.3.1)

are primitive in $H_{Q}$, where $N(d)$ is the number of elements of
$\mathbb{KP}^{1}$ of degree d. In paper [3], the authors proved this
conjecture to be correct. In Hall algebra of Jordan quiver, we have our
formula that,

$p_{n}=\frac{n}{1-q^{-n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}z_{\lambda}$

so we can substitute our formula into the formula (6.3.1) for $P_{n}(\rho)$.
For a fixed $d$, $N(d)=\frac{1}{d}\sum_{k|d}\mu(\frac{d}{k})q^{k}$, where
$\mu(r)$ is the Möbius function. So we have that primitive elements have
formula that

######  Corollary 1.

For a tame acyclic quiver, elements $p_{n}$ in the following form are
primitive,

$\displaystyle\begin{split}p_{n}=\frac{n}{1-q^{-n}}\sum_{\lambda\vdash
n}(-1)^{l(\lambda)+1}\frac{1}{l(\lambda)}{l(\lambda)\choose
m_{1}(\lambda),...,m_{n}(\lambda)}\big{(}z_{\lambda}(\rho)-\frac{1}{N(deg\rho)}\sum_{\rho^{\prime}}z_{\lambda}(\rho^{\prime})\big{)}\end{split}$
(7.3.2)

###  7.4 Primitive elements for Hall algebra of cyclic quiver for all case

Representation of cyclic quiver over finite field has the property that all
representations are regular. So we have that

$\displaystyle\mathbf{H}_{\hat{A}_{N}}$
$\displaystyle=\mathbf{H}_{\hat{A}_{N}}^{homo}\times\mathbf{H}_{\hat{A}_{N}}^{nonhomo}$
(7.4.1)

Here $\mathbf{H}_{\hat{A}_{N}}^{homo}$ is isomorphic to
$\mathbf{H}_{\hat{A}_{0}}^{nil}$, $\mathbf{H}_{\hat{A}_{N}}^{nonhomo}$ is
isomorphic to $H_{\hat{A}_{N}}^{nil}$. So we have (7.7) as

$\displaystyle\mathbf{H}_{Q}$
$\displaystyle=\mathbf{H}_{Q}^{homo}\times\mathbf{H}_{Q}^{nonhomo}$
$\displaystyle=\mathbf{H}_{Q}^{homo}\times Z(Q)\times C(Q)$

So the center is $\mathbf{H}_{Q}^{homo}\times Z(Q)$ and the primitive elements
are $(p_{n},p_{m},[0])$ where $p_{n}$ is a primitive element of
$\mathbf{H}_{\hat{A}_{0}}^{nil}$ and $p_{m}$ is a primitive element of
$\mathbf{H}_{\hat{A}_{N}}^{nil}$.

###  7.5 Primitive elements for Hall algebra of 2-periodic complexes

Bridgeland defined a twisted Hall algebra $\mathbf{DH}(\mathcal{A})$ over
$\mathbb{Z}_{2}-graded$ complexes for any finitary category $\mathcal{A}$ in
[2]. Bridgeland proves that when $\mathcal{A}$ is finitary, hereditary, there
is an isomorphism of algebra $\Phi:$ Drinfeld double
$\tilde{\mathbf{H}}_{\mathcal{A}}\rightarrow\mathbf{DH}(\mathcal{A})$. Later,
Yanagida [11] proves that $\mathbf{DH}(\mathcal{A})$ is actually a bialgebra
and $\mathbf{DH}(\mathcal{A})$ is isomorphic to
$\tilde{\mathbf{H}}_{\mathcal{A}}$ as a bialgebra. Then we have for cyclic
quiver case, we can use our method to prove primitive elements by maps
$\Lambda[t]\rightarrow$ center of
$\tilde{\mathbf{H}}_{\mathcal{A}}\hookrightarrow\mathbf{DH}(\mathcal{A}).$

###  7.6 Examples of $z_{n}$

For the case of $\mathbf{H}_{\hat{A}_{1}}^{nil}$, since

$\displaystyle z_{n}=(-1)^{n}q^{-2n}\sum_{\begin{aligned} &dim[M]=n\delta\\\
&soc(M)\text{ is square-free}\\\ \end{aligned}}(-1)^{dimEnd(M)}a_{M}[M].$
(7.6.1)

In order for $[M]$ to be square-free, $[M]=I_{[1;l_{1}]}\oplus I_{[0;l_{0}]}$
with $dimI_{[1;l_{1}]}=(a_{1},a_{2})$, $dimI_{[0;l_{0}]}=(b_{1},b_{2})$. $[M]$
only exists for

  * case 1

$a_{1}=a_{2}$
$[M]=I_{[1;2a]}\oplus I_{[0;2(n-a)]}$, $a_{M}=|Aut(I_{[1;2a]}\oplus
I_{[0;2b]})|=(q-1)^{2}q^{n-2}$

  * case 2

$|a_{1}-a_{2}|=1$
$[M]=I_{[1;2a-1]}\oplus I_{[0;2(n-a)+1]}$, $a_{M}=|Aut(I_{[1;2a-1]}\oplus
I_{[0;2(n-a)+1]})|=(q-1)^{2}q^{n-1}$

So

$\displaystyle\begin{split}z_{n}&=q^{-2n}[\sum_{a=1}^{n}(-1)^{n}(q-1)^{2}q^{n-2}I_{[1;2a]}\oplus
I_{[0;2(n-a)]}+\sum_{a=1}^{n}(-1)^{n+1}(q-1)^{2}q^{n-1}I_{[1;2a-1]}\oplus
I_{[0;2(n-a)+1]}]\\\
&=(-q^{-1})^{n}(1-q^{-1})^{2}\sum_{a=1}^{n}I_{[1;2a]}\oplus
I_{[0;2(n-a)]}-qI_{[1;2a-1]}\oplus I_{[0;2(n-a)+1]}\end{split}$

## Bibliography

  * [1] Arkady Berenstein and Jacob Greenstein  “Primitively Generated Hall Algebras”  In _Pacific Journal of Mathematics_ 281.2, 2016, pp. 287–330  DOI: 10.2140/pjm.2016.281.287
  * [2] Tom Bridgeland  “Quantum groups via Hall algebras of complexes”  In _Annals of Mathematics_ 177, 2013, pp. 739–759  DOI: http://dx.doi.org/10.4007/annals.2013.177.2.9
  * [3] Bangming Deng and Shiquan Ruan  “Hall polynomials for tame type”, 2015  arXiv:1512.03504 [math.RT]
  * [4] James A. Green  “Hall algebras, hereditary algebras and quantum groups”  In _Inventiones mathematicae_ 120, 1995, pp. 361–377
  * [5] Andrew Hubery  “Ringel-Hall Algebras of Cyclic Quivers”  In _Sao Paulo Journal of Mathematical Sciences 4_ 3, 2010
  * [6] I.G.Macdonald  “Symmetric Functions and Hall Polynomials”  Oxford University Press, 2015
  * [7] Xiao J  “Drinfeld double and Ringel-Green theory of Hall algebras”  In _J.Algebra_ 190.1, 1997, pp. 100–144
  * [8] Claus Michael Ringel  “Hall algebra and quantum groups”  In _Inventiones mathematicae_ 101, 1990, pp. 583–592
  * [9] Olivier Schiffmann  “Lecturs on Hall Algebras”, 2009  arXiv:math.RT/0611617 [math.RT]
  * [10] Olivier Schiffmann  “The Hall algebra of a cyclic quiver and canonical bases of Fock spaces”  In _International Mathematics Research Notices_ 2000.8, 2000, pp. 413–440  DOI: 10.1155/S1073792800000234
  * [11] Shintarou Yanagida  “A note on Bridgeland’s Hall algebra of two-periodic complexes”  arXiv, 2012  DOI: 10.48550/ARXIV.1207.0905


10

# Primordial black hole formation from a nonspherical
density profile with a misaligned deformation tensor

Chul-Moon Yoo<EMAIL_ADDRESS>Division of Particle and
Astrophysical Science, Graduate School of Science,
Nagoya University, Nagoya 464-8602, Japan

###### Abstract

We perform the numerical simulation of primordial black hole formation from a
nonspherical profile of the initial curvature perturbation $\zeta$. We
consider the background expanding universe filled with the perfect fluid with
the linear equation of state $p=w\rho$ ($w=1/3$ or $1/5$), where $p$ and
$\rho$ are the pressure and the energy density, respectively. The initial
condition is set in a way such that the principal directions of the second
derivatives of $\zeta$ and $\triangle\zeta$ at the central peak are
misaligned, where $\triangle$ is the Laplacian. In this setting, since the
linearized density is proportional to $\triangle\zeta$, the inertia tensor and
deformation tensor $\partial_{i}\partial_{j}\zeta$ are misaligned. Thus tidal
torque may act and the spin of a resultant primordial black hole would be non-
zero in general, although it is estimated to be very small from previous
perturbative analyses. As a result, we do not find a finite value of the spin
within our numerical precision, giving support for the negligibly small value
of the black hole spin for $1/5\lesssim w\lesssim 1/3$. More specifically, our
results suggest that the dimensionless PBH spin $s$ is typically so small that
$s\ll 0.1$ for $w\gtrsim 0.2$.

##  I Introduction

The possibility of black hole formation in the early universe has been
proposed in Refs. 1967SvA….10..602Z ; Hawking:1971ei ; Carr:1974nx  more than
half a century ago. Since then, it had been a fascinating but relatively minor
possible scenario until the first detection of the gravitational waves from a
binary black hole Abbott:2016blz . However, after the possibility for the
binary to originate from primordial black holes (PBHs) is pointed out
Sasaki:2016jop , people started to realize the utility of PBHs for many kinds
of subjects in cosmology and astrophysics.

PBHs are the remnants of primordial inhomogeneity and there is no doubt that
black holes can be formed if sufficiently over-dense regions exist in the
early universe. They may play the role of dark matter Carr:2009jm ;
Carr:2020gox ; Carr:2020xqk  or could be the source of micro-lensing events
Niikura:2019kqi  and/or black hole binaries observed by the gravitational
waves Bird:2016dcv ; Sasaki:2016jop ; Clesse:2017bsw ; Sasaki:2018dmp . In
addition, they may be the seeds of supermassive black holes and cosmic
structures Kawasaki:2012kn ; Kohri:2014lza ; Nakama:2016kfq ; Carr:2018rid ;
Serpico:2020ehh ; Unal:2020mts ; Kohri:2022wzp . Another attractive observable
associated with large primordial perturbations is the gravitational waves
induced by the primordial curvature perturbations Ananda:2006af ;
Baumann:2007zm ; Saito:2008jc ; Saito:2009jt ; Assadullahi:2009nf ;
Bugaev:2009zh ; Bugaev:2010bb ; Espinosa:2018eve ; Kohri:2018awv ;
Domenech:2021ztg . Since the induced gravitational waves and PBHs share the
primordial curvature perturbations as those origins, combination of those
observations are attractive tools to probe the early universe.

In this paper, we focus on the PBH formation in a nonspherical setting and
possible generation of the black hole spin. Although spherically symmetric
dynamics for PBH formation has been mainly considered Niemeyer:1997mt ;
Niemeyer:1999ak ; Shibata:1999zs ; 2002CQGra..19.3687H ; Musco:2004ak ;
Harada:2013epa ; Nakama:2013ica ; Nakama:2014fra ; Harada:2015yda ;
Musco:2018rwt ; Escriva:2019nsa ; Escriva:2019phb ; Escriva:2020tak ;
Escriva:2021aeh ; Musco:2021sva ; Escriva:2022bwe ; Franciolini:2022tfm ;
Papanikolaou:2022cvo ; Escriva:2023qnq ; Uehara:2024yyp  for many years, we
can also find several works in which nonspherical PBH formation is considered
Khlopov:1980mg ; Harada:2016mhb ; Chiba:2017rvs ; Harada:2017fjm ;
DeLuca:2019buf ; Harada:2020pzb ; Yoo:2020lmg ; deJong:2021bbo ; Saito:2023fpt
; deJong:2023gsx . Some of them Khlopov:1980mg ; Harada:2016mhb ;
Harada:2017fjm  focus on the PBH formation in a matter-dominated universe, in
which the non-spherical collapse would be essential for understanding the
criterion of the PBH formation. In the case of a radiation-dominated universe
Chiba:2017rvs ; DeLuca:2019buf ; Harada:2020pzb ; Yoo:2020lmg , the pressure
gradient is the main obstacle to PBH formation and a large amplitude of the
curvature perturbation is required. Then, according to the peak theory
1986ApJ…304…15B , the system approaches a spherically symmetric configuration
in the high peak limit, so that the deviation from the spherical configuration
becomes relatively ineffective. The situation would not change much for the
cases of somewhat softer equations of states Saito:2023fpt .

The previous works DeLuca:2019buf ; Harada:2020pzb ; Saito:2023fpt  suggest
that, for the case of a perfect fluid with non-negligible pressure, the PBH
spin is negligibly small immediately after the formation. Nevertheless, the
PBH spin has been estimated based on the perturbative analyses and non-linear
simulation with a standard initial setting has not been done yet (see Refs.
deJong:2021bbo ; deJong:2023gsx  for simulations with massive and massless
scalar fields). The purpose of this work is to check the validity of the
results obtained in Refs. DeLuca:2019buf ; Harada:2020pzb ; Saito:2023fpt
through fully non-linear numerical simulations. Therefore, following the
setting in Refs. DeLuca:2019buf ; Harada:2020pzb ; Saito:2023fpt , we consider
only one field variable, which corresponds to the growing mode solution of the
curvature perturbation in the long-wavelength limit, characterizing the
initial condition. Then we suppose the tidal torque as the mechanism to
generate angular momentum transfer. That is, we do not explicitly introduce
the initial angular momentum in an ad-hoc manner, and check the efficiency of
the tidal torque in the PBH formation.

Throughout this paper, we use the geometrized units in which both the speed of
light and Newton’s gravitational constant are set to unity, $c=G=1$.

##  II Basic setups

First, let us introduce the characteristic comoving length scale $1/k$ for the
initial perturbation. Since we are interested in the perturbations which are
initially super-horizon scale, we assume
$\epsilon:=k/(a_{\mathrm{i}}H_{\mathrm{i}})\ll 1$ with $a_{\mathrm{i}}$ and
$H_{\mathrm{i}}$ being the scale factor and the Hubble expansion rate at the
initial time, respectively. For simplicity, hereafter we assume that the
matter field is given by the perfect fluid with the linear equation of states
$p=w\rho$, where $p$ and $\rho$ are the pressure and the energy density of the
fluid. In this paper, we consider the two specific values $1/3$ and $1/5$ for
$w$. Growing mode solutions for super-horizon scale perturbations can be
obtained by performing the gradient expansion Shibata:1999zs ; Harada:2015yda
and it turns out that, at the leading order, the solution can be characterized
by the conformal factor of the spatial metric given as an arbitrary function
of spatial coordinates $x^{i}$. Following a convention, we write the spatial
metric of the initial condition as ${\rm
e}^{-2\zeta(\bm{x})}a_{\mathrm{i}}^{2}\eta_{ij}$, where $\eta_{ij}$ is the
reference flat metric and $\zeta$ is an arbitrary function of the spatial
coordinates $\bm{x}$. We simply call $\zeta$ the curvature perturbation
hereafter.

In practice, setting the functional form of $\zeta$, we calculate all
geometrical variables following Ref. Harada:2015yda  with the uniform Hubble
(constant-mean-curvature) slicing and the normal coordinates (vanishing shift
vectors). Then the fluid variables are calculated by using the exact form of
the Hamiltonian and momentum constraint equations. That is, the fluid
configuration is set in a way such that the constraint equations are satisfied
within the machine’s precision. The configuration is also consistent with the
long-wavelength approximation as long as the typical length scale
$a_{\mathrm{i}}/k$ of the perturbation is sufficiently larger than the Hubble
length $1/H_{\mathrm{i}}$.

For the numerical simulation, we only consider the numerical domain given by
$-L\leq X\leq L$, $0\leq Y\leq L$ and $0\leq Z\leq L$, where $X$, $Y$ and $Z$
are the reference Cartesian coordinates. We note that the Cartesian
coordinates can be different from the reference spatial coordinates $x^{i}$ in
general. Indeed, we will introduce scale-up coordinates as $x^{i}$ later. We
identify the boundary surface $(0<X\leq L,Y=0,0\leq Z\leq L)$ as $(-L\leq
X<0,Y=0,0\leq Z\leq L)$ with the $\pi$ rotation around $Z$ axis (see Fig. 1).
The reflection symmetries are assumed for other boundary surfaces. The
numerical region and the boundary conditions are summarized in Fig. 1.

Figure 1: The numerical region and boundary conditions.

In this paper, we use the following specific functional form of $\zeta$:

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle-\mu\Bigl{[}1+\frac{1}{2}\left(k_{1}^{2}(X+Y)^{2}/2+k_{2}^{2}(X-Y)^{2}/2+k_{3}^{2}Z^{2}\right)$
(1) $\displaystyle+$
$\displaystyle\frac{1}{4}\left(k_{1}^{2}(X+Y)^{2}/2+k_{2}^{2}(X-Y)^{2}/2+k_{3}^{2}Z^{2}\right)^{2}$
(2) $\displaystyle+$
$\displaystyle\frac{1}{280}k^{2}R^{2}\left(9\kappa_{1}^{2}-\kappa_{2}^{2}-\kappa_{3}^{2}\right)X^{2}+(\kappa_{1}^{2}-9\kappa_{2}^{2}+\kappa_{3}^{2})Y^{2}+(\kappa_{1}^{2}+\kappa_{2}^{2}-9\kappa_{3}^{2})Z^{2}\Bigr{]}^{-1}$
(4) $\displaystyle\exp\left[-\frac{1}{2880}k^{6}R^{6}\right],$

where

$\displaystyle R^{2}$ $\displaystyle=$ $\displaystyle X^{2}+Y^{2}+Z^{2},$ (5)
$\displaystyle k_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\hat{\xi}_{1}+3\hat{\xi}_{2}+\hat{\xi}_{3}\right),$
(6) $\displaystyle k_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\hat{\xi}_{1}-3\hat{\xi}_{2}+\hat{\xi}_{3}\right),$
(7) $\displaystyle k_{3}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\hat{\xi}_{1}-2\hat{\xi}_{3}\right),$ (8)
$\displaystyle\kappa_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\widetilde{\xi}_{1}+3\widetilde{\xi}_{2}+\widetilde{\xi}_{3}\right),$
(9) $\displaystyle\kappa_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\widetilde{\xi}_{1}-3\widetilde{\xi}_{2}+\widetilde{\xi}_{3}\right),$
(10) $\displaystyle\kappa_{3}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\widetilde{\xi}_{1}-2\widetilde{\xi}_{3}\right)$
(11)

with

$\displaystyle
k^{2}=\sum_{i=1,2,3}k_{i}^{2}=\hat{\xi}_{1}=\sum_{i=1,2,3}\kappa_{i}^{2}=\widetilde{\xi}_{1}=100/L^{2},$
(12)
$\displaystyle\hat{\xi}_{2}=10/L^{2},~{}\widetilde{\xi}_{2}=15/L^{2},~{}\hat{\xi}_{3}=\widetilde{\xi}_{3}=0.$
(13)

There are some reasons for using this specific form of the curvature
perturbation $\zeta$.

First, by expanding $\zeta$ around the origin, we can find

$\displaystyle\frac{\zeta}{\mu}$ $\displaystyle\simeq$
$\displaystyle-1+\frac{1}{2}\left(k_{1}^{2}(X+Y)^{2}/2+k_{2}^{2}(X-Y)^{2}/2+k_{3}^{2}Z^{2}\right)+\mathcal{O}(R^{4}),$
(14) $\displaystyle\frac{\triangle\zeta}{\mu k^{2}}$ $\displaystyle\simeq$
$\displaystyle
1-\frac{1}{2}\left(\kappa_{1}^{2}X^{2}+\kappa_{2}^{2}Y^{2}+\kappa_{3}^{2}Z^{2}\right)+\mathcal{O}(R^{4}),$
(15)

where $\triangle$ is the Laplacian for the reference flat metric. Since the
linear density perturbation $\delta$ is proportional to $\triangle\zeta$, the
principal directions of the deformation tensor $\partial_{i}\partial_{j}\zeta$
and the inertia tensor are misaligned with each other with $\pi/4$. Because of
this misalignment, we may expect the tidal torque acts and the gravitational
collapse is accompanied by the rotation DeLuca:2019buf ; Harada:2020pzb . The
last factor $\exp\left[-\frac{1}{2880}k^{6}R^{6}\right]$ is multiplied to
realize $\zeta\simeq 0$ around the outer boundary. In practice, we also
introduce a window function as in Ref. Yoo:2018pda  to regularize $\zeta$ in
the vicinity of the outer boundary (see Eq. (24) in Ref. Yoo:2018pda  for the
specific form of the window function).

The eigenvalues of the inertia and deformation tensors $k_{i}$ and
$\kappa_{i}$ are characterized by $\hat{\xi}_{i}$ and $\widetilde{\xi}_{i}$.
The reason for using this expression is in the probability distribution based
on the peak theory for those parameters 1986ApJ…304…15B . Let us introduce the
$n$-th order gradient moment $\sigma_{n}$ 1986ApJ…304…15B . According to the
peak theory, the values of $-\zeta|_{\rm peak}/\sigma_{0}$,
$\triangle\zeta|_{\rm peak}/\sigma_{2}$ and $-\triangle\triangle\zeta|_{\rm
peak}/\sigma_{4}$ are significantly correlated with each other (see Eq. (A7)
in Ref. Yoo:2020dkz  for the probability distribution function), where
$~{}|_{\rm peak}$ indicates the value at the peak. From these correlations, we
expect that the orders of magnitude for those variables are similar to each
other. Here we simply assumed $-\zeta
k^{2}\kappa^{2}=\triangle\zeta\kappa^{2}=-\triangle\triangle\zeta=\mu
k^{2}\kappa^{2}$ at the peak with
$k^{2}=\hat{\xi}_{1}=\kappa^{2}=\widetilde{\xi}_{1}$. The probability
distribution of $\xi_{2}:=\hat{\xi}_{2}/\sigma_{2}$ and
$\xi_{3}:=\hat{\xi}_{3}/\sigma_{2}$ are given by

$P(\xi_{2},\xi_{3})=\frac{5^{5/2}3^{2}}{\sqrt{2\pi}}\xi_{2}(\xi_{2}^{2}-\xi_{3}^{2})\exp\left[-\frac{5}{2}(3\xi_{2}^{2}+\xi_{3}^{2})\right]$
(16)

without correlation with $-\zeta|_{\rm peak}/\sigma_{0}$ and
$\triangle\zeta|_{\rm peak}/\sigma_{2}$ (see, e.g., Eqs.(2-6) in Ref.
Yoo:2020lmg  with $\lambda_{2}\leftrightarrow\lambda_{3}$). The parameter
region of $\xi_{2}$ and $\xi_{3}$ are restricted to
$-\xi_{2}\leq\xi_{3}\leq-\xi_{2}$ and $0\leq\xi_{2}$. Then we choose the most
probable value for $\xi_{3}$, namely, $\xi_{3}=0$. We can make a similar
argument for $\widetilde{\xi}_{3}$ assigning $\triangle\zeta$ to the role of
$\zeta$ in the above discussion. Thus we set
$\hat{\xi}_{3}=\widetilde{\xi}_{3}=0$. Then the probability distribution for
$\xi_{2}$ takes the maximum value at $\xi_{2}=1/\sqrt{5}\simeq 0.45$.
Therefore we can roughly estimate the typical value of $\hat{\xi}_{2}$ as
$\hat{\xi}_{2}=\xi_{2}\sigma_{2}\sim
k^{2}\sigma/\sqrt{5}=100\sigma/(\sqrt{5}L^{2})$ with $\sigma^{2}$ being the
typical amplitude of the curvature power spectrum. Since, for PBH formation
scenario, the value of $\sigma$ is typically given by $\sigma<0.1$, we may
evaluate the value of $\hat{\xi}_{2}$ as $\hat{\xi}_{2}\lesssim 5/L^{2}$.
Therefore the values of $\hat{\xi}_{2}=10/L^{2}$ and
$\widetilde{\xi}_{2}=15/L^{2}$ are unexpectedly large. These large values of
$\hat{\xi}_{2}$ and $\widetilde{\xi}_{2}$ together with the finely tuned
misalignment angle $\pi/4$ are assumed to make the tidal torque more
effective. That is, the setting is optimized for the generation of the PBH
spin, and the value of the PBH spin is expected to be larger than the typical
value.

##  III time evolution

Our simulation code is based on the COSMOS code developed in Refs. Yoo:2013yea
; Okawa:2014nda . For the simulation, we follow the numerical schemes adopted
in Ref. Yoo:2020lmg , newly implementing a mesh-refinement procedure in the
central region. The 4th-order Runge-Kutta method with the BSSN (Baumgarte-
Shapiro-Shibata-Nakamura) formalism is used for solving the Einstein equations
Shibata:1995we ; Baumgarte:1998te . For the spatial coordinates, we employ the
scale-up coordinates introduced in Ref. Yoo:2018pda  with the parameter
$\eta=10$, where the ratio between the scale-up coordinate interval $\Delta x$
and the Cartesian coordinate interval $\Delta X$ is given by $\Delta x/\Delta
X=1+\eta$ at the center. For the mesh refinement, we introduce two upper
layers to resolve the gravitational collapse around the center. Therefore 4
times finer resolution is realized near the center when they are introduced.
We start the calculation with the single layer, and the upper layers are
introduced when the value of the lapse function gets smaller than 0.1 and 0.05
at the origin111The values at newly introduced grid points are evaluated by
the Lagrange interpolation. In the upper layer, 7 grid points on the boundary
are regarded as buffer points, and 3 of them (every two) are shared with the
lower layer. In an iteration step of the time evolution, first, the values on
the buffer grid points are evaluated by the interpolation from the values on
the lower layer. Then the values at the inner 4 buffer points are evaluated by
solving the evolution equations while the values at the outer 3 points are
evaluated by the interpolation from the values on the lower layer. In the
inside bulk region, the values obtained by the time evolution are kept and
taken to the lower layer grid points while, for the buffer grid points, they
are discarded just before the next evolution step of the lower layer. . For
the initial condition, we set $a_{\mathrm{i}}=1$ and $H_{\mathrm{i}}=50/L=5k$.
The number of grids for each side is taken as 60, 70 and 80 for the initial
lowest layer, and the convergence is checked.

First, let us show the existence of the threshold value of the amplitude $\mu$
for the black hole formation. In order to explicitly show the existence of the
threshold value, we show the time evolution of the value of the lapse function
at the origin. In our gauge condition for the numerical simulation, the value
of the lapse function significantly decreases and approaches zero before the
horizon formation. On the other hand, if the value of the amplitude $\mu$ is
not sufficiently large, the value of the lapse function at the origin bounces
back and no horizon formation is observed. In Fig. 2, we show the value of the
lapse function at the origin as a function of the coordinate time.

Figure 2:  The time evolution of the lapse function at the origin. The
detection of the horizon triggers the excision procedure inside the horizon
and the lapse function at the origin is fixed after that.

From the behavior of the lapse function, we can see that there are threshold
values of $\mu$ between 0.91 and 0.92 for $w=1/3$ and between 0.72 and 0.73
for the $w=1/5$ case.

Hereafter we show the results for the case I:$(w,\mu)=(1/3,0.92)$ and case
II:$(w,\mu)=(1/5,0.73)$, that is, the cases slightly above the threshold
values. First, let us show the violation of the Hamiltonian constraint in Fig.
3.

Figure 3: Time evolution of the normalized max-norm of the Hamiltonian
constraint violation.

The violation of the Hamiltonian constraint is evaluated at each grid point
with an appropriate normalization, and the maximum value $H_{\rm max}$ is
taken including the grid points on the higher layers if they exist. We
excluded the grid points well inside the horizon for the evaluation of the
constraint violation. A discontinuous change can happen when a higher layer is
introduced or the horizon is formed. In addition, depending on how to exclude
the grid points inside the horizon, the value of $H_{\rm max}$ slightly
changes. Due to this dependence, a discontinuous change also may be observed
when the calculation is terminated once and continued with the data at the
termination (it can be seen at around $t=210L$ for the red solid line in Fig.
3). Overall, the constraint violation is acceptably small outside the horizon
and reasonable convergent behaviors with different resolutions can be found.

Let us show the snapshots of the contour map of the fluid comoving density on
the $z=Z=0$ plane in Figs. 4 and 5.

Figure 4: Snapshots of the contour map of the fluid comoving density on the
$z=Z=0$ plane for the case I:$(w,\mu)=(1/3,0.92)$. The figures are described
in the Cartesian coordinates $X$ and $Y$. The green meshes in the last two
panels describe the apparent horizon. The horizon is first found at the time
$t=91.7L$.  Figure 5:  Snapshots of the contour map of the fluid comoving
density on the $z=Z=0$ plane for the case II:$(w,\mu)=(1/5,0.73)$. The figures
are described in the Cartesian coordinates $X$ ad $Y$. The green meshes in the
last two panels describe the apparent horizon. The horizon is first found at
the time $t=201.4L$.

It can be seen that the system is highly non-spherical until the apparent
horizon forms. The time evolutions of the contour map show oscillating
behaviors rather than rotation. The two cases (I and II) are qualitatively
similar to each other if the time scale is appropriately tuned.

##  IV evaluation of spin and asphericity of the horizon

In this section, we try to estimate the spin and asphericity of the black hole
by using geometrical quantities of the horizon. In a realistic situation, if
we can perform the numerical simulation for a sufficiently long time. The
black hole would approach a stationary asymptotically flat black hole, namely,
a Kerr black hole, because the size of the cosmological horizon becomes much
larger than the black hole horizon scale and the effect of accretion becomes
negligible. First, let us check the accretion effect by plotting the time
evolution of the horizon area $A$. We plot the value
$\sqrt{A/(4\pi)}/(2M_{H})$ rather than $A$ itself noting that
$\sqrt{A/(4\pi)}$ has the mass dimension, where $M_{H}$ is the horizon mass
defined in Eq. (29).

Figure 6: Time evolution of the value of $\sqrt{A/(4\pi)}/(2M_{H})$, where
$M_{H}$ is the horizon mass defined in Eq. (29).

For case I, the mass accretion almost ends around $t=150L$. For case II,
although the area of the horizon is gradually increasing even around $t=400L$,
the contribution of the accreting mass is expected to be subdominant for the
black hole.

In order to evaluate the spin of the black hole, let us refer to the Kerr
black hole with the mass $M$ and the spin parameter $a$. The area of the
horizon $A_{\rm Kerr}$ is given by

$A_{\rm Kerr}=8\pi(M^{2}+\sqrt{M^{4}-a^{2}M^{2}}).$ (17)

The equatorial circumference $d_{\rm Kerr}$ and meridional circumference
$\ell_{\rm Kerr}$ of the Kerr black hole can be calculated as

$\displaystyle d_{\rm Kerr}$ $\displaystyle=$ $\displaystyle 4\pi M,$ (18)
$\displaystyle\ell_{\rm Kerr}$ $\displaystyle=$ $\displaystyle
4\sqrt{2Mr_{+}}E\left(\frac{a^{2}}{2Mr_{+}}\right),$ (19)

where $r_{+}=M+\sqrt{M^{2}-a^{2}}$ and $E()$ is the complete elliptic integral
of the 2nd kind. Combining Eq. (17) and (18), we find

$\frac{a^{2}}{M^{2}}=\frac{4\pi A_{\rm Kerr}\left(d_{\rm Kerr}^{2}-\pi A_{\rm
Kerr}\right)}{d_{\rm Kerr}^{4}}.$ (20)

Therefore, by using the horizon area $A$ and the equatorial circumference $d$,
we define the effective dimensionless spin parameter $s$ as

$s^{2}:=\frac{4\pi A\left(d^{2}-\pi A\right)}{d^{4}}.$ (21)

We also define the following indicators for the asphericity of the horizon:

$\displaystyle\alpha$ $\displaystyle:=$ $\displaystyle\frac{\ell_{x=0}}{d},$
(22) $\displaystyle\beta$ $\displaystyle:=$
$\displaystyle\frac{\ell_{y=0}}{d},$ (23)

where $\ell_{X=0}$ and $\ell_{Y=0}$ are the meridional circumference measured
on the $X=0$ and $Y=0$ planes, respectvely. The deviation of $\alpha$ or
$\beta$ from 1 indicates that the horizon is non-spherical.

In Fig. 7, we plot the effective dimensionless spin parameter $s^{2}$ as a
function of the time $t$ for each parameter set. It can be found that the
value of $s^{2}$ seems to converge to a negatively small value at a late time.
The absolute value of $s^{2}$ tends to be a smaller value for a finer
resolution and seems to converge to a very small value $|s^{2}|\lesssim
10^{-3}$ for a sufficiently fine resolution. Therefore we conclude that our
result is consistent with $s^{2}=0$ within the numerical precision.

Figure 7: Time evolution of the effective dimensionless spin $s^{2}$.

Since our setting is not exactly vacuum and not asymptotically flat, the value
of $s^{2}$, which is defined in analogy with the Kerr black hole solution, may
not be perfectly suitable for our purpose. Thus we also check the values of
$\alpha$ and $\beta$, which are the indicators of the asphericity of the
horizon. If the horizon is highly spherically symmetric, we may not expect the
large value of the spin parameter. As is shown in Fig. 8, the horizon is
highly spherically symmetric at a late time. We also show the value of
$\ell_{\rm Kerr}/d_{\rm Kerr}$ for the case $s_{\rm
Kerr}^{2}:=a^{2}/M^{2}=10^{-2}$ in Fig. 8. We can see that the dimensionless
spin parameter is expected to be much smaller than $10^{-2}$.

Figure 8:  Time evolution of the values of $\alpha$ and $\beta$ for each
parameter set. The solid black line shows the value of $\ell_{\rm Kerr}/d_{\rm
Kerr}$ for the case $s_{\rm Kerr}^{2}:=a^{2}/M^{2}=10^{-2}$. The results for
the highest resolution (80 grid points for each side in the lowest layer) are
used for this figure.

##  V summary and discussions

We performed numerical simulations of primordial black hole formation from a
non-spherical initial density profile with a misaligned deformation tensor.
The equation of states is assumed to be the linear relation between the energy
density $\rho$ and the pressure $p$, namely, $p=w\rho$ ($w=1/3$ or $1/5$). We
have optimized the initial profile for the generation of PBH spin due to the
tidal torque taking the probability estimation based on the peak theory into
account. That is, the initial setting is finely tuned to make the tidal torque
more effective. Then the probability of obtaining such a situation is highly
suppressed based on the peak theory for PBH formation, and the PBH spin is
expected to be larger than the typical value. Nevertheless, our results are
consistent with non-rotating PBH formation, that is, the spin parameter of the
resultant PBH is so small that we cannot detect the non-zero value of the spin
within our numerical precision. This result is qualitatively consistent with
the prediction provided by the analyses based on perturbative approaches
DeLuca:2019buf ; Harada:2020pzb ; Saito:2023fpt .

It should be mentioned that there are several caveats to our results. First,
we only performed the numerical simulation for a specific initial
inhomogeneity. Other initial profiles could give different results. Due to the
limitation of the computer resources, the support of the initial inhomogeneity
could not be much smaller than the numerical domain to keep a sufficient
resolution. Therefore, essentially, in our simulation, the system has been
characterized by only one scale of $1/k$. The contribution of multiple scales
might promote angular momentum transfer.

Even if we stick to the initial configuration treated in this paper, for a
value of the amplitude $\mu$ very close to the threshold for the black hole
formation, we would get a very small black hole associated with the critical
behaviorChoptuik:1992jv ; Koike:1995jm . Since a larger spin parameter has
been predicted in this critical scaling regime from the perturbative
analysesDeLuca:2019buf ; Harada:2020pzb ; Saito:2023fpt , we may expect to
detect a non-zero spin in the case near the threshold.

Our results suggest that, for $p/\rho=w\gtrsim 0.2$, the dimensionless PBH
spin $s$ is typically so small that $s\ll 0.1$. Since the softening due to the
QCD crossover is expected to satisfy $w>0.2$, the crossover seems not to
contribute much to spinning up PBHs. One interesting possibility is
considering a much softer equation of states $w\ll 0.2$ with a beyond-standard
model in the universe earlier than electroweak phase transition
Escriva:2022yaf ; Escriva:2023nzn . For the PBH formation in a matter-
dominated universe, since the pressure gradient force is absent, asphericity
is essential in the consideration of the PBH formation criterion
Harada:2017fjm ; Harada:2020pzb . Then we can expect highly spinning PBH
formation Harada:2020pzb . The simulations of the spinning PBH formation in
the early universe with the extremely soft equation of states and an early
matter-dominated universe would be interesting future works.

## Acknowledgements

We would like to thank T. Harada for useful comments on the draft. This work
was supported by JSPS KAKENHI Grant Numbers JP20H05850 and JP20H05853.

##  Appendix A Horizon entry time and horizon mass

We calculate the horizon entry time for the inhomogeneity whose length scale
is given by $1/k$ in the universe filled with the perfect fluid with the
equation of states given by $p=w\rho$. The scale factor and the Hubble
parameter behave as functions of the cosmological time $t$ as

$\displaystyle a$ $\displaystyle=$ $\displaystyle
a_{\mathrm{i}}\left(\frac{t}{t_{\mathrm{i}}}\right)^{\frac{2}{3(1+w)}},$ (24)
$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{2}{3(1+w)}\frac{1}{t},$
(25)

where $t_{\mathrm{i}}$ and $a_{\mathrm{i}}$ are the initial time and the scale
factor, respectively. The value of $aH$ is given by

$aH=a_{\mathrm{i}}H_{\mathrm{i}}\left(\frac{3}{2}(1+w)H_{\mathrm{i}}t\right)^{-\frac{1+3w}{3(1+w)}},$
(26)

where $H_{\mathrm{i}}$ is the initial Hubble parameter. Then the horizon entry
condition $a_{\rm ent}H_{\rm ent}=k$ can be rewritten as

$a_{\rm ent}H_{\rm
ent}=a_{\mathrm{i}}H_{\mathrm{i}}\left(\frac{3}{2}(1+w)H_{\mathrm{i}}t_{\rm
ent}\right)^{-\frac{1+3w}{3(1+w)}}=k.$ (27)

The horizon entry time $t_{\rm ent}$ is given by

$t_{\rm
ent}=\left(\frac{k}{a_{\mathrm{i}}H_{\mathrm{i}}}\right)^{-\frac{3(1+w)}{1+3w}}\frac{2}{3(1+w)H_{\mathrm{i}}}.$
(28)

The horizon mass $M_{\rm H}$ can be estimated as follows:

$\displaystyle M_{\rm H}=\frac{1}{2}H^{-1}_{\rm ent}$ $\displaystyle=$
$\displaystyle\frac{3}{4}(1+w)t_{\rm
ent}=\frac{1}{2H_{\mathrm{i}}}\left(\frac{k}{a_{\mathrm{i}}H_{\mathrm{i}}}\right)^{-\frac{3(1+w)}{1+3w}}.$
(29)

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