# Deformations of Hopf algebras by partial actions

Grasiela Martini, Antonio Paques and Leonardo Duarte Silva  Universidade
Federal do Rio Grande, Brazil<EMAIL_ADDRESS>Universidade Federal
do Rio Grande do Sul, Brazil<EMAIL_ADDRESS>Universidade Federal do Rio
Grande do Sul, Brazil<EMAIL_ADDRESS>
###### Abstract.

In this work we study deformations of a Hopf algebra $H$ by partial actions of
$H$ on its base field $\Bbbk$, via partial smash product algebras. We
introduce the concept of a $\lambda$-Hopf algebra as a Hopf algebra obtained
as a partial smash product algebra, and show that every Hopf algebra is a
$\lambda$-Hopf algebra. Moreover, a method to compute partial actions of a
given Hopf algebra on its base field is developed and, as an application, we
exhibit all partial actions of such type for some families of Hopf algebras.

MSC 2020: 16T05, 16T99, 16S40, 16S99, 16W99.

Key words and phrases: Deformation of a Hopf algebra, $\lambda$-Hopf algebra,
partial action, partial smash product algebra.

The third author was partially supported by CNPq, Brazil.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Partial actions of a Hopf algebra on its base field
    1. 3.1 Properties of a partial action
    2. 3.2 About the computations of a partial action of $H$ on $\Bbbk$
    3. 3.3 Partial actions of the pointed non-semisimple Hopf algebras $H$ with $dim_{\Bbbk}(H)=8,16$
  4. 4 $\lambda$-Hopf algebras
    1. 4.1 Definition and Properties
    2. 4.2 Taft’s algebra as a $\lambda$-Hopf algebra
    3. 4.3 Examples of $\lambda$-Hopf algebras

##  1\. Introduction

The first notion of a partial action that appeared in the literature was a
partial action of groups on algebras within the theory of operator algebras.
Precisely, in the Exel’s paper [17] whose aim was to describe the structure of
suitable $C^{*}$-algebras with the unitary circle $\mathbb{S}_{1}$ acting by
automorphisms. In [15], Dokuchaev and Exel presented the definition of a group
acting partially on an algebra, carrying it to a purely algebraic context.
From this previous work, the partial group actions arose the attention of many
other researchers. In particular, a Galois theory for commutative ring
extensions was developed using partial actions [16]. Such a work motivated
Caenepeel and Janssen to extend it to the context of Hopf algebras [12].

The notion of a partial action have several applications and extensions to
other settings, as groupoids [8], categories [2] and among many others, as can
be noticed in [14] and references therein. In the partial Hopf actions theory,
examples constitute a fundamental part of its understanding and development.

An interesting example of partial action is given by scalars, that is, a Hopf
algebra $H$ acting partially on a $\Bbbk$-algebra $A$ via a partial action of
$H$ on $\Bbbk$. Precisely, if $\cdot:H\otimes\Bbbk\longrightarrow\Bbbk$ is a
partial action of $H$ on $\Bbbk$, then $H$ acts partially on the algebra $A$
via $\rightharpoonup:H\otimes A\longrightarrow A$, $h\rightharpoonup a=(h\cdot
1_{\Bbbk})a$. In fact, many authors have obtained a characterization for this
particular case of partial action, even in more general settings. For
instance, see [13, Lemma 4.1], [2, Section 3.2] and [3, Examples 2.3 - 2.5].
However, even with this characterization, few concrete examples are known.

Another point of interest, which has always been involved with the concept of
actions, are the deformations, and they expand their horizons when carried to
the partial context. Given a global action of $H$ on $A$, the corresponding
smash product algebra $A\\#H$ is actually a deformation of the algebra
$A\otimes H$ by such an action. In particular, since the unique global action
of $H$ on $\Bbbk$ is given by the counit of $H$, the corresponding smash
product algebra $\Bbbk\\#H$ coincides with the algebra $\Bbbk\otimes H\simeq
H$, that is, there are no deformations of $H$ using global actions in this
way. However, in the partial setting, that is, with a partial (not global)
action of $H$ on $\Bbbk$, the partial smash product algebra
$\underline{\Bbbk\\#H}$ coincides no more with $H$, that is, using a partial
(not global) action of $H$ on $\Bbbk$ one can produce a non-trivial
deformation of $H$.

In this work we deal with deformations of this type, that is, deformations of
a Hopf algebra $H$ by partial actions of $H$ over its base field $\Bbbk$. To
do so, we investigate partial actions of $H$ on $\Bbbk$ and partial smash
product algebras. In particular, we have increased the source of examples as
well as characterized some specific properties of such partial actions.

This paper is organized as follows. First we recall in Section 2 some
definitions and results about partial actions of a Hopf algebra $H$ on an
algebra $A$. In particular, the partial smash product algebra
$\underline{A\\#H}$ is presented. We focus in the particular case when $A$ is
the base field $\Bbbk$ of $H$. Some guiding examples are presented and
important notations are fixed. Then, we dedicate Section 3 to study partial
actions of a Hopf algebra over its base field. First, some properties and a
method to compute such partial actions are developed, then examples are
exhibited. Finally, in Section 4 we investigate partial smash product
algebras. We obtain a characterization of the conditions under which such an
algebra results in a Hopf algebra. The concept of a $\lambda$-Hopf algebra is
introduced, some results concerning it are proved and several examples are
calculated using the previously computed partial actions.

### Conventions

In this work, we deal only with algebras over a fixed field $\Bbbk$. Unadorned
$\otimes$ means $\otimes_{\Bbbk}$. We write $G(H)=\\{g\in H\setminus\\{0\\}\
|\ \Delta(g)=g\otimes g\\}$ for the group of the group-like elements of a Hopf
algebra $H$. Given $g,h\in G(H)$, an element $x\in H$ is called a _$(g,h)$
-primitive element_ if $\Delta(x)=x\otimes g+h\otimes x$, and the linear space
of all $(g,h)$-primitive elements of $H$ is denoted by $P_{g,h}(H)$. If no
emphasis on the elements $g,h\in G(H)$ is needed, an element $x\in P_{g,h}(H)$
is called simply of a _skew-primitive element_. Clearly,
$\Bbbk\\{g-h\\}\subseteq P_{g,h}(H)$; these are called the trivial ones.
Finally, given two algebras $A,B,$ we recall that there exist natural
inclusions $A,B\hookrightarrow A\otimes B;$ then, for $a\in A$ and $b\in B$,
by $a,b\in A\otimes B$ we mean $a\otimes 1_{B},1_{A}\otimes b\in A\otimes B$.

##  2\. Preliminaries

We present here some definitions and preliminaries results about partial
actions of a Hopf algebra $H$ on an algebra $A$. Some examples are presented
and some notations are fixed. For details we refer to [2, 4, 12].

A _left partial action of a Hopf algebra $H$ on an algebra $A$_ is a linear
map $\cdot:H\otimes A\longrightarrow A$, denoted by $\cdot(h\otimes a)=h\cdot
a$, such that:

  1. (i)

$1_{H}\cdot a=a$;

  2. (ii)

$h\cdot ab=(h_{1}\cdot a)(h_{2}\cdot b)$;

  3. (iii)

$h\cdot(k\cdot a)=(h_{1}\cdot 1_{A})(h_{2}k\cdot a)$,

for all $h,k\in H$ and $a,b\in A$. In this case, $A$ is called a _partial
$H$-module algebra_.

A left partial action is _symmetric_ if in addition we have

  1. (iv)

$h\cdot(k\cdot a)=(h_{1}k\cdot a)(h_{2}\cdot 1_{A})$,

for all $h,k\in H$ and $a\in A$.

Clearly, every global action is a partial action. The definition of a _right
partial action_ is given in an analogous way. Since throughout this work we
deal only with left partial actions, from now on by a _partial action_ we mean
a left partial action.

There exists a characterization for partial actions of $H$ on $\Bbbk$, where
$\Bbbk$ is the base field of the Hopf algebra $H$, given by a linear map
$\lambda$ of $H^{*}=Hom_{\Bbbk}(H,\Bbbk)$. In fact, this map gives rise to an
important source of examples for the partial action theory in several
settings, as for example in partial actions of group algebras on (co)algebras,
partial actions of weak Hopf algebras on (co)algebras and partial actions of
Hopf algebras on categories (see [3, 4, 9, 13] for details).

The characterization given in the next result is more general. In fact, in
[13] the authors suppose that $H$ is a weak Hopf algebra. However, we present
this result with the assumption that $H$ is a Hopf algebra, since we deal only
with such algebras in this work.

######  Proposition 2.1.

[13, Lemma 4.1] Let $H$ be a Hopf algebra. Then, $\Bbbk$ is a partial
$H$-module algebra if and only if there exists a linear map $\lambda\in H^{*}$
such that $\lambda(1_{H})=1_{\Bbbk}$ and

$\displaystyle\lambda(h)\lambda(k)=\lambda(h_{1})\lambda(h_{2}k),$ (2.1)

for all $h,k\in H$. Moreover, the corresponding partial action is symmetric if
and only if $\lambda$ satisfies the additional condition

$\displaystyle\lambda(h)\lambda(k)=\lambda(h_{1}k)\lambda(h_{2}),$ (2.2)

for all $h,k\in H$.

From now on, by a partial action of a Hopf algebra $H$ on its base field
$\Bbbk$ we mean a linear map $\lambda\in H^{*}$ such that
$\lambda(1_{H})=1_{\Bbbk}$ and satisfies condition (2.1).

Observe that if $\lambda\in H^{*}$ is a partial action of $H$ on $\Bbbk$, then
clearly $\lambda$ is an idempotent element in the convolution algebra $H^{*}$.

The next examples, to the best of the authors’ knowledge, constitute all the
explicit examples of partial actions of Hopf algebras on the base field known
in the literature:

######  Example 2.2.

[2, Example 3.7] Let $G$ be a group. Partial actions of $\Bbbk G$ on $\Bbbk$
are parametrized by subgroups of $G$. Indeed, if $N$ is a subgroup of $G$,
then $\delta_{N}$ is a partial action of $\Bbbk G$ on $\Bbbk$, where
$\delta_{N}:\Bbbk G\longrightarrow\Bbbk$ is given by
$\delta_{N}(g)=\begin{cases}1_{\Bbbk},&\textrm{if }g\in N\\\ 0,&\textrm{if
}g\notin N\end{cases},$ for all $g\in G$. Conversely, let $\lambda$ be a
partial action of $\Bbbk G$ on $\Bbbk$. Then, $\lambda(g)\in\\{0,1_{\Bbbk}\\}$
and the subset $N=\\{g\in G\,|\,\lambda(g)\neq 0\\}=\\{g\in
G\,|\,\lambda(g)=1_{\Bbbk}\\}$ is a subgroup of $G$. Thus,
$\lambda=\delta_{N}$.

Using the correspondence of the previous example, given $N$ a subgroup of $G$,
we denote by $\lambda_{N}$ the partial action of $\Bbbk G$ on $\Bbbk$, given
as $\lambda_{N}=\delta_{N}$. In particular, $\lambda_{G}=\varepsilon_{\Bbbk
G}$ is the global action of $\Bbbk G$ on $\Bbbk$.

######  Example 2.3.

[2, Subsection 3.2] Let $G$ be a finite group. Then, partial actions of
$(\Bbbk G)^{*}$ on $\Bbbk$ are in one-to-one correspondence with

$\\{N\ |\ N\textrm{ is a subgroup of }G\textrm{ and }char(\Bbbk)\nmid|N|\\}.$

Indeed, let $\\{g^{*}\ |\ g\in G\\}$ be the dual basis for the canonical basis
$\\{g\ |\ g\in G\\}$ of $\Bbbk G$. Then, for each subgroup $N$ of $G$ such
that $char(\Bbbk)\nmid|N|$, the linear map $\lambda^{N}:(\Bbbk
G)^{*}\longrightarrow\Bbbk$ given by
$\lambda^{N}(g^{*})=\begin{cases}1_{\Bbbk}/|N|,&\textrm{ if }g\in N\\\
0,&\textrm{ if }g\notin N\end{cases},$ for all $g\in G$, is a partial action
of $(\Bbbk G)^{*}$ on $\Bbbk$. Conversely, let $\lambda$ be a partial action
of $(\Bbbk G)^{*}$ on $\Bbbk$. Then, the subset $N=\\{g\in
G\,|\,\lambda(g^{*})\neq 0\\}$ is a subgroup of $G$. In this case, it is known
that $\lambda(g^{*})=1_{\Bbbk}/|N|$, for all $g\in N$, and so
$\lambda=\lambda^{N}$.

######  Example 2.4.

[12, Proposition 4.10] Let $\mathfrak{g}$ be a Lie algebra and
$U(\mathfrak{g})$ its universal enveloping algebra. Then, every partial
$U(\mathfrak{g})$-module algebra $A$ is in fact a global one. That is, there
is no genuine partial action (_i.e._ , a partial action that is not a global
one) of $U(\mathfrak{g})$ on any algebra $A$. In particular, if $\lambda$ is a
partial action of $U(\mathfrak{g})$ on $\Bbbk$, then $\lambda$ is necessarily
the counit of $U(\mathfrak{g})$.

######  Example 2.5.

[2, Example 3.8] Consider $\mathbb{H}_{4}$ the Sweedler’s Hopf algebra.
Precisely, $\mathbb{H}_{4}$ is the algebra over the field $\Bbbk$,
$char(\Bbbk)\neq 2$, generated by the letters $g$ and $x$ satisfying the
relations $g^{2}=1,x^{2}=0$ and $xg=-gx$. The set $\\{1,g,x,gx\\}$ is a basis
for $\mathbb{H}_{4}$, $g$ is a group-like element and $x$ is a
$(1,g)$-primitive element. Thus, it is straightforward to check that for any
$\alpha\in\Bbbk$ the map $\lambda_{\alpha}:\mathbb{H}_{4}\longrightarrow\Bbbk$
given by $\lambda_{\alpha}(1)=1_{\Bbbk},\lambda_{\alpha}(g)=0$ and
$\lambda_{\alpha}(x)=\lambda_{\alpha}(gx)=\alpha$ is a partial action of
$\mathbb{H}_{4}$ on $\Bbbk$. Moreover, these are all partial actions of
$\mathbb{H}_{4}$ on $\Bbbk$, _i.e._ , if $\lambda$ is a partial action of
$\mathbb{H}_{4}$ on $\Bbbk$, then $\lambda=\varepsilon$ (and so the global
one) or $\lambda=\lambda_{\alpha}$ for some $\alpha\in\Bbbk$.

In this work we are interested in deformations of a Hopf algebra $H$ through
partial actions of $H$ on $\Bbbk$. Precisely, given a partial action of $H$ on
$\Bbbk$, we want to study the partial smash product algebra
$\underline{\Bbbk\\#H}$. For this purpose, we start recalling that given any
global $H$-module algebra $A$, via an action $\triangleright:H\otimes
A\longrightarrow A$, we can endow the tensor product $A\otimes H$ with an
algebra structure induced by

$(a\otimes h)(b\otimes g)=a(h_{1}\triangleright b)\otimes h_{2}g,$

for $a,b\in A$ and $h,g\in H$. This structure is denoted by $A\\#H$ and called
_the smash product algebra of $A$ with $H$_. We typically denote the element
$a\otimes h$ by $a\\#h\in A\\#H$. In particular, note that $\Bbbk\\#H=H$.

The same construction can be performed in the partial case. But, in this case,
it turns that $A\\#H$ is not a unital algebra since $1_{A}\\#1_{H}$ is only a
left unit.

######  Definition 2.6.

[12] Let $A$ be a partial $H$-module algebra via $\cdot:H\otimes
A\longrightarrow A$. Then, the vector subspace
$\underline{A\\#H}=(A\\#H)(1_{A}\\#1_{H})$ is a unital algebra with the
multiplication induced by

$\underline{(x\\#h)}\ \underline{(y\\#g)}=\underline{x(h_{1}\cdot
y)\\#h_{2}g},$

for all $x,y\in A$, $h,g\in H$, where
$\underline{(x\\#h)}=(x\\#h)(1_{A}\\#1_{H})=x(h_{1}\cdot 1_{A})\\#h_{2}$ is a
typical element of $\underline{A\\#H}$ and the unit is given by
$\underline{1_{A}\\#1_{H}}$. The unital algebra $\underline{A\\#H}$ is called
the _partial smash product algebra of $A$ with $H$_.

For a partial action of $H$ on $\Bbbk$ given by
$\lambda:H\longrightarrow\Bbbk$, the corresponding partial smash product
algebra $\underline{\Bbbk\\#H}$ is generated by
$\left\\{\underline{1_{\Bbbk}\\#h}=1_{\Bbbk}\\#\lambda(h_{1})h_{2}\,|\,h\in
H\right\\}$. Also notice
$\underline{1_{\Bbbk}\\#h}\,\,\underline{1_{\Bbbk}\\#k}=\underline{1_{\Bbbk}\\#\lambda(h_{1})h_{2}k}$,
for all $h,k\in H$.

For example, let $\lambda_{N}$ be the partial action of $\Bbbk G$ on $\Bbbk$
given in Example 2.2. In this case, it follows that $\underline{\Bbbk\\#\Bbbk
G}\simeq\Bbbk N$.

##  3\. Partial actions of a Hopf algebra on its base field

So far we know, only the partial actions on the base field presented in
Examples 2.2 \- 2.5 are known in the literature. For this reason, in this
section some properties and a method to compute such partial actions are
developed, and then we exhibit these partial actions for some families of Hopf
algebras.

###  3.1. Properties of a partial action

This subsection is dedicated to present some properties of a partial action
$\lambda:H\longrightarrow\Bbbk$. Most of the results are about the values that
the map $\lambda$ can admit for group-like and skew-primitive elements.

The following proposition generalizes to any Hopf algebra $H$ an important
fact observed in Example 2.2 just for group algebras.

######  Proposition 3.1.

Let $\lambda:H\longrightarrow\Bbbk$ be a partial action. Then, for any $g\in
G(H)$, $\lambda(g)\in\\{0,1_{\Bbbk}\\}$ and $N=\\{g\in
G(H)\,|\,\lambda(g)=1_{\Bbbk}\\}$ is a subgroup of $G(H)$. In particular, if
$g\in N$, then $\lambda(g^{i})=1_{\Bbbk},$ for each $i\in\mathbb{Z}$.
Moreover, if $g\in G(H)$ has prime order $p$ and $\lambda(g)=0$, then
$\lambda(g^{i})=0,$ for each $i\in\mathbb{Z}$ such that $gcd(i,p)\\!=\\!1$.

###### Proof.

First, for any $g\in G(H)$,
$\lambda(g)\lambda(1_{H})=\lambda(g)\lambda(g1_{H})$ by (2.1). Since
$\lambda(1_{H})=1_{\Bbbk}$, it follows that $\lambda(g)=\lambda(g)^{2}$ and so
$\lambda(g)\in\\{0,1_{\Bbbk}\\}$. Now, clearly $1_{H}\in N$. Let $g,h\in N$,
then we obtain that $\lambda(h)\lambda(h^{-1})=\lambda(h)$ by (2.1), and
consequently $h^{-1}\in N$. Again, using (2.1), we have that
$\lambda(g)\lambda(h)=\lambda(g)\lambda(gh)$, and hence $gh\in N$. Thus, $N$
is a subgroup of $G(H)$. If $g\in N$, then clearly $\lambda(g^{i})=1_{\Bbbk}$,
since $\langle g\rangle\subseteq N$, where $\langle g\rangle$ stands for the
subgroup generated by $g$.

Now suppose that $g^{p}=1$ and $\lambda(g)=0$, where $p\in\mathbb{Z}$ is a
prime number. In this case, $g\notin N$ and $\langle
g\rangle=\\{1,g,g^{2},...,g^{p-1}\\}$. Consequently, we have that
$N\cap\langle g\rangle=\\{1\\}$ and thus $\lambda(g^{i})=0$, whenever
$gcd(i,p)=1$. ∎

The next result shows the behavior of a partial action
$\lambda:H\longrightarrow\Bbbk$ for certain elements.

######  Lemma 3.2.

Let $\lambda:H\longrightarrow\Bbbk$ be a partial action. For $g,t\in G(H)$ and
$x\in P_{g,t}(H)$, it holds that:

  * a)

If $\lambda(g)=1$, then $\lambda(gu)=\lambda(u)$, for all $u\in H$;

  * b)

If $\lambda(g)=\lambda(t)$, then $\lambda(x)=0$;

  * c)

If $\lambda(x)=0$ and $\lambda(t)=1$, then $\lambda(xu)=0$, for all $u\in H$;

  * d)

If $\lambda(g)=1$ and $\lambda(t)=0$, then $\lambda(xt^{-1})=-\lambda(x)$;

  * e)

If $\lambda(g)=0$ and $\lambda(t)=1$, then $\lambda(xg^{-1})=-\lambda(x)$.

Moreover, if $gt=tg$, then, for all $i\in\mathbb{Z}$, we also have that:

  * f)

If $\lambda(g)=1$ and $\lambda(t)=0$, then $\lambda(t^{i}x)=0$ or
$\lambda(t^{i})+\lambda(t^{i+1})=1$;

  * g)

If $\lambda(g)=0$ and $\lambda(t)=1$, then $\lambda(g^{i}x)=0$ or
$\lambda(g^{i})+\lambda(g^{i+1})=1$;

  * h)

If $\lambda(g)=\lambda(t)=0$, then $\lambda(g^{-1}t^{-1}x)=0$.

###### Proof.

It follows directly from the application of equality (2.1) for good choices of
elements $h,k\in H$ and Proposition 3.1. ∎

###  3.2. About the computations of a partial action of $H$ on $\Bbbk$

Let $H$ be a Hopf algebra. In this subsection a method to compute partial
actions of $H$ on its base field is given.

Recall by Proposition 2.1 that $\lambda\in H^{*}$ is a partial action of $H$
on $\Bbbk$ if and only if the map $\lambda$ satisfies
$\lambda(1_{H})=1_{\Bbbk}$ and
$\lambda(u)\lambda(v)=\lambda(u_{1})\lambda(u_{2}v)$, for all $u,v\in H$.

######  Definition 3.3.

Let $\mathcal{B}$ be a basis of $H$ and consider the linear map
$\Lambda:H\longrightarrow\Bbbk[X_{b}\ |\ b\in\mathcal{B}]$,
$\Lambda(b)=X_{b}$. We define by _partial system associated with
$\mathcal{B}$_ the following system of equations:

$\displaystyle\left\\{\begin{array}[]{ll}\Lambda(1_{H})=1_{\Bbbk}\\\
\Lambda(a)\Lambda(b)=\Lambda(a_{1})\Lambda(a_{2}b)\end{array}\right.\qquad
a,b\in\mathcal{B}.$ (3.3)

Hence, the map $\lambda:H\longrightarrow\Bbbk$ is a partial action of $H$ on
$\Bbbk$ if and only if $(\lambda(b))_{b\in\mathcal{B}}$ is a solution of
(3.3). In particular, the partial system associated with $\mathcal{B}$ always
has at least one solution: the global action of $H$ on $\Bbbk$, namely
$(\varepsilon(b))_{b\in\mathcal{B}}$.

Now, extend $G(H)$ to a basis $\mathcal{B}=G(H)\sqcup\mathcal{B}^{\prime}$ of
$H$, where $\sqcup$ denotes the disjoint union. Then, (3.3) is rewritten as

$\displaystyle\left\\{\begin{array}[]{ll}\Lambda(1_{H})=1_{\Bbbk}\\\
\Lambda(g)\Lambda(v)=\Lambda(g)\Lambda(gv)\\\
\Lambda(u)\Lambda(v)=\Lambda({u}_{1})\Lambda({u}_{2}v)\end{array}\right.\qquad
g\in G(H),u\in\mathcal{B}^{\prime},v\in\mathcal{B}.$

Let $\lambda:H\longrightarrow\Bbbk$ be a partial action. Proposition 3.1
ensures that $\lambda(h)\in\\{0,1_{\Bbbk}\\}$, for all $h\in G(H)$, and there
exists a subgroup $N$ of $G(H)$ such that $\lambda|_{\Bbbk G(H)}=\lambda_{N}$
(see Example 2.2). Then, by Lemma 3.2 (a), we have that
$\lambda(v)=\lambda(gv)$ for all $g\in N,v\in\mathcal{B}$. Therefore,
$(\lambda(b))_{b\in\mathcal{B}}$ is a solution of the following system:

$\displaystyle\left\\{\begin{array}[]{ll}\Lambda(g)=1_{\Bbbk}\\\
\Lambda(h)=0\\\ \Lambda(u)=\Lambda(gu)\\\
\Lambda(u)\Lambda(v)=\Lambda({u}_{1})\Lambda({u}_{2}v)\end{array}\right.\qquad
g\in N,h\in G(H)\setminus N,u\in\mathcal{B}^{\prime},v\in\mathcal{B}.$ (3.8)

On the other hand, let $N$ be a subgroup of $G(H)$ and consider a system of
equations as (3.8). If there exists a solution
$(\alpha_{b})_{b\in\mathcal{B}}$ to the latter system, then it clearly is also
a solution of the partial system associated with $\mathcal{B}$.

Fix a subgroup $N$ of $G(H)$. For $x,y\in\mathcal{B}^{\prime}$, define the
relation

$x\sim_{N}y\quad\textrm{ if and only if}\quad\exists\,\,g\in N\textrm{ such
that }y=gx.$

Notice that it is an equivalence relation on the set $\mathcal{B}^{\prime}$.
Given $x\in\mathcal{B}^{\prime}$, $[x]=\\{y\in\mathcal{B}^{\prime}\ |\
y\sim_{N}x\\}$ denotes the _equivalence class of $x$_. In particular, for any
$x\in\mathcal{B}^{\prime}$, $[x]=\\{x\\}\sqcup[x]^{\perp}$, where
$[x]^{\perp}=\\{y\in\mathcal{B}^{\prime}\ |\ y\sim_{N}x,y\neq x\\}$.

Write $\widetilde{N}$ for a _transversal set of the relation $\sim_{N}$ on
$\mathcal{B}^{\prime}$_, that is, $\widetilde{N}$ is a subset of
$\mathcal{B}^{\prime}$ consisting of exactly one representative from each
equivalence class. Then, we have a partition of $\mathcal{B}^{\prime}$ given
by $\mathcal{B}^{\prime}=\widetilde{N}\sqcup N^{\perp}$, where
$N^{\perp}=\mathcal{B}^{\prime}\setminus\widetilde{N}$. Observe that
$N^{\perp}=\cup_{x\in\widetilde{N}}[x]^{\perp}$.

Consider now the following system of equations:

$\displaystyle\left\\{\begin{array}[]{ll}\Lambda(g)=1_{\Bbbk}\\\
\Lambda(h)=0\\\ \Lambda(u)=\Lambda(gu)\end{array}\right.\qquad g\in N,h\in
G(H)\setminus N,u\in\mathcal{B}^{\prime}.$ (3.12)

Then, $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of (3.12) if and only if
$\alpha_{g}=1_{\Bbbk}$, $\alpha_{h}=0$, $\alpha_{y}=\alpha_{x}$, and
$\alpha_{x}\in\Bbbk$ is a free parameter, for all $g\in N$, $h\in
G(H)\setminus N$, $x\in\widetilde{N}$ and $y\in[x]$. In this case, we say that
$(\alpha_{b})_{b\in\mathcal{B}}$ is an _initial $N$-condition for the partial
system associated with $\mathcal{B}$_.

Finally, we define the following sets: $\mathcal{B}_{t,s}=\left\\{x\in
P_{t,s}(H)\ |\ t\in N,s\in G(H)\setminus N\right\\}$ and
$\widetilde{\mathcal{B}}=\widetilde{N}\setminus\left(\cup_{t,s\in
G(H)}\mathcal{B}_{t,s}\right).$ With these notations, the system

$\displaystyle\left\\{\begin{array}[]{ll}\Lambda(u)\Lambda(v)=\Lambda({u}_{1})\Lambda({u}_{2}v)\end{array}\right.\qquad
u\in\widetilde{\mathcal{B}},v\in\mathcal{B},$ (3.14)

is called of the _$N$ -reduced partial system associated with $\mathcal{B}$_.

######  Theorem 3.4.

Let $\mathcal{B}=G(H)\sqcup\mathcal{B}^{\prime}$ be a basis of $H$ and
$\lambda:H\longrightarrow\Bbbk$ a linear map. Then, $\lambda$ is a partial
action of $H$ on $\Bbbk$ if and only if $N=\\{g\in G(H)\ |\
\lambda(g)=1_{\Bbbk}\\}$ is a subgroup of $G(H)$ and
$(\lambda(b))_{b\in\mathcal{B}}$ is both: an initial $N$-condition for the
partial system associated with $\mathcal{B}$ and also a solution of the
$N$-reduced partial system associated with $\mathcal{B}$.

###### Proof.

If $\lambda$ is a partial action of $H$ on $\Bbbk$, then we have that $N$ is a
subgroup of $G(H)$ by Proposition 3.1, and $(\lambda(b))_{b\in\mathcal{B}}$ is
a solution of the system (3.8). Then, clearly $(\lambda(b))_{b\in\mathcal{B}}$
is both: an initial $N$-condition for the partial system associated with
$\mathcal{B}$ and a solution of the $N$-reduced partial system associated with
$\mathcal{B}$.

For the converse, it is sufficient to check that
$(\lambda(b))_{b\in\mathcal{B}}$ satisfies each equation
$\Lambda(u)\Lambda(v)=\Lambda({u}_{1})\Lambda({u}_{2}v)$, for
$u\in\mathcal{B}^{\prime}\setminus\widetilde{\mathcal{B}},v\in\mathcal{B}.$
Then, in this case, $(\lambda(b))_{b\in\mathcal{B}}$ is also a solution of the
system given in (3.8) and therefore $\lambda$ is a partial action.

First, suppose $u\in N^{\perp}.$ Note that $u\in[x]^{\perp}$ for some
$x\in\widetilde{N}$, and so there exists $g\in N$ such that $u=gx$. Then,
$\Lambda(u)\Lambda(v)=\Lambda({u}_{1})\Lambda({u}_{2}v)$ means
$\Lambda(gx)\Lambda(v)=\Lambda(gx_{1})\Lambda(gx_{2}v)$. Since
$(\lambda(b))_{b\in\mathcal{B}}$ is an initial $N$-condition for the partial
system associated with $\mathcal{B}$, we have $\lambda(gw)=\lambda(w)$ for any
$w\in H$, and the equality
$\lambda(x)\lambda(v)=\lambda(x_{1})\lambda(x_{2}v)$ holds because
$(\lambda(b))_{b\in\mathcal{B}}$ is solution of (3.14). Hence,

$\lambda(gx)\lambda(v)=\lambda(x)\lambda(v)=\lambda(x_{1})\lambda(x_{2}v)=\lambda(gx_{1})\lambda(gx_{2}v).$

Finally, if $u\in\mathcal{B}_{t,s}\cap\widetilde{N}$, then the equation
$\Lambda(u)\Lambda(v)=\Lambda(u_{1})\Lambda(u_{2}v)$ is the same as
$\Lambda(u)\Lambda(v)=\Lambda(u)\Lambda(tv)+\Lambda(s)\Lambda(uv)$, for any
$v\in\mathcal{B}$. Since $(\lambda(b))_{b\in\mathcal{B}}$ is a solution of
(3.12), then $\lambda(v)=\lambda(tv)$ and $\lambda(s)=0$. Thus, the equality
$\lambda(u)\lambda(v)=\lambda(u)\lambda(tv)+\lambda(s)\lambda(uv)$ holds. ∎

Similar to Lemma 3.2, we have the following result.

######  Lemma 3.5.

Let $\mathcal{B}=G(H)\sqcup\mathcal{B}^{\prime}$ be a basis of $H$, $g,t\in
G(H)$, $x\in P_{g,t}(H)\cap\mathcal{B}^{\prime}$ and
$(\alpha_{b})_{b\in\mathcal{B}}$ an initial $N$-condition for the partial
system associated with $\mathcal{B}$. If $(\alpha_{b})_{b\in\mathcal{B}}$ is
also a solution of the $N$-reduced partial system associated with
$\mathcal{B}$, then:

  * a)

If $\alpha_{g}=\alpha_{t}$, then $\alpha_{x}=0$;

  * b)

If $\alpha_{t}=1_{\Bbbk}$ and $\alpha_{x}=0$, then $\alpha_{xu}=0$ for all
$u\in H$ such that $xu\in\mathcal{B}^{\prime}$;

  * c)

If $\alpha_{g}=1_{\Bbbk}$, $\alpha_{t}=0$ and
$xt^{-1}\in\mathcal{B}^{\prime}$, then $\alpha_{xt^{-1}}=-\alpha_{x}$;

  * d)

If $\alpha_{g}=0$, $\alpha_{t}=1_{\Bbbk}$ and
$xg^{-1}\in\mathcal{B}^{\prime}$, then $\alpha_{xt^{-1}}=-\alpha_{x}$.

###### Proof.

Straightforward from Theorem 3.4 and Lemma 3.2. ∎

Given a partial action $\lambda:H\longrightarrow\Bbbk$, we say that $\lambda$
has _initial condition $N$_ if $N=\\{g\in G(H)\ |\ \lambda(g)=1_{\Bbbk}\\}$.

Now, based on Theorem 3.4, we present our method to calculate partial actions
of $H$ on $\Bbbk$.

The Method: Let $H$ be a given Hopf algebra. To obtain a partial action of $H$
on $\Bbbk$, we perform the following steps:

  * Step 1.

Extend $G(H)$ to a basis $\mathcal{B}$ of $H$;

  * Step 2.

Consider $N$ a subgroup of $G(H)$;

  * Step 3.

Regard the solutions $(\alpha_{b})_{b\in\mathcal{B}}$ of the system (3.12);

  * Step 4.

Investigate the $N$-reduced partial system associated with $\mathcal{B}$;

  * Step 5.

Conclusion: If $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the
$N$-reduced partial system associated with $\mathcal{B}$, then the linear map
$\lambda:H\longrightarrow\Bbbk$ given by $\lambda(b)=\alpha_{b}$, for all
$b\in\mathcal{B}$, is a partial action of $H$ on $\Bbbk$; Otherwise, there is
not a partial action of $H$ on $\Bbbk$ with initial condition $N$.

Remarks. 1) For Step 4, Lemma 3.5 is useful to determine if
$(\alpha_{b})_{b\in\mathcal{B}}$ can be a solution of (3.14) or not, since it
imposes necessary conditions on the values $\alpha_{u}\in\Bbbk$ for some
elements $u\in\widetilde{N}$.

2) Since every partial action of $H$ on $\Bbbk$ has an initial condition, if
we apply the method to all subgroups of $G(H)$, then we obtain all partial
actions of $H$ on $\Bbbk$. Moreover, the method gives an explicit
characterization of each partial action $\lambda:H\longrightarrow\Bbbk$.

######  Example 3.6.

To illustrate the method, we reobtain the partial actions of the Sweedler’s
Hopf algebra $\mathbb{H}_{4}$ on its base field, previously exhibited in
Example 2.5. Recall that $G(\mathbb{H}_{4})=\\{1,g\\}$ and
$\mathcal{B}=\\{1,g,x,gx\\}$ is a basis of $\mathbb{H}_{4}$. Thus, we have two
subgroups of $G(H)$, namely $N=\\{1,g\\}$ and $N=\\{1\\}$.

  1. (1)

$N=\\{1,g\\}$. In Step 3 we assume $\alpha_{1}=\alpha_{g}=1_{\Bbbk}$ and
$\alpha_{gx}=\alpha_{x}\in\Bbbk$. Now, Step 4 says to study (3.14). But, using
Lemma 3.5 (a), if $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of (3.14),
then necessarily $\alpha_{x}=0$. In this case, the only possibility for the
associated partial action $\lambda:\mathbb{H}_{4}\longrightarrow\Bbbk$,
$\lambda(b)=\alpha_{b}$, is the counit $\varepsilon$. Thus, as conclusion,
$\lambda$ is a partial action of $\mathbb{H}_{4}$ on $\Bbbk$ with initial
condition $\\{1,g\\}$ if and only if $\lambda=\varepsilon$;

  2. (2)

$N=\\{1\\}$. In this case, by Step 3 we suppose $\alpha_{1}=1_{\Bbbk}$,
$\alpha_{g}=0$ and $\alpha_{x},\alpha_{gx}\in\Bbbk$. Now, we move to Step 4.
Note that $\widetilde{N}=\\{x,gx\\}$. As $x\in\mathcal{B}_{1,g}$, it follows
that $\widetilde{\mathcal{B}}=\\{gx\\}$ and the system (3.14) has only $4$
equations:

$\displaystyle\left\\{\Lambda(gx)\Lambda(v)=\Lambda(gx)\Lambda(gv)+\Lambda(1)\Lambda(gxv)\right.\qquad
v\in\mathcal{B}.$

In order to verify if $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the
system above, we have

$\left\\{\begin{array}[]{ll}\alpha_{gx}=\alpha_{gx}\\\
0=\alpha_{gx}-\alpha_{x}\\\ \alpha_{gx}\alpha_{x}=\alpha_{gx}\alpha_{gx}\\\
\alpha_{gx}\alpha_{gx}=\alpha_{gx}\alpha_{x}.\end{array}\right.$

Thus, $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the $N$-reduced
partial system associated with $\mathcal{B}$ if and only if
$\alpha_{x}=\alpha_{gx}$.

Consequently, for any $\alpha\in\Bbbk$, the linear map
$\lambda_{\alpha}:\mathbb{H}_{4}\longrightarrow\Bbbk$ given by
$\lambda(1)=1_{\Bbbk}$, $\lambda(g)=0$ and $\lambda(x)=\lambda(gx)=\alpha$, is
a partial action of $\mathbb{H}_{4}$ on $\Bbbk$.

Therefore, the partial actions of $\mathbb{H}_{4}$ on $\Bbbk$ are exactly
$\varepsilon$ and $\lambda_{\alpha}$, $\alpha\in\Bbbk$.

We point out the benefit of our method: since a partial action of $H$ on
$\Bbbk$ corresponds to a linear map $\lambda\in H^{*}$ satisfying
$\lambda(1_{H})=1_{\Bbbk}$ and (2.1), to calculate explicitily such a map is a
task attached to solve the system (3.3), for some basis $\mathcal{B}$ of $H$.
However, with our method we deal only with the potentially smaller system
(3.14). In the Sweedler’s Hopf algebra, for instance, the system (3.3) has
$4^{2}=16$ equations, while we deal only with the system of 4 equations
(3.14), in the case $N=\\{1\\}$.

Since our developments deal with group-like and skew-primitive elements, it is
expected that the method will be effective mainly for a Hopf algebra generated
by elements of these types, as illustrated through the example above.

Using the method described, we were able to determine all partial actions on
the base field of some particular families of Hopf algebras, as can be seen in
the following subsection.

###  3.3. Partial actions of the pointed non-semisimple Hopf algebras $H$ with
$dim_{\Bbbk}(H)=8,16$

In this subsection, for each pointed non-semisimple Hopf algebra $H$ with
$dim_{\Bbbk}(H)=8,16$, we exhibit all partial actions of $H$ on its base field
$\Bbbk$. Throughout this subsection $\Bbbk$ is assumed to be an algebraically
closed field of characteristic zero.

The 8-dimensional Hopf algebras were classified independently by [21] and
[19]. There exist five (up to isomorphism) 8-dimensional pointed non-
semisimple Hopf algebras. We use the notations and presentations for such Hopf
algebras as given in [10]:

$\mathcal{A}_{2}=\langle
g,\,x,\,y\,|\,g^{2}=1,\,\,x^{2}=y^{2}=0,\,\,xg=-gx,\,\,yg=-gy,\,\,yx=-xy\rangle,$
where $g\in G(\mathcal{A}_{2})$ and $x,y\in P_{1,g}(\mathcal{A}_{2})$;

$\mathcal{A}_{4}^{\prime}=\langle
g,\,x\,|\,g^{4}=1,\,\,x^{2}=0,\,\,xg=-gx\rangle,$ where $g\in
G(\mathcal{A}_{4}^{\prime})$ and $x\in P_{1,g}(\mathcal{A}_{4}^{\prime})$;

$\mathcal{A}_{4}^{\prime\prime}=\langle
g,\,x\,|\,g^{4}=1,\,x^{2}=g^{2}-1,\,xg=-gx\rangle,$ where $g\in
G(\mathcal{A}_{4}^{\prime\prime})$ and $x\in
P_{1,g}(\mathcal{A}_{4}^{\prime\prime})$;

$\mathcal{A}_{4,q}^{\prime\prime\prime}=\langle
g,\,x\,|\,g^{4}=1,\,\,x^{2}=0,\,\,gx=qxg\rangle,$ where $q$ is a primitive
root of unity of order $4$, $g\in G(\mathcal{A}_{4,q}^{\prime\prime\prime})$
and $x\in P_{1,g^{2}}(\mathcal{A}_{4,q}^{\prime\prime\prime})$;

$\mathcal{A}_{2,2}=\langle
g,\,h,\,x\,|\,g^{2}=h^{2}=1,\,x^{2}=0,\,xg=-gx,\,xh=-hx,\,hg=gh\rangle,$ where
$g,h\in G(\mathcal{A}_{2,2})$ and $x\in P_{1,g}(\mathcal{A}_{2,2})$.

We observe that the subscript index in each Hopf algebra above described
represents its group-like subalgebra.

######  Theorem 3.7.

For each pointed non-semisimple Hopf algebra of dimension 8, all partial
actions on its base field are computed. They are presented in Tables 1 \- 5,
where $\alpha,\beta,\gamma\in\Bbbk$ and $\gamma^{2}=-1$.

Table 1. Partial actions of $\mathcal{A}_{2}$ on $\Bbbk$ | $1$ | $g$ | $x$ | $y$ | $gx$ | $gy$ | $xy$ | $gxy$
---|---|---|---|---|---|---|---|---
$\lambda_{G(\mathcal{A}_{2})}=\varepsilon_{\mathcal{A}_{2}}$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $1$ | $0$ | $\alpha$ | $\beta$ | $\alpha$ | $\beta$ | $0$ | $0$
Table 2. Partial actions of $\mathcal{A}_{4}^{\prime}$ on $\Bbbk$ | $1$ | $g$ | $g^{2}$ | $g^{3}$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G(\mathcal{A}_{4}^{\prime})}=\varepsilon_{\mathcal{A}_{4}^{\prime}}$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $1$ | $0$ | $1$ | $0$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1\\}}$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 3. Partial actions of $\mathcal{A}_{4}^{\prime\prime}$ on $\Bbbk$ $(\gamma^{2}=-1)$ | $1$ | $g$ | $g^{2}$ | $g^{3}$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G(\mathcal{A}_{4}^{\prime\prime})}=\varepsilon_{\mathcal{A}_{4}^{\prime\prime}}$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $1$ | $0$ | $1$ | $0$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1\\}}$ | $1$ | $0$ | $0$ | $0$ | $\gamma$ | $0$ | $0$ | $\gamma$

Table 4. Partial actions of $\mathcal{A}_{4,q}^{\prime\prime\prime}$ on $\Bbbk$ | $1$ | $g$ | $g^{2}$ | $g^{3}$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G(\mathcal{A}_{4,q}^{\prime\prime\prime})}=\varepsilon_{\mathcal{A}_{4,q}^{\prime\prime\prime}}$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $1$ | $0$ | $0$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $1$ | $0$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 5. Partial actions of $\mathcal{A}_{2,2}$ on $\Bbbk$ | $1$ | $g$ | $h$ | $gh$ | $x$ | $gx$ | $hx$ | $ghx$
---|---|---|---|---|---|---|---|---
$\lambda_{G(\mathcal{A}_{2,2})}=\varepsilon_{\mathcal{A}_{2,2}}$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $1$ | $0$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $0$ | $0$
$\lambda_{\\{1,gh\\}}$ | $1$ | $0$ | $0$ | $1$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,h\\}}$ | $1$ | $0$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g\\}}$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$

###### Proof.

To obtain the partial actions presented in Tables 1 \- 5, one does routine
computations applying the method presented. We will do just some computations
to obtain all partial actions of $\mathcal{A}_{4}^{\prime\prime}$ on $\Bbbk$,
presented in Table 3.

Step 1: $G(\mathcal{A}_{4}^{\prime\prime})=\\{1,g,g^{2},g^{3}\\}$ and
$\mathcal{B}=\\{1,g,g^{2},g^{3},x,gx,g^{2}x,g^{3}x\\}$.

For Step 2, we note that $G(\mathcal{A}_{4}^{\prime\prime})$ has 3 subgroups:
$\\{1,g,g^{2},g^{3}\\}$, $\\{1,g^{2}\\}$ and $\\{1\\}$.

  1. (1)

$N=\\{1,g,g^{2},g^{3}\\}$. Here, if we proceed as in Example 3.6 (1), we
obtain that $\lambda$ is a partial action of $\mathcal{A}_{4}^{\prime\prime}$
on $\Bbbk$ with initial condition $\\{1,g,g^{2},g^{3}\\}$ if and only if
$\lambda=\varepsilon_{\mathcal{A}_{4}^{\prime\prime}}$;

  2. (2)

$N=\\{1,g^{2}\\}$. By Step 3, we consider $\alpha_{1}=\alpha_{g^{2}}=1$,
$\alpha_{g}=\alpha_{g^{3}}=0$, $\alpha_{g^{2}x}=\alpha_{x}$ and
$\alpha_{g^{3}x}=\alpha_{gx}$, where $\alpha_{x},\alpha_{gx}\in\Bbbk$. For
Step 4, since $x\in\mathcal{B}_{1,g}$ we have
$\widetilde{\mathcal{B}}=\\{gx\\}$. Then, the system (3.14) has 8 equations

$\displaystyle\left\\{\Lambda(gx)\Lambda(v)=\Lambda(gx)\Lambda(gv)+\Lambda(g^{2})\Lambda(gxv)\right.\qquad
v\in\mathcal{B}.$

If we replace $(\alpha_{b})_{b\in\mathcal{B}}$ in the system above, we get

$\left\\{\begin{array}[]{ll}\alpha_{gx}=\alpha_{gx}\\\
0=\alpha_{gx}-\alpha_{x}\\\ \alpha_{gx}=\alpha_{gx}\\\
0=\alpha_{gx}-\alpha_{x}\\\ \alpha_{gx}\alpha_{x}=\alpha_{gx}\alpha_{gx}\\\
\alpha_{gx}\alpha_{gx}=\alpha_{gx}\alpha_{x}\\\
\alpha_{gx}\alpha_{x}=\alpha_{gx}\alpha_{gx}\\\
\alpha_{gx}\alpha_{gx}=\alpha_{gx}\alpha_{x}.\end{array}\right.$

Clearly, $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the system above if
and only if $\alpha_{x}=\alpha_{gx}$.

Consequently, for any $\alpha\in\Bbbk$, the linear map
$\lambda_{N}:\mathcal{A}_{4}^{\prime\prime}\longrightarrow\Bbbk$ given by
$\lambda(1)=\lambda(g^{2})=1$, $\lambda(g)=\lambda(g^{3})=0$ and
$\lambda(x)=\lambda(gx)=\lambda(g^{2}x)=\lambda(g^{3}x)=\alpha$, is a partial
action of $\mathbb{H}_{4}$ on $\Bbbk$.

  3. (3)

$N=\\{1\\}$. By Step 3, we consider $\alpha_{1}=1$,
$\alpha_{g}=\alpha_{g^{2}}=\alpha_{g^{3}}=0$ and
$\alpha_{x},\alpha_{gx},\alpha_{g^{2}x}\alpha_{g^{3}x}\in\Bbbk$.

Now we move to Step 4. As $x\in\mathcal{B}_{1,g}$,
$\widetilde{\mathcal{B}}=\\{gx,g^{2}x,g^{3}x\\}$. Then, the system (3.14) has
3 blocks of 8 equations each:

$\displaystyle\left\\{\begin{array}[]{ll}\Lambda(gx)\Lambda(v)=\Lambda(gx)\Lambda(gv)+\Lambda(g^{2})\Lambda(gxv)\\\
\Lambda(g^{2}x)\Lambda(v)=\Lambda(g^{2}x)\Lambda(g^{2}v)+\Lambda(g^{3})\Lambda(g^{2}xv)\\\
\Lambda(g^{3}x)\Lambda(v)=\Lambda(g^{3}x)\Lambda(g^{3}v)+\Lambda(1)\Lambda(g^{3}xv)\end{array}\right.\qquad
v\in\mathcal{B}.$

Suppose that $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the system
above. Then, by Lemma 3.5 (a), we obtain $\alpha_{gx}=\alpha_{g^{2}x}=0$.
Thus, we can discard the first and second block of equations in the system
above, and deal only with the third. That is,

$\left\\{\begin{array}[]{ll}\alpha_{g^{3}x}=\alpha_{g^{3}x}\\\
0=\alpha_{g^{3}x}-\alpha_{x}\\\ 0=\alpha_{gx}\\\ 0=-\alpha_{g^{2}x}\\\
\alpha_{g^{3}x}\alpha_{x}=\alpha_{g^{3}x}\alpha_{g^{3}x}\\\
0=\alpha_{g^{3}x}\alpha_{x}+1\\\ 0=\alpha_{g^{3}x}\alpha_{gx}\\\
\alpha_{g^{3}x}\alpha_{g^{3}x}=-1.\end{array}\right.$

Hence, $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the system above if
and only if $\alpha_{x}=\alpha_{g^{3}x}$ and $\alpha_{x}^{2}=-1$.

Therefore, all partial actions of $\mathcal{A}_{4}^{\prime\prime}$ on $\Bbbk$
are given in Table 3. ∎

The 16-dimensional pointed non-semisimple Hopf algebras were classified by
[11], and there exist 29 isomorphism classes, all of them listed below. We
will adopt the notation and presentation for these Hopf algebras given by
generators and relations in [1], and we write $G(H)$, $P_{g,h}(H)$ and
$\varepsilon_{H}$ shortly by $G$, $P_{g,h}$ and $\varepsilon$, respectively,
since the Hopf algebra $H$ is obvious on each case:

$\mathbf{H_{1}}=\langle
g,\,x,\,y,\,z\,\,|\,\,g^{2}=1,\,\,x^{2}=y^{2}=z^{2}=0,\,\,ab=-ba\rangle,$
where $a,b\in\\{g,x,y,z\\},a\neq b$, $g\in G$ and $x,y,z\in P_{1,g}$;

$\mathbf{H_{2}}=\langle
g,\,x,\,y\,\,|\,\,g^{4}=1,\,\,x^{2}=y^{2}=0,\,\,xg=qgx,\,\,yg=qgy,\,\,yx=-xy\rangle,$
where $q$ is a primitive root of unity of order $4$, $g\in G$ and $x,y\in
P_{1,g^{2}}$;

$\mathbf{H_{3}}=\langle
g,\,x,\,y\,\,|\,\,g^{4}=1,\,\,x^{2}=y^{2}=0,\,\,xg=qgx,\,\,yg=-qgy,\,\,yx=-xy\rangle,$
where $q$ is a primitive root of unity of order $4$, $g\in G$ and $x,y\in
P_{1,g^{2}}$;

$\mathbf{H_{4}}=\langle
g,\,x,\,y\,\,|\,\,g^{4}=1,\,\,x^{2}=y^{2}=0,\,\,xg=-gx,yg=-gy,\,\,yx=-xy\rangle,$
where $g\in G$ and $x,y\in P_{1,g}$;

$\mathbf{H_{5}}=\langle
g,x,y|g^{4}=1,x^{2}=0,y^{2}=g^{2}-1,xg=-gx,yg=-gy,yx=-xy\rangle,$ where $g\in
G$ and $x,y\in P_{1,g}$;

$\mathbf{H_{6}}=\langle
g,x,y\,|\,g^{4}=1,x^{2}=0,y^{2}=g^{2}-1,xg=-gx,yg=-gy,yx=-xy+g^{2}-1\rangle,$
where $g\in G$ and $x,y\in P_{1,g}$;

$\mathbf{H_{7}}=\langle
g,x,y\,\,|\,\,g^{4}=1,\,x^{2}=0,\,y^{2}=0,\,xg=-gx,\,yg=-gy,\,yx=-xy\rangle,$
where $g\in G$, $x\in P_{1,g}$ and $y\in P_{1,g^{3}}$;

$\mathbf{H_{8}}=\langle
g,x,y|g^{4}=1,x^{2}=0,y^{2}=g^{2}-1,xg=-gx,yg=-gy,yx=-xy\rangle,$ where $g\in
G$, $x\in P_{1,g}$ and $y\in P_{1,g^{3}}$;

$\mathbf{H_{9}}=\langle
g,x,y\,|\,g^{4}=1,\,x^{2}=y^{2}=g^{2}-1,\,xg=-gx,\,yg=-gy,\,yx=-xy\rangle,$
where $g\in G$, $x\in P_{1,g}$ and $y\in P_{1,g^{3}}$;

$\mathbf{H_{10}}=\langle
g,\;\;x\;\;|\;\;g^{4}=1,\;\;x^{4}=0,\;\;xg=qgx\rangle,$ where $q$ is a
primitive root of unity of order $4$, $g\in G$ and $x\in P_{1,g}$;

$\mathbf{H_{11}}=\langle
g,\;\;x\;\;|\;\;g^{4}=1,\;\;x^{4}=0,\;\;xg=-qgx\rangle,$ where $q$ is a
primitive root of unity of order $4$, $g\in G$ and $x\in P_{1,g}$;

$\mathbf{H_{12}}=\mathcal{A}_{2}\otimes\Bbbk C_{2}$, where we will consider
$\mathbf{H_{12}}$ as

$\langle
g,x,y,|g^{2}=1,x^{2}=y^{2}=0,xg=-gx,yg=-gy,yx=-xy\rangle\otimes\langle
h|h^{2}=1\rangle;$

$\mathbf{H_{13}}=\langle
g,\,h,\,x,\,y\,|\,g^{2}=h^{2}=(gh)^{2}=1,\,x^{2}=y^{2}=0,\,xg=-gx,yg=-gy,\,xh=hx,\,yh=-hy,\,yx=-xy\rangle,$
where $g,h\in G$ and $x,y\in P_{1,g}$;

$\mathbf{H_{14}}=\mathbb{H}_{4}\otimes\mathbb{H}_{4}$, where $\mathbb{H}_{4}$
is the Sweedler’s Hopf algebra. Precisely, consider

$\mathbf{H_{14}}=\langle
g,x\,\,|\,\,g^{2}=1,x^{2}=0,xg=-gx\rangle\otimes\langle
h,y\,\,|\,\,h^{2}=1,y^{2}=0,yh=-hy\rangle;$

$\mathbf{H_{15}}=\langle
g,h,x,y|g^{2}=h^{2}=(gh)^{2}=1,x^{2}=y^{2}=0,yx=-xy,ab=-ba\rangle,$ where
$a\in\\{x,y\\},b\in\\{g,h\\},$ $g,h\in G$, $x\in P_{1,g}$ and $y\in P_{1,h}$;

$\mathbf{H_{16}}=\langle
g,h,x,y|g^{2}=h^{2}=(gh)^{2}=1,x^{2}=y^{2}=0,yx=-xy+gh-1,ab=-ba\rangle,$ where
$a\in\\{x,y\\},b\in\\{g,h\\},$ $g,h\in G$, $x\in P_{1,g}$ and $y\in P_{1,h}$;

$\mathbf{H_{17}}=\mathbb{H}_{4}\otimes\Bbbk(C_{2}\times C_{2})$, where we will
consider $\mathbf{H_{17}}$ as

$\langle
g,x\,\,|\,\,g^{2}=1,x^{2}=0,xg=-gx\rangle\otimes\langle\,a,b\,\,|\,\,a^{2}=b^{2}=(ab)^{2}=1\,\rangle;$

$\mathbf{H_{18}}=\langle g,x\,|\,g^{8}=1,x^{2}=0,xg=-gx\rangle,$ where $g\in
G$ and $x\in P_{1,g}$;

$\mathbf{H_{19}}=\langle g,x\,|\,g^{8}=1,x^{2}=0,xg=qgx\rangle,$ where $q$ is
a primitive root of unity of order $8$, $g\in G$ and $x\in P_{1,g^{4}}$;

$\mathbf{H_{20}}=\langle g,x\,|\,g^{8}=1,x^{2}=0,xg=qgx\rangle,$ where $q$ is
a primitive root of unity of order $4$, $g\in G$ and $x\in P_{1,g^{2}}$;

$\mathbf{H_{21}}=\langle g,x\,|\,g^{8}=1,x^{2}=0,xg=qgx\rangle,$ where $q$ is
a primitive root of unity of order $4$, $g\in G$ and $x\in P_{1,g^{6}}$;

$\mathbf{H_{22}}=\langle g,x\,|\,g^{8}=1,x^{2}=g^{2}-1,xg=-gx\rangle,$ where
$g\in G$ and $x\in P_{1,g}$;

$\mathbf{H_{23}}=\langle
g,\,h,\,x\;\,|\;\,g^{4}=h^{2}=(gh)^{4}=1,\,\;x^{2}=0,\,\;xg=-gx,\,\;\;xh=hx\rangle,$
where $g,h\in G$ and $x\in P_{1,g}$;

$\mathbf{H_{24}}=\langle
g,\,h,\,x\;\,|\;\,g^{4}=h^{2}=(gh)^{4}=1,\,\;x^{2}=0,\,\;xg=gx,\,\;\;xh=-hx\rangle,$
where $g,h\in G$ and $x\in P_{1,gh}$;

$\mathbf{H_{25}}=\langle
g,\,h,\,x\;\,|\;\,g^{4}=h^{2}=(gh)^{4}=1,\,\;x^{2}=0,\,\;xg=qgx,\,\;\;xh=hx\rangle,$
where $q$ is a primitive root of unity of order $4$, $g,h\in G$ and $x\in
P_{1,g^{2}}$;

$\mathbf{H_{26}}=\langle
g,\,h,\,x\;\,|\;\,g^{4}=h^{2}=(gh)^{4}=1,\,\;x^{2}=0,\,\;xg=gx,\,\;\;xh=-hx\rangle,$
where $g,h\in G$ and $x\in P_{1,h}$;

$\mathbf{H_{27}}=\langle
g,\,h,\,x\;\,|\;\,g^{4}=h^{2}=(gh)^{4}=1,\,\;x^{2}=0,\,\;xg=qgx,\,\;\;xh=hx\rangle,$
where $q$ is a primitive root of unity of order $4$, $g,h\in G$ and $x\in
P_{1,g^{2}h}$;

$\mathbf{H_{28}}=\langle
g,\,h,\,x\,|\,g^{4}=h^{2}=(gh)^{4}=1,\,x^{2}=g^{2}-1,\,xg=-gx,\,xh=hx\rangle,$
where $g,h\in G$ and $x\in P_{1,g}$;

$\mathbf{H_{29}}=\langle
g,\,h,\,x\,|\,g^{4}=h^{2}=(gh)^{4}=1,\,\,x^{2}=g^{2}-1,\,\,xg=gx,\,\,xh=-hx\rangle,$
where $g,h\in G$ and $x\in P_{1,gh}$.

In the following theorem, we present the partial actions of the Hopf algebras
listed above. In order to reduce the size of the tables, we will omit
$\lambda_{N}(h)$ for all $h\in G(H)$, since they are determined uniquely by
$N$.

######  Theorem 3.8.

For each pointed non-semisimple Hopf algebra of dimension 16, all partial
actions on its base field are computed. They are presented in Tables 6 \- 34,
where $\alpha,\beta,\theta,\gamma,\omega,\delta,\sigma,\Omega,\zeta\in\Bbbk$
are such that $\gamma^{2}=\omega^{2}=-1,$ $\delta\sigma=0$ and
$\Omega\zeta=-\frac{1}{2}$.

Table 6. Partial actions of $\mathbf{H_{1}}$ on $\Bbbk$ | $x$ | $gx$ | $y$ | $gy$ | $z$ | $gz$ | $xy$ | $gxy$ | $xz$ | $gxz$ | $yz$ | $gyz$ | $xyz$ | $gxyz$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}\\!=\\!\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\theta$ | $\theta$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 7. Partial actions of $\mathbf{H_{2}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\beta$ | $0$ | $\beta$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 8. Partial actions of $\mathbf{H_{3}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\beta$ | $0$ | $\beta$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 9. Partial actions of $\mathbf{H_{4}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 10. Partial actions of $\mathbf{H_{5}}$ on $\Bbbk$ $(\gamma^{2}=-1)$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $\gamma$ | $0$ | $0$ | $\gamma$ | $0$ | $0$ | $0$ | $0$
Table 11. Partial actions of $\mathbf{H_{6}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
Table 12. Partial actions of $\mathbf{H_{7}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 13. Partial actions of $\mathbf{H_{8}}$ on $\Bbbk$ $(\gamma^{2}=-1)$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $\gamma$ | $\gamma$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 14. Partial actions of $\mathbf{H_{9}}$ on $\Bbbk$ $(\gamma^{2}=\omega^{2}=-1)$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $y$ | $gy$ | $g^{2}y$ | $g^{3}y$ | $xy$ | $gxy$ | $g^{2}xy$ | $g^{3}xy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\gamma$ | $0$ | $0$ | $\gamma$ | $\omega$ | $\omega$ | $0$ | $0$ | $0$ | $-\gamma\omega$ | $0$ | $\gamma\omega$
Table 15. Partial actions of $\mathbf{H_{10}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $x^{2}$ | $gx^{2}$ | $g^{2}x^{2}$ | $g^{3}x^{2}$ | $x^{3}$ | $gx^{3}$ | $g^{2}x^{3}$ | $g^{3}x^{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}\\!=\\!\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $0$ | $-q\alpha$ | $\alpha^{2}$ | $0$ | $-q\alpha^{2}$ | $(1\\!-\\!q)\alpha^{2}$ | $\alpha^{3}$ | $\alpha^{3}$ | $\alpha^{3}$ | $\alpha^{3}$
Table 16. Partial actions of $\mathbf{H_{11}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $x^{2}$ | $gx^{2}$ | $g^{2}x^{2}$ | $g^{3}x^{2}$ | $x^{3}$ | $gx^{3}$ | $g^{2}x^{3}$ | $g^{3}x^{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}\\!=\\!\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $0$ | $q\alpha$ | $\alpha^{2}$ | $0$ | $q\alpha^{2}$ | $(1\\!+\\!q)\alpha^{2}$ | $\alpha^{3}$ | $\alpha^{3}$ | $\alpha^{3}$ | $\alpha^{3}$
Table 17. Partial actions of $\mathbf{H_{12}}$ on $\Bbbk$ | $x$ | $hx$ | $y$ | $hy$ | $gx$ | $ghx$ | $gy$ | $ghy$ | $xy$ | $hxy$ | $gxy$ | $ghxy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,gh\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $\beta$ | $0$ | $\alpha$ | $0$ | $\beta$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 18. Partial actions of $\mathbf{H_{13}}$ on $\Bbbk$ | $x$ | $gx$ | $hx$ | $ghx$ | $y$ | $gy$ | $hy$ | $ghy$ | $xy$ | $gxy$ | $hxy$ | $ghxy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,gh\\}}$ | $0$ | $0$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 19. Partial actions of $\mathbf{H_{14}}$ on $\Bbbk$ | $y$ | $hy$ | $x$ | $gx$ | $gy$ | $ghy$ | $hx$ | $ghx$ | $xy$ | $gxy$ | $hxy$ | $ghxy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g\\}}$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,gh\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\beta$ | $\beta$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$ | $\alpha\beta$ | $\alpha\beta$ | $\alpha\beta$ | $\alpha\beta$
Table 20. Partial actions of $\mathbf{H_{15}}$ on $\Bbbk$ $(\sigma\delta=0)$ | $x$ | $gx$ | $hx$ | $ghx$ | $y$ | $gy$ | $hy$ | $ghy$ | $xy$ | $gxy$ | $hxy$ | $ghxy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,gh\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\sigma$ | $\sigma$ | $0$ | $0$ | $\delta$ | $0$ | $\delta$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 21. Partial actions of $\mathbf{H_{16}}$ on $\Bbbk$ $\left(\Omega\zeta=-\frac{1}{2}\right)$ | $x$ | $gx$ | $hx$ | $ghx$ | $y$ | $gy$ | $hy$ | $ghy$ | $xy$ | $gxy$ | $hxy$ | $ghxy$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,gh\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\beta$ | $\beta$ | $\beta$ | $\beta$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\Omega$ | $\Omega$ | $0$ | $0$ | $\zeta$ | $0$ | $\zeta$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
Table 22. Partial actions of $\mathbf{H_{17}}$ on $\Bbbk$ | $x$ | $ax$ | $bx$ | $abx$ | $gx$ | $agx$ | $bgx$ | $abgx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,a,g,ag\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,b,g,bg\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,ab,g,abg\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,a,gb,agb\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,b,ag,abg\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,ab,ag,bg\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,a,b,ab\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,ag\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,bg\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,abg\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,a\\}}$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $0$ | $0$
$\lambda_{\\{1,b\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$
$\lambda_{\\{1,ab\\}}$ | $\alpha$ | $0$ | $0$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $\alpha$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $0$ | $0$ | $\alpha$ | $0$ | $0$ | $0$
Table 23. Partial actions of $\mathbf{H_{18}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $g^{4}x$ | $g^{5}x$ | $g^{6}x$ | $g^{7}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g^{4}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 24. Partial actions of $\mathbf{H_{19}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $g^{4}x$ | $g^{5}x$ | $g^{6}x$ | $g^{7}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{4}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $0$ | $0$ | $\alpha$ | $0$ | $0$ | $0$
Table 25. Partial actions of $\mathbf{H_{20}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $g^{4}x$ | $g^{5}x$ | $g^{6}x$ | $g^{7}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{4}\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 26. Partial actions of $\mathbf{H_{21}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $g^{4}x$ | $g^{5}x$ | $g^{6}x$ | $g^{7}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{4}\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 27. Partial actions of $\mathbf{H_{22}}$ on $\Bbbk$ $(\gamma^{2}=-1)$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $g^{4}x$ | $g^{5}x$ | $g^{6}x$ | $g^{7}x$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g^{4}\\}}$ | $\gamma$ | $0$ | $0$ | $\gamma$ | $\gamma$ | $0$ | $0$ | $\gamma$
$\lambda_{\\{1\\}}$ | $\gamma$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\gamma$
Table 28. Partial actions of $\mathbf{H_{23}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $hx$ | $ghx$ | $g^{2}hx$ | $g^{3}hx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g,g^{2},g^{3}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},h,g^{2}h\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},gh,g^{3}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 29. Partial actions of $\mathbf{H_{24}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $hx$ | $ghx$ | $g^{2}hx$ | $g^{3}hx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g,g^{2},g^{3}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g^{2},h,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},gh,g^{3}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $0$ | $\alpha$ | $0$ | $\alpha$
$\lambda_{\\{1\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 30. Partial actions of $\mathbf{H_{25}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $hx$ | $ghx$ | $g^{2}hx$ | $g^{3}hx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g,g^{2},g^{3}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},h,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$
$\lambda_{\\{1,g^{2},gh,g^{3}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 31. Partial actions of $\mathbf{H_{26}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $hx$ | $ghx$ | $g^{2}hx$ | $g^{3}hx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g,g^{2},g^{3}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g^{2},h,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},gh,g^{3}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $0$ | $0$ | $\alpha$ | $0$ | $0$ | $0$
Table 32. Partial actions of $\mathbf{H_{27}}$ on $\Bbbk$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $hx$ | $ghx$ | $g^{2}hx$ | $g^{3}hx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g,g^{2},g^{3}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},h,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,h\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $\alpha$ | $0$
$\lambda_{\\{1,g^{2},gh,g^{3}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\alpha$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\alpha$ | $0$
Table 33. Partial actions of $\mathbf{H_{28}}$ on $\Bbbk$ $(\gamma^{2}=-1)$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $hx$ | $ghx$ | $g^{2}hx$ | $g^{3}hx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g,g^{2},g^{3}\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},h,g^{2}h\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g^{2}h\\}}$ | $\gamma$ | $0$ | $0$ | $\gamma$ | $0$ | $\gamma$ | $\gamma$ | $0$
$\lambda_{\\{1,h\\}}$ | $\gamma$ | $0$ | $0$ | $\gamma$ | $\gamma$ | $0$ | $0$ | $\gamma$
$\lambda_{\\{1,g^{2},gh,g^{3}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1\\}}$ | $\gamma$ | $0$ | $0$ | $\gamma$ | $0$ | $0$ | $0$ | $0$

Table 34. Partial actions of $\mathbf{H_{29}}$ on $\Bbbk$ $(\gamma^{2}=-1)$ | $x$ | $gx$ | $g^{2}x$ | $g^{3}x$ | $hx$ | $ghx$ | $g^{2}hx$ | $g^{3}hx$
---|---|---|---|---|---|---|---|---
$\lambda_{G}=\varepsilon$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g,g^{2},g^{3}\\}}$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
$\lambda_{\\{1,g^{2},h,g^{2}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2},gh,g^{3}h\\}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\lambda_{\\{1,g^{2}\\}}$ | $\alpha$ | $0$ | $\alpha$ | $0$ | $0$ | $\alpha$ | $0$ | $\alpha$
$\lambda_{\\{1\\}}$ | $\gamma$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\gamma$

###### Proof.

To obtain the partial actions presented in Tables 6 \- 34, routine
computations using the method described in Subsection 3.2 are performed.

We only make some considerations for $\mathbf{H_{6}}$ (Table 11), as this is
the first case where there exists a subgroup $N$ of $G(\mathbf{H_{6}})$ such
that none of the initial $N$-conditions for the partial system associated with
$\mathcal{B}$ is also a solution of the $N$-reduced partial system associated
with $\mathcal{B}$.

Step 1: let
$\mathcal{B}=\\{1,g,g^{2},g^{3},x,gx,g^{2}x,g^{3}x,y,gy,g^{2}y,g^{3}y,xy,gxy,g^{2}xy,g^{3}xy\\}$
be a basis of $\mathbf{H}_{6}$.

Step 2: since $G(\mathbf{H}_{6})=\\{1,g,g^{2},g^{3}\\}$, we have 3
possibilities for $N$: $\\{1,g,g^{2},g^{3}\\}$, $\\{1,g^{2}\\}$ and $\\{1\\}$.

  1. (1)

$N=\\{1,g,g^{2},g^{3}\\}$. By Step 3 we assume
$\alpha_{1}=\alpha_{g}=\alpha_{g^{2}}=\alpha_{g^{3}}=1$,
$\alpha_{g^{3}x}=\alpha_{g^{2}x}=\alpha_{gx}=\alpha_{x}$,
$\alpha_{g^{3}y}=\alpha_{g^{2}y}=\alpha_{gy}=\alpha_{y}$ and
$\alpha_{g^{3}xy}=\alpha_{g^{2}xy}=\alpha_{gxy}=\alpha_{xy}$, where
$\alpha_{x},\alpha_{y},\alpha_{xy}\in\Bbbk$.

Step 4: investigate the $N$-reduced partial system associated with
$\mathcal{B}$. First, if $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of
(3.14), then $\alpha_{x}=\alpha_{y}=0$ and also $\alpha_{xy}=0$, by Lemma 3.5
items (a) and (b), respectively. Hence, in this case, the only possibility for
the associated partial action $\lambda:\mathbf{H}_{6}\longrightarrow\Bbbk$,
$\lambda(b)=\alpha_{b}$, is the counit $\varepsilon$.

Step 5: $\lambda$ is a partial action of $\mathbf{H}_{6}$ on $\Bbbk$ with
initial condition $\\{1,g,g^{2},g^{3}\\}$ if and only if
$\lambda=\varepsilon$;

  2. (2)

$N=\\{1,g^{2}\\}$. Step 3: consider $\alpha_{1}=\alpha_{g^{2}}=1$,
$\alpha_{g}=\alpha_{g^{3}}=0$, $\alpha_{g^{2}x}=\alpha_{x}$,
$\alpha_{g^{3}x}=\alpha_{gx}$, $\alpha_{g^{2}y}=\alpha_{y}$,
$\alpha_{g^{3}y}=\alpha_{gy}$, $\alpha_{g^{2}xy}=\alpha_{xy}$ and
$\alpha_{g^{3}xy}=\alpha_{gxy}$, where
$\alpha_{x},\alpha_{gx},\alpha_{y},\alpha_{gy},\alpha_{xy},\alpha_{gxy}\in\Bbbk$.

Step 4: since $x,y\in\mathcal{B}_{1,g}$ we have
$\widetilde{\mathcal{B}}=\\{gx,gy,xy,gxy\\}$. Then, the system (3.14) has 4
blocks of 16 equations each:

$\displaystyle\left\\{\begin{array}[]{rl}\Lambda(gx)\Lambda(v)=&\\!\\!\\!\\!\Lambda(gx)\Lambda(gv)+\Lambda(g^{2})\Lambda(gxv)\\\
\Lambda(gy)\Lambda(v)=&\\!\\!\\!\\!\Lambda(gy)\Lambda(gv)+\Lambda(g^{2})\Lambda(gyv)\\\
\Lambda(xy)\Lambda(v)=&\\!\\!\\!\\!\Lambda(xy)\Lambda(v)-\Lambda(gx)\Lambda(yv)\\\
&\\!\\!\\!\\!+\Lambda(gy)\Lambda(xv)+\Lambda(g^{2})\Lambda(xyv)\\\
\Lambda(gxy)\Lambda(v)=&\\!\\!\\!\\!\Lambda(gxy)\Lambda(gv)-\Lambda(g^{2}x)\Lambda(gyv)\\\
&\\!\\!\\!\\!+\Lambda(g^{2}y)\Lambda(gxv)+\Lambda(g^{3})\Lambda(gxyv)\end{array}\right.\qquad
v\in\mathcal{B}.$

Assume that $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the system
above.

If we replace $(\alpha_{b})_{b\in\mathcal{B}}$ in the first two blocks of
equations, we get:

$\left\\{\begin{array}[]{ll}\alpha_{gx}=\alpha_{gx}\\\
0=\alpha_{gx}-\alpha_{x}\\\ \alpha_{gx}=\alpha_{gx}\\\
0=\alpha_{gx}-\alpha_{x}\\\ \alpha_{gx}\alpha_{x}=\alpha_{gx}\alpha_{gx}\\\
\alpha_{gx}\alpha_{gx}=\alpha_{gx}\alpha_{x}\\\
\alpha_{gx}\alpha_{x}=\alpha_{gx}\alpha_{gx}\\\
\alpha_{gx}\alpha_{gx}=\alpha_{gx}\alpha_{x}\\\
\alpha_{gx}\alpha_{y}=\alpha_{gx}\alpha_{gy}+\alpha_{gxy}\\\
\alpha_{gx}\alpha_{gy}=\alpha_{gx}\alpha_{y}-\alpha_{xy}\\\
\alpha_{gx}\alpha_{y}=\alpha_{gx}\alpha_{gy}+\alpha_{gxy}\\\
\alpha_{gx}\alpha_{gy}=\alpha_{gx}\alpha_{y}-\alpha_{xy}\\\
\alpha_{gx}\alpha_{xy}=\alpha_{gx}\alpha_{gxy}\\\
\alpha_{gx}\alpha_{gxy}=\alpha_{gx}\alpha_{xy}\\\
\alpha_{gx}\alpha_{xy}=\alpha_{gx}\alpha_{gxy}\\\
\alpha_{gx}\alpha_{gxy}=\alpha_{gx}\alpha_{xy}\end{array}\right.\textrm{ \ \
and \ \ }\left\\{\begin{array}[]{ll}\alpha_{gy}=\alpha_{gy}\\\
0=\alpha_{gy}-\alpha_{y}\\\ \alpha_{gy}=\alpha_{gy}\\\
0=\alpha_{gy}-\alpha_{y}\\\
\alpha_{gy}\alpha_{x}=\alpha_{gy}\alpha_{gx}-\alpha_{gxy}\\\
\alpha_{gy}\alpha_{gx}=\alpha_{gy}\alpha_{x}+\alpha_{xy}\\\
\alpha_{gy}\alpha_{x}=\alpha_{gy}\alpha_{gx}-\alpha_{gxy}\\\
\alpha_{gy}\alpha_{gx}=\alpha_{gy}\alpha_{x}+\alpha_{xy}\\\
\alpha_{gy}\alpha_{y}=\alpha_{gy}\alpha_{gy}\\\
\alpha_{gy}\alpha_{gy}=\alpha_{gy}\alpha_{y}\\\
\alpha_{gy}\alpha_{y}=\alpha_{gy}\alpha_{gy}\\\
\alpha_{gy}\alpha_{gy}=\alpha_{gy}\alpha_{y}\\\
\alpha_{gy}\alpha_{xy}=\alpha_{gy}\alpha_{gxy}\\\
\alpha_{gy}\alpha_{gxy}=\alpha_{gy}\alpha_{xy}\\\
\alpha_{gy}\alpha_{xy}=\alpha_{gy}\alpha_{gxy}\\\
\alpha_{gy}\alpha_{gxy}=\alpha_{gy}\alpha_{xy}.\end{array}\right.$

From the second line of each block, we obtain $\alpha_{x}=\alpha_{gx}$ and
$\alpha_{y}=\alpha_{gy}$. Consequently one concludes that
$\alpha_{gxy}=\alpha_{xy}=0$, and with these parameters the two blocks of
conditions above hold.

Now, one needs analyze the third and fourth blocks of equations, that is,

$\displaystyle\left\\{\begin{array}[]{rl}\Lambda(xy)\Lambda(v)=&\\!\\!\\!\\!\Lambda(xy)\Lambda(v)-\Lambda(gx)\Lambda(yv)\\\
&\\!\\!\\!\\!+\Lambda(gy)\Lambda(xv)+\Lambda(g^{2})\Lambda(xyv)\\\
\Lambda(gxy)\Lambda(v)=&\\!\\!\\!\\!\Lambda(gxy)\Lambda(gv)-\Lambda(g^{2}x)\Lambda(gyv)\\\
&\\!\\!\\!\\!+\Lambda(g^{2}y)\Lambda(gxv)+\Lambda(g^{3})\Lambda(gxyv)\end{array}\right.\qquad
v\in\mathcal{B}.$

However, is an easy task to verify that $(\alpha_{b})_{b\in\mathcal{B}}$, with
the parameters already obtained, satisfies these blocks of equations, and no
additional conditions are required.

Step 5: As conclusion, for any $\alpha,\beta\in\Bbbk$, the linear map
$\lambda:\mathbf{H}_{6}\longrightarrow\Bbbk$ given by
$\lambda(1)=\lambda(g^{2})=1$, $\lambda(g)=\lambda(g^{3})=0$,
$\lambda(x)=\lambda(gx)=\lambda(g^{2}x)=\lambda(g^{3}x)=\alpha$,
$\lambda(y)=\lambda(gy)=\lambda(g^{2}y)=\lambda(g^{3}y)=\beta$ and
$\lambda(xy)=\lambda(gxy)=\lambda(g^{2}xy)=\lambda(g^{3}xy)=0$, is a partial
action of $\mathbf{H}_{6}$ on $\Bbbk$.

  3. (3)

$N=\\{1\\}$. Step 3: we consider $(\alpha_{b})_{b\in\mathcal{B}}$ a solution
of (3.12). Now, in Step 4, we investigate the system (3.14). As
$gx,g^{3}x,g^{3}y\in\widetilde{\mathcal{B}}$, consider the particular next 5
equations of the system (3.14):

$\displaystyle\left\\{\begin{array}[]{lr}\Lambda(gx)\Lambda(v)=\Lambda(gx)\Lambda(gv)+\Lambda(g^{2})\Lambda(gxv)&\qquad\qquad
v=1\\\
\Lambda(g^{3}x)\Lambda(v)=\Lambda(g^{3}x)\Lambda(g^{3}v)+\Lambda(1)\Lambda(g^{3}xv)&v=g\\\
\Lambda(g^{3}x)\Lambda(v)=\Lambda(g^{3}x)\Lambda(g^{3}v)+\Lambda(1)\Lambda(g^{3}xv)&v=gx\\\
\Lambda(g^{3}x)\Lambda(v)=\Lambda(g^{3}x)\Lambda(g^{3}v)+\Lambda(1)\Lambda(g^{3}xv)&v=gy\\\
\Lambda(g^{3}y)\Lambda(v)=\Lambda(g^{3}y)\Lambda(g^{3}v)+\Lambda(1)\Lambda(g^{3}yv)&v=gx\end{array}\right.$

If $(\alpha_{b})_{b\in\mathcal{B}}$ is a solution of the system (3.14), then
for the particular equations above, we get:

$\left\\{\begin{array}[]{ll}\alpha_{gx}=0\\\ 0=\alpha_{g^{3}x}-\alpha_{x}\\\
\alpha_{g^{3}x}\alpha_{gx}=\alpha_{g^{3}x}\alpha_{x}\\\
\alpha_{g^{3}x}\alpha_{gy}=\alpha_{g^{3}x}\alpha_{y}-\alpha_{xy}\\\
\alpha_{g^{3}y}\alpha_{gx}=\alpha_{g^{3}y}\alpha_{x}+\alpha_{xy}+1.\end{array}\right.$

The first 3 conditions imply that $\alpha_{x}=\alpha_{gx}=\alpha_{g^{3}x}=0$.
Then, the fourth condition imposes $\alpha_{xy}=0$. But, in this case, the
fifth condition states $0=1$, a contradiction.

Hence, $(\alpha_{b})_{b\in\mathcal{B}}$ cannot be a solution to the both
systems: (3.12) and (3.14).

Therefore, all partial actions of $\mathbf{H}_{6}$ on $\Bbbk$ are given in
Table 11. ∎

In particular, Theorem 3.8 guarantees that given a Hopf algebra $H$ and $N$ a
subgroup of $G(H)$, it is not true that there always exists a partial action
$\lambda:H\longrightarrow\Bbbk$ with initial condition $N$. See, for instance,
Table 11.

##  4\. $\lambda$-Hopf algebras

In this section, the concept of a $\lambda$-Hopf algebra is introduced. Some
results and examples are developed. Subsection 4.2 is dedicated to present the
Taft’s algebra as a $\lambda$-Hopf algebra. The deformations produced with the
partial actions of pointed non-semisimple Hopf algebras with dimension 8,16 on
its base field, calculated in the previous section, are analyzed in the last
subsection.

###  4.1. Definition and Properties

In this subsection, a $\lambda$-Hopf algebra is defined: a Hopf algebra
obtained by a deformation of a Hopf algebra $L$ through a partial action
$\lambda:L\longrightarrow\Bbbk$. Some guiding examples are presented and
results concerning such types of deformations and partial actions are
developed. In particular, every Hopf algebra $H$ is actually a $\lambda$-Hopf
algebra.

Recall from Definition 2.6 that, given a partial action
$\lambda:H\longrightarrow\Bbbk$, the corresponding partial smash product
algebra $\underline{\Bbbk\\#H}$ is the algebra generated by the set
$\left\\{\underline{1_{\Bbbk}\\#h}\,|\,h\in
H\right\\}=\left\\{1_{\Bbbk}\\#\lambda(h_{1})h_{2}\,|\,h\in H\right\\}$.
Moreover $\underline{1_{\Bbbk}\\#h}=1_{\Bbbk}\\#\lambda(h_{1})h_{2}$ and
$\underline{1_{\Bbbk}\\#h}\,\,\underline{1_{\Bbbk}\\#k}=\underline{1_{\Bbbk}\\#\lambda(h_{1})h_{2}k}$
for all $h,k\in H$. Then, $\varphi:\underline{\Bbbk\\#H}\longrightarrow H$
given by
$\varphi\left(1_{\Bbbk}\\#\lambda(h_{1})h_{2}\right)=\lambda(h_{1})h_{2}$ is
an injective homomorphism of algebras. We denote by $H_{\lambda}$ the
subalgebra of $H$ given by the image of $\varphi$, that is,
$H_{\lambda}=\varphi\left(\underline{\Bbbk\\#H}\right)\subseteq H$. Hence,
$H_{\lambda}=\\{\lambda(h_{1})h_{2}\,|\,h\in H\\}$ and so
$\underline{\Bbbk\\#H}\simeq H_{\lambda}$ is a subalgebra of $H$. However, it
does not always occur that this subalgebra is a Hopf subalgebra, neither a
subbialgebra nor a subcoalgebra of $H$. We exhibit two examples to illustrate
this situation:

######  Example 4.1.

Let $G$ be a group and $\lambda_{N}$ as in Example 2.2. Then, $(\Bbbk
G)_{\lambda_{N}}\simeq\Bbbk N$ and consequently a Hopf subalgebra of $\Bbbk
G$.

######  Example 4.2.

Assume $char(\Bbbk)\neq 2$ and let $\mathbb{H}_{4}$ be the Sweedler’s Hopf
algebra. Consider the partial action $\lambda_{\alpha}$ as in Example 2.5,
_i.e._ , $\lambda_{\alpha}:\mathbb{H}_{4}\longrightarrow\Bbbk$,
$\lambda_{\alpha}(1)=1_{\Bbbk},$ $\lambda_{\alpha}(g)=0$ and
$\lambda_{\alpha}(x)=\lambda_{\alpha}(gx)=\alpha$, where $\alpha\in\Bbbk$.

The subalgebra
$(\mathbb{H}_{4})_{\lambda_{\alpha}}=\Bbbk\\{1\\}\oplus\Bbbk\\{\alpha g+gx\\}$
is not a subcoalgebra of $\mathbb{H}_{4}$, since

$\Delta(\alpha g+gx)=(\alpha g+gx)\otimes g+1\otimes
gx\notin(\mathbb{H}_{4})_{\lambda_{\alpha}}\otimes(\mathbb{H}_{4})_{\lambda_{\alpha}}.$

The previous examples show that for a given Hopf algebra $L$ and a partial
action $\lambda\in L^{*}$, $L_{\lambda}$ is not always a Hopf subalgebra of
$L$. Thus, we ask when a Hopf algebra $H$ is obtained in this way, that is,
when $H\simeq L_{\lambda}$, for some Hopf algebra $L$ and partial action
$\lambda\in L^{*}$. In this sense, we define:

######  Definition 4.3.

We say that a Hopf algebra $H$ is a _(left) $\lambda$-Hopf algebra_ if there
exists a pair $(L,\lambda)$ where $L$ is a Hopf algebra and
$\lambda:L\longrightarrow\Bbbk$ is a (left) partial action such that
$L_{\lambda}\simeq H$.

Since $\varepsilon$ is a partial action (global, in fact) and
$\varepsilon(h_{1})h_{2}=h$, for all $h\in H$, we have that
$H_{\varepsilon}=H$. Then, every Hopf algebra $H$ is an $\varepsilon$-Hopf
algebra in the usual way.

In the next example we prove that $\Bbbk G$ is a $\lambda$-Hopf algebra.

######  Example 4.4.

Let $G$ be a group. Consider a group $F$ that contains $G$ as subgroup and the
partial action $\lambda_{G}:\Bbbk F\longrightarrow\Bbbk$. It is clear that
$(\Bbbk F)_{\lambda_{G}}\simeq\Bbbk G$. Moreover, since $F$ can be chosen
arbitrarily, we do not have uniqueness for a Hopf algebra $L$ such that
$L_{\lambda}\simeq\Bbbk G$. In particular, we can consider $F$ as the group
$G\times C_{\ell}$, where $C_{\ell}$ denotes the cyclic group of order $\ell$,
$\ell\geq 1$. In this case, $dim_{\Bbbk}(\Bbbk F)=\ell\ dim_{\Bbbk}(\Bbbk G)$.

Now our goal is to determine necessary and sufficient conditions to conclude
whether $H_{\lambda}$ is a Hopf subalgebra of $H$ or not. First, we have the
following lemma:

######  Lemma 4.5.

Let $\lambda:H\longrightarrow\Bbbk$ be a partial action of $H$ on $\Bbbk$.
Then, $\lambda{|_{H_{\lambda}}}=\varepsilon{|_{H_{\lambda}}}$.

###### Proof.

Consider $\lambda(h_{1})h_{2}\in H_{\lambda}$. Then,
$\lambda(\lambda(h_{1})h_{2})=\lambda(h_{1})\lambda(h_{2})=\lambda(h).$ On the
other hand,
$\varepsilon(\lambda(h_{1})h_{2})=\lambda(h_{1})\varepsilon(h_{2})=\lambda(h_{1}\varepsilon(h_{2}))=\lambda(h).$
Thus, $\lambda{|_{H_{\lambda}}}=\varepsilon{|_{H_{\lambda}}}$. ∎

Thus, in the subalgebra $H_{\lambda}$, the maps $\varepsilon$ and $\lambda$
coincide. In particular, $\lambda$ is multiplicative on $H_{\lambda}$, and so
if $H_{\lambda}$ is a subcoalgebra (or subbialgebra or Hopf subalgebra) of
$H$, then the counit of $H_{\lambda}$ is given exactly by
$\lambda{|_{H_{\lambda}}}=\varepsilon{|_{H_{\lambda}}}$. Hence, we obtain the
following characterization:

######  Theorem 4.6.

Let $H$ be a finite-dimensional Hopf algebra and
$\lambda:H\longrightarrow\Bbbk$ a partial action. Then, $H_{\lambda}$ is a
Hopf subalgebra of $H$ if and only if

$\displaystyle\lambda(h_{1})h_{2}=\lambda(h_{1})h_{2}\lambda(h_{3})$ (4.1)

holds for all $h\in H$.

###### Proof.

First, suppose $\lambda(h_{1})h_{2}=\lambda(h_{1})h_{2}\lambda(h_{3})$ for all
$h\in H$. We already have that $H_{\lambda}$ is a subalgebra of $H$. Consider
the restriction of the comultiplication $\Delta_{H}$ of $H$ to $H_{\lambda}$,
that is, $\Delta_{H}{|_{H_{\lambda}}}:H_{\lambda}\longrightarrow H\otimes H$.
We denote $\Delta=\Delta_{H}{|_{H_{\lambda}}}$. We shall see
$\Delta(H_{\lambda})\subseteq H_{\lambda}\otimes H_{\lambda}$. Note that

$\Delta(\lambda(h_{1})h_{2})=\lambda(h_{1})\Delta(h_{2})=\lambda(h_{1})h_{2}\otimes
h_{3}=\lambda(h_{11})h_{12}\otimes h_{2}.$

Now, using the hypothesis, we get $\lambda(h_{11})h_{12}\otimes
h_{2}=\lambda(h_{11})h_{12}\lambda(h_{13})\otimes h_{2},$ and so

$\displaystyle\lambda(h_{11})h_{12}\lambda(h_{13})\otimes h_{2}$
$\displaystyle=\lambda(h_{1})h_{2}\lambda(h_{3})\otimes h_{4}$
$\displaystyle=\lambda(h_{1})h_{2}\otimes\lambda(h_{3})h_{4}$
$\displaystyle=\lambda(h_{11})h_{12}\otimes\lambda(h_{21})h_{22}.$

Therefore, $H_{\lambda}$ is closed under the comultiplication $\Delta$ and,
since we are dealing with restriction maps, we obtain that
$(H_{\lambda},\Delta,\varepsilon|{{}_{H_{\lambda}}})$ is a subcoalgebra of
$H$. Moreover, $H_{\lambda}$ is a subbialgebra of $H$ and, since $H$ is finite
dimensional, it follows that $H_{\lambda}$ is a Hopf subalgebra of $H$ (_cf._
[18, Proposition 7.6.1]).

Conversely, suppose that $H_{\lambda}$ is a Hopf subalgebra of $H$, that is,
$\Delta_{H_{\lambda}}=\Delta_{H}{|_{H_{\lambda}}}$ and
$\varepsilon_{H_{\lambda}}=\varepsilon_{H}{|_{H_{\lambda}}}$. Then,
$H_{\lambda}$ is a Hopf algebra and, in particular,
$\psi\circ(Id_{H_{\lambda}}\otimes\varepsilon_{H_{\lambda}})\circ\Delta_{H_{\lambda}}=Id_{H_{\lambda}}$
holds, where $\psi:H_{\lambda}\otimes\Bbbk\longrightarrow H_{\lambda}$ is the
canonical isomorphism. Using Lemma 4.5, we have that
$\lambda{|_{H_{\lambda}}}=\varepsilon_{H_{\lambda}}$ and then
$\psi\circ(Id_{H_{\lambda}}\otimes\lambda{|_{H_{\lambda}}})\circ\Delta_{H_{\lambda}}=Id_{H_{\lambda}}$
holds. Thus, given $\lambda(h_{1})h_{2}\in H_{\lambda}$, it follows that

$\displaystyle\lambda(h_{1})h_{2}=$ $\displaystyle
Id_{H_{\lambda}}(\lambda(h_{1})h_{2})$ $\displaystyle=$
$\displaystyle[\psi\circ(Id_{H_{\lambda}}\otimes\lambda{|_{H_{\lambda}}})\circ\Delta_{H_{\lambda}}](\lambda(h_{1})h_{2})$
$\displaystyle=$ $\displaystyle\lambda(h_{1})h_{2}\lambda(h_{3}).$

Therefore, $\lambda(h_{1})h_{2}=\lambda(h_{1})h_{2}\lambda(h_{3})$ holds for
all $h\in H$. ∎

Remark. The characterization above can be obtained through the theory of
partial matched pairs of Hopf algebras. In such a setting, the algebra
$H_{\lambda}$ is the same algebra $\Bbbk\underline{\overline{\\#}}H$
constructed in [7], and the compatibility
$\lambda(h_{1})h_{2}=\lambda(h_{1})h_{2}\lambda(h_{3})$, for all $h\in H$, is
exactly the condition needed for $(\Bbbk,H)$ be a partial matched pair of Hopf
algebras. Therefore, $\Bbbk\underline{\overline{\\#}}H$ is a Hopf algebra if
and only if $\lambda(h_{1})h_{2}=\lambda(h_{1})h_{2}\lambda(h_{3})$, for all
$h\in H$ (_cf._ [7, Corollary 4.5]).

######  Corollary 4.7.

Let $H$ be a finite-dimensional Hopf algebra and
$\lambda:H\longrightarrow\Bbbk$ a partial action. Suppose $x\in
P_{g,h}(H)\setminus\Bbbk\\{g-h\\}$ with $\lambda(g)=0$ and $\lambda(h)=1$.
Then, $\lambda(x_{1})x_{2}\neq\lambda(x_{1})x_{2}\lambda(x_{3})$ and,
consequently, $H_{\lambda}$ is not a Hopf subalgebra of $H$.

###### Proof.

Since $\Delta(x)=x\otimes g+h\otimes x$ and $\Delta_{2}(x)=x\otimes g\otimes
g+h\otimes x\otimes g+h\otimes h\otimes x$, where $\Delta_{2}=(\Delta\otimes
Id_{H})\circ\Delta=(Id_{H}\otimes\Delta)\circ\Delta$, we obtain
$\lambda(x_{1})x_{2}=\lambda(x)g+\lambda(h)x=\lambda(x)g+x$. On the other
hand,
$\lambda(x_{1})x_{2}\lambda(x_{3})=\lambda(x)g\lambda(g)+\lambda(h)x\lambda(g)+\lambda(h)h\lambda(x)=\lambda(x)h.$
As $x\neq\lambda(x)(h-g)$, we conclude
$\lambda(x_{1})x_{2}\neq\lambda(x_{1})x_{2}\lambda(x_{3})$.

Thus, Theorem 4.6 implies that $H_{\lambda}$ is not a Hopf subalgebra of $H$.
∎

The most important feature in the previous corollary is that $x\in H$ is a
non-trivial $(g,h)$-primitive element such that $\lambda(g)\neq\lambda(h)$. We
obtain a complementary result for the case when $x$ is a non-trivial
$(g,h)$-primitive element, but $\lambda(g)=1$ and $\lambda(h)=0$.

######  Corollary 4.8.

Let $H$ be a finite-dimensional Hopf algebra and
$\lambda:H\longrightarrow\Bbbk$ a partial action. If $x\in
P_{g,h}(H)\setminus\Bbbk\\{g-h\\}$ with $\lambda(g)=1$ and $\lambda(h)=0$,
then $H_{\lambda}$ is not a Hopf subalgebra of $H$.

###### Proof.

Consider $y=xh^{-1}$. Note that $y\in
P_{gh^{-1},1}(H)\setminus\Bbbk\\{gh^{-1}-1\\}$ and $\lambda(gh^{-1})=0$ by
Proposition 3.1. Hence, the result follows by Corollary 4.7. ∎

######  Corollary 4.9.

Let $H$ be a finite-dimensional Hopf algebra and
$\lambda:H\longrightarrow\Bbbk$ a partial action. If there exists $x\in
P_{g,h}(H)\setminus\Bbbk\\{g-h\\}$ such that $\lambda(g)\neq\lambda(h)$, then
$H_{\lambda}$ is not a Hopf subalgebra of $H$.

###### Proof.

Straightforward from Corollaries 4.7 and 4.8. ∎

The results above deal with $x\in P_{g,h}(H)$ such that
$\lambda(g)\neq\lambda(h)$. The next one deals when $\lambda(g)=\lambda(h)$.

######  Corollary 4.10.

Let $H$ be a Hopf algebra and $\lambda:H\longrightarrow\Bbbk$ a partial
action. Then,

  1. (1)

if $g\in G(H)$ then $\lambda(g_{1})g_{2}=\lambda(g_{1})g_{2}\lambda(g_{3});$

  2. (2)

if $x\in P_{g,h}(H)$ such that $\lambda(g)=\lambda(h)$, then
$\lambda(x_{1})x_{2}=\lambda(x_{1})x_{2}\lambda(x_{3}).$

In particular, if $H$ is finite-dimensional and has a basis given only by
group-like and $(g_{i},h_{i})$-primitive elements, with $g_{i},h_{i}\in G(H)$
such that $\lambda(g_{i})=\lambda(h_{i})$ for all $i$, then $H_{\lambda}$ is a
Hopf subalgebra of $H$.

###### Proof.

Recall from Proposition 3.1 that $\lambda(g)\in\\{0,1_{\Bbbk}\\}$. Then, item
(1) is clear. Now, note that $\lambda(x)=0$ by Lemma 3.2 (b), then
$\lambda(x_{1})x_{2}=\lambda(g)x$ and
$\lambda(x_{1})x_{2}\lambda(x_{3})=\lambda(g)^{2}x$. Thus, (2) holds. Finally,
the last statement in the corollary is clear by Theorem 4.6. ∎

Using these results, sometimes we can quickly check whether $H_{\lambda}$ is a
Hopf subalgebra of $H$ or not. For instance, consider
$(\mathbb{H}_{4})_{\lambda_{\alpha}}$ as in Example 4.2. Since $x\in
P_{1,g}(\mathbb{H}_{4})$ and $\lambda_{\alpha}(1)\neq\lambda_{\alpha}(g)$, it
follows that $(\mathbb{H}_{4})_{\lambda_{\alpha}}$ is not a Hopf subalgebra of
$\mathbb{H}_{4}$. We present in Subsection 4.3 the whole setting for the
partial actions computed in Subsection 3.3.

We exhibit now a condition stronger than (4.1).

######  Proposition 4.11.

Let $\lambda:H\longrightarrow\Bbbk$ be a partial action. If
$\lambda(h_{1})h_{2}=h_{1}\lambda(h_{2})$ for all $h\in H$, then
$\lambda(h_{1})h_{2}=\lambda(h_{1})h_{2}\lambda(h_{3})$ for all $h\in H$. In
this case, if $H$ is finite-dimensional, then $H_{\lambda}$ is a Hopf
subalgebra of $H$.

###### Proof.

Suppose that $\lambda(h_{1})h_{2}=h_{1}\lambda(h_{2})$, for all $h\in H$. Then

$\displaystyle\lambda(h_{1})h_{2}\lambda(h_{3})$
$\displaystyle=\lambda(h_{1})(h_{2})_{1}\lambda((h_{2})_{2})$
$\displaystyle=\lambda(h_{1})\lambda((h_{2})_{1})(h_{2})_{2}$
$\displaystyle=\lambda((h_{1})_{1})\lambda((h_{1})_{2})h_{2}$
$\displaystyle=\lambda(h_{1})h_{2},$

where the last equality follows from (2.1). Thus, if $H$ is finite-
dimensional, it follows by Theorem 4.6 that $H_{\lambda}$ is a Hopf subalgebra
of $H$. ∎

The proposition above has a stronger condition than Theorem 4.6. However,
checking $\lambda(h_{1})h_{2}=h_{1}\lambda(h_{2})$ sometimes it is easier than
checking $\lambda(h_{1})h_{2}\lambda(h_{3})=\lambda(h_{1})h_{2}$, for $h\in
H$. It happens because for the former equality is necessary to use $\Delta$
only once, while the latter needs $\Delta_{2}$.

Recall from Definition 4.3 that a Hopf algebra $H$ is a $\lambda$-Hopf algebra
when there exists a Hopf algebra $L$ and a partial action
$\lambda:L\longrightarrow\Bbbk$ such that $L_{\lambda}\simeq H$. Thus, we ask
when a Hopf algebra $H$ is a $\lambda$-Hopf algebra. We shall see that every
Hopf algebra is a $\lambda$-Hopf algebra. For this purpose, we recall how
partial actions of $H\otimes L$ on $A\otimes B$ are canonically obtained from
partial actions of $H$ and $L$ on $A$ and $B$, respectively.

######  Remark 4.12.

Let $H$ and $L$ be two Hopf algebras. Consider $\cdot_{H}$ and $\cdot_{L}$
partial actions of $H$ and $L$ on algebras $A$ and $B$, respectively. Then,
the linear map
$\rightharpoonup=(\cdot_{H}\otimes\cdot_{L})\circ(Id_{H}\otimes\tau_{L,A}\otimes
Id_{B})$ is a partial action of $H\otimes L$ on $A\otimes B$, where
$\tau_{L,A}:L\otimes A\longrightarrow A\otimes L$ is the canonical twist
isomorphism between $L$ and $A$. Moreover, if $\cdot_{H}$ and $\cdot_{L}$ are
symmetric, then so is $\rightharpoonup$. In particular, since
$\Bbbk\simeq\Bbbk\otimes\Bbbk$, we obtain partial actions of $H\otimes L$ on
$\Bbbk$ from partial actions of $H$ and $L$ on $\Bbbk$.

We emphasize that the partial actions of $H\otimes L$ on $A\otimes B$,
constructed as above, in general do not cover all partial actions of $H\otimes
L$ on $A\otimes B$, even when $A=B=\Bbbk$. As a very simple example, take
$A=B=\Bbbk$ and $H=L=\Bbbk C_{2}$. Then, $H\otimes L=\Bbbk C_{2}\otimes\Bbbk
C_{2}\simeq\Bbbk(C_{2}\times C_{2})$. Recall, by Example 2.2, that a partial
action of $\Bbbk(C_{2}\times C_{2})$ on $\Bbbk$ is parametrized by a subgroup
of $C_{2}\times C_{2}$. Since for $C_{2}=\\{1,g\\}$ we have only the trivial
subgroups, we only have two partial actions of $\Bbbk C_{2}$ on $\Bbbk$,
namely, $\lambda_{\\{1,g\\}}=\varepsilon_{\Bbbk C_{2}}$ and
$\lambda_{\\{1\\}}$. Hence, by Remark 4.12, we obtain only $4$ partial actions
of $\Bbbk C_{2}\otimes\Bbbk C_{2}\simeq\Bbbk(C_{2}\times C_{2})$ on $\Bbbk$:
$\lambda_{\\{1\\}}\otimes\lambda_{\\{1\\}}=\lambda_{\\{(1,1)\\}}$,
$\lambda_{\\{1\\}}\otimes\varepsilon_{\Bbbk
C_{2}}=\lambda_{\\{(1,1),(1,g)\\}}$, $\varepsilon_{\Bbbk
C_{2}}\otimes\lambda_{\\{1\\}}=\lambda_{\\{(1,1),(g,1)\\}}$ and
$\varepsilon_{\Bbbk C_{2}}\otimes\varepsilon_{\Bbbk
C_{2}}=\varepsilon_{\Bbbk(C_{2}\times C_{2})}.$ However, since $C_{2}\times
C_{2}$ has $5$ subgroups, there is a partial action that is not covered by
this construction. Precisely the partial action $\lambda_{\\{(1,1),(g,g)\\}}.$

This fact can also be observed in the examples given in Subsection 3.3. For
instance, we obtain 4 partial actions of $\mathbf{H_{12}}$ on $\Bbbk$ using
the corresponding partial actions of $\mathcal{A}_{2}$ and $\Bbbk C_{2}$ on
$\Bbbk$. Namely, $\varepsilon_{\mathcal{A}_{2}}\otimes\varepsilon_{\Bbbk
C_{2}}=\varepsilon_{\mathbf{H_{12}}}$,
$\varepsilon_{\mathcal{A}_{2}}\otimes\lambda_{\\{1\\}}^{\Bbbk
C_{2}}=\lambda_{\\{1,g\\}}$,
$\lambda_{\\{1\\}}^{\mathcal{A}_{2}}\otimes\varepsilon_{\Bbbk
C_{2}}=\lambda_{\\{1,h\\}}$ and
$\lambda_{\\{1\\}}^{\mathcal{A}_{2}}\otimes\lambda_{\\{1\\}}^{\Bbbk
C_{2}}=\lambda_{\\{1\\}}$. However, note that the partial action
$\lambda_{\\{1,gh\\}}$ is not obtained using Remark 4.12.

We use Remark 4.12 to conclude that every Hopf algebra is a $\lambda$-Hopf
algebra, where $\lambda$ is a genuine partial action, that is, distinct from
the counit, as follows.

######  Theorem 4.13.

Let $H$ be a Hopf algebra. Then, there exists a Hopf algebra $L$ and a partial
action $\lambda:L\longrightarrow\Bbbk$ such that $L_{\lambda}\simeq H$.

###### Proof.

Let $G$ be group, $G\neq\\{1_{G}\\}$, where $1_{G}$ denotes the identity
element of $G$. Consider the partial action of $\Bbbk G$ on $\Bbbk$
parametrized by the subgroup $\\{1_{G}\\}$, that is,
$\lambda_{\\{1_{G}\\}}:\Bbbk G\longrightarrow\Bbbk$ is given by
$\lambda_{\\{1_{G}\\}}(g)=\delta_{1_{G},g}$, that is,
$\lambda_{\\{1_{G}\\}}(1_{G})=1_{\Bbbk}$ and $\lambda_{\\{1_{G}\\}}(g)=0$, for
all $g\in G$, $g\neq 1_{G}$. By Remark 4.12,
$\lambda=\varepsilon_{H}\otimes\lambda_{\\{1_{G}\\}}$ is a partial action of
$H\otimes\Bbbk G$ on $\Bbbk$. Moreover, an easy calculation shows that
$(H\otimes\Bbbk G)_{\lambda}=H\otimes 1_{\Bbbk G}\simeq H$. Indeed, for
$h\otimes g\in H\otimes\Bbbk G$, we have

$\displaystyle\lambda((h\otimes g)_{1})(h\otimes
g)_{2}=\varepsilon(h_{1})\lambda_{\\{1_{G}\\}}(g)(h_{2}\otimes
g)=h\otimes\lambda_{\\{1_{G}\\}}(g)g=h\otimes\delta_{1_{G},g}g.$

Then, for $L=H\otimes\Bbbk G$ and $\lambda$ as above, it follows that

$L_{\lambda}=\langle\lambda((h\otimes g)_{1})(h\otimes g)_{2}\ |\ h\in H,g\in
G\rangle=\langle h\otimes 1_{G}\ |\ h\in H\rangle\simeq H,$

as Hopf algebras. Hence, $H$ is a $\lambda$-Hopf algebra. ∎

The result above is constructive, however this is not the only way to obtain a
Hopf algebra $L$ and a partial action of $L$ on $\Bbbk$ such that
$L_{\lambda}\simeq H$. We shall see in the next subsection that Taft’s algebra
is a $\lambda$-Hopf algebra by another way. Furthermore, since the group $G$
can be chosen arbitrarily in the proof of Theorem 4.13, we obtain that $H$ can
be embedded into the Hopf algebra $L$, where $dim_{\Bbbk}(L)$ is quite
arbitrary, and there exists a partial action $\lambda:L\longrightarrow\Bbbk$
such that $L_{\lambda}\simeq H$. See Example 4.4.

###  4.2. Taft’s algebra as a $\lambda$-Hopf algebra

Throughout this subsection, we assume that $\Bbbk$ is an algebraically closed
field of characteristic zero.

The Taft’s algebra of order $n$, here denoted by $T_{n}(q)$, is an important
Hopf algebra presented by Taft in [20]. We briefly recall its definition. Let
$n\geq 2$ be a positive integer and $q$ a primitive root of unity of order
$n$. As algebra $T_{n}(q)=\langle
g,\;x,\;|\;g^{n}=1,\;x^{n}=0,\;xg=qgx\rangle.$ Thus,
$\mathcal{B}=\\{g^{i}x^{j}\,|\,0\leq i,j\leq n-1\\}$ is the canonical basis
for $T_{n}(q)$, and consequently $dim_{\Bbbk}(T_{n}(q))=n^{2}$. For the
coalgebra structure, $g$ is a group-like element and $x$ is a
$(1,g)$-primitive element. In general, we have
$\Delta(g^{i}x^{j})=\sum_{\ell=0}^{j}{j\choose\ell}_{q}g^{i+\ell}x^{j-\ell}\otimes
g^{i}x^{\ell}$. So, completing the Hopf structure of $T_{n}(q)$, set
$S(g)=g^{n-1}$ and $S(x)=-g^{n-1}x.$ Also, note that $G(T_{n}(q))=\langle
g\rangle=\\{1,g,\cdots,g^{n-1}\\}=C_{n}.$

In the sequel, we present $T_{n}(q)$ as a $\lambda$-Hopf algebra, where the
Hopf algebra $L$ such that $L_{\lambda}\simeq T_{n}(q)$ is not simply
$T_{n}(q)$ tensorized by a group algebra. See Theorem 4.13. Aiming this, we
recall a family of Hopf algebras, here denoted by $T_{n}^{k}(\omega)$, and
calculate a suitable partial action
$\lambda_{k}:T_{n}^{k}(\omega)\longrightarrow\Bbbk$ such that
$(T_{n}^{k}(\omega))_{\lambda_{k}}\simeq T_{n}(\omega^{k})$.

The Hopf algebra $T_{n}^{k}(\omega)$ is a generalization of the Taft’s algebra
$T_{n}(q)$ (see [5] and [6, Appendix]). Let $k$ and $n$ be positive integers,
$n\geq 2$, and $\omega$ a primitive root of unity of order $kn$. As algebra,
$T_{n}^{k}(\omega)$ is generated by the letters $g$ and $x$ with the relations
$g^{kn}=1,$ $x^{n}=0$ and $xg=\omega gx$. Thus,
$\mathcal{B}=\\{g^{i}x^{j}\,|\,0\leq j\leq n-1,\,0\leq i\leq kn-1\\}$ is the
canonical basis of $T_{n}^{k}(\omega)$, and consequently
$dim_{\Bbbk}(T_{n}^{k}(\omega))=kn^{2}$. For the coalgebra structure, $g$ is a
group-like element and $x$ is a $(1,g^{k})$-primitive element. In general,
$\Delta(g^{i}x^{j})=\sum_{\ell=0}^{j}{j\choose\ell}_{\omega^{k}}g^{i+\ell
k}x^{j-\ell}\otimes g^{i}x^{\ell}$. To complete the Hopf structure of
$T_{n}^{k}(\omega)$, set $S(g)=g^{kn-1}$ and $S(x)=-g^{kn-k}x.$ Also, notice
that $G(T_{n}^{k}(\omega))=\langle g\rangle=\\{1,g,\cdots,g^{kn-1}\\}=C_{kn}.$

######  Remark 4.14.

Notice that $T_{n}^{k}(\omega)$ generalizes the Taft’s algebra, since when
$k=1$, $T_{n}^{1}(\omega)$ is exactly the Taft’s algebra of order $n$.
Moreover, for any positive integer $k$, the Taft’s algebra $T_{n}(q)$ is
embedded as a Hopf subalgebra of $T_{n}^{k}(\omega)$, where $q=\omega^{k}$.
Indeed, consider $T_{n}(q)=\langle
h,\;y,\;|\;h^{n}=1,\;y^{n}=0,\;yh=qhy\rangle,$ where $q=\omega^{k}$. Then,
$\varphi:T_{n}(q)\longrightarrow T_{n}^{k}(\omega)$ given by $\varphi(y)=x$
and $\varphi(h)=g^{k}$ is an injective homomorphism of Hopf algebras.

Now, we want to calculate a suitable partial action of $T_{n}^{k}(\omega)$ on
$\Bbbk$. Consider the subgroup $N=\langle g^{k}\rangle$ of
$G(T_{n}^{k}(\omega))$ and the linear map
$\lambda_{k}:T_{n}^{k}(\omega)\longrightarrow\Bbbk$ given by
$\lambda(g^{i}x^{j})=\delta_{j,0}\delta_{N}(g^{i})$ for all
$i,j\in\mathbb{Z}$, $j\geq 0$, where the symbol $\delta_{j,0}$ stands for the
Kronecker delta and $\delta_{N}$ defined as in Example 2.2.

We use Proposition 2.1 to prove that $\lambda_{k}$ as defined above is a
partial action of $T_{n}^{k}(\omega)$ on $\Bbbk$. Denotes $\lambda_{k}$ simply
by $\lambda$. Clearly $\lambda(1)=1_{\Bbbk}$. Now, it remains only to verify
condition (2.1), that is, if

$\lambda(u)\lambda(v)=\lambda(u_{1})\lambda(u_{2}v)$

holds for all $u,v\in T_{n}^{k}(\omega)$.

First, if $u=g^{i}$, $i\in\mathbb{Z}$, then (2.1) is satisfied for all $v\in
T_{n}^{k}(\omega)$. Indeed, if $v=g^{r}x^{s}$, with $r,s\in\mathbb{Z}$, $s\geq
0$, then

$\lambda(u)\lambda(v)=\lambda(g^{i})\lambda(g^{r}x^{s})=\delta_{0,0}\delta_{N}(g^{i})\delta_{s,0}\delta_{N}(g^{r})=\delta_{N}(g^{i})\delta_{s,0}\delta_{N}(g^{r}).$

On the other hand,

$\displaystyle\lambda(u_{1})\lambda(u_{2}v)$
$\displaystyle=\lambda(g^{i})\lambda(g^{i}g^{r}x^{s})=\lambda(g^{i})\lambda(g^{i+r}x^{s})$
$\displaystyle=\delta_{0,0}\delta_{N}(g^{i})\delta_{s,0}\delta_{N}(g^{i+r})=\delta_{N}(g^{i})\delta_{s,0}\delta_{N}(g^{i+r}).$

Thus, if $g^{i}\notin N$, then $\lambda(g^{i})\lambda(g^{i+r}x^{s})=0$ and
$\delta_{N}(g^{i})\delta_{s,0}\delta_{N}(g^{i+r})=0$, since
$\delta_{N}(g^{i})=0$. Otherwise, if $g^{i}\in N$, then
$\delta_{N}(g^{r})=\delta_{N}(g^{i+r})$. Hence, the equality desired holds.

Now, assume $u=g^{i}x^{j}$ with $i,j\in\mathbb{Z}$, $j\geq 1$. Then, for all
$v\in T_{n}^{k}(\omega)$, (2.1) means

$\lambda(g^{i}x^{j})\lambda(v)=\sum_{\ell=0}^{j}{j\choose\ell}_{\omega^{k}}\lambda(g^{i+\ell
k}x^{j-\ell})\lambda(g^{i}x^{\ell}v).$

Since $j\geq 1$, we get $\lambda(g^{i}x^{j})=\delta_{j,0}\delta_{N}(g^{i})=0$
and $\lambda(g^{i+\ell k}x^{j-\ell})=\delta_{j-\ell,0}\delta_{N}(g^{i+\ell
k})=0,$ for all $0\leq\ell\leq(j-1)$. Then, the previous sum is reduced only
when $\ell=j$, and results in

$0={j\choose
j}_{\omega^{k}}\lambda(g^{i+jk}x^{j-j})\lambda(g^{i}x^{j}v)=\lambda(g^{i+jk})\lambda(g^{i}x^{j}v)$

for all $v\in T_{n}^{k}(\omega)$.

For $v=g^{s}x^{t}$ with $s,t\in\mathbb{Z}$, $t\geq 0$, it follows that
$j+t\geq 1$ and consequently
$\lambda(g^{i+s}x^{j+t})=\delta_{j+t,0}\delta_{N}(g^{i+s})=0$. Hence,

$\displaystyle\lambda(g^{i+jk})\lambda(g^{i}x^{j}v)$
$\displaystyle=\lambda(g^{i+jk})\lambda(g^{i}x^{j}g^{s}x^{t})=\omega^{js}\lambda(g^{i+jk})\lambda(g^{i+s}x^{j+t})=0.$

Thus, (2.1) holds for all $u,v\in T_{n}^{k}(\omega)$, and so $\lambda$ is a
partial action of $T_{n}^{k}(\omega)$ on $\Bbbk$.

Finally, it remains to check that
$(T_{n}^{k}(\omega))_{\lambda}=T_{n}(\omega^{k})$. Recall that
$(T_{n}^{k}(\omega))_{\lambda}=\\{\lambda(u_{1})u_{2}\ |\ u\in
T_{n}^{k}(\omega)\\}$. Thus, for $u=g^{i}x^{j}$, $i,j\in\mathbb{Z}$, $j\geq
0$, we have

$\lambda((g^{i}x^{j})_{1})(g^{i}x^{j})_{2}=\sum_{\ell=0}^{j}{j\choose\ell}_{\omega^{k}}\lambda(g^{i+\ell
k}x^{j-\ell})g^{i}x^{\ell}.$

Since $\lambda(g^{i+\ell k}x^{j-\ell})=\delta_{j-\ell,0}\delta_{N}(g^{i+\ell
k})=0$, for any $0\leq\ell\leq(j-1)$, we conclude that
$\lambda((g^{i}x^{j})_{1})(g^{i}x^{j})_{2}={j\choose
j}_{\omega^{k}}\lambda(g^{i+jk}x^{0})g^{i}x^{j}=\lambda(g^{i+jk})g^{i}x^{j}=\delta_{N}(g^{i+jk})g^{i}x^{j},$
for all $i,j\in\mathbb{Z}$, $j\geq 0$.

As $g^{jk}\in N$, it follows that $\delta_{N}(g^{i+jk})=\delta_{N}(g^{i})$ and
consequently
$\lambda((g^{i}x^{j})_{1})(g^{i}x^{j})_{2}=\delta_{N}(g^{i})g^{i}x^{j}$.
Therefore,

$\displaystyle(T_{n}^{k}(\omega))_{\lambda}$
$\displaystyle=\\{\lambda(u_{1})u_{2}\ |\ u\in T_{n}^{k}(\omega)\\}$
$\displaystyle=\langle\lambda(u_{1})u_{2}\ |\ u\in\mathcal{B}\rangle$
$\displaystyle=\langle\lambda((g^{i}x^{j})_{1})(g^{i}x^{j})_{2}\ |\ 0\leq
i\leq(kn-1),0\leq j\leq(n-1)\rangle$
$\displaystyle=\langle\delta_{N}(g^{i})g^{i}x^{j}\ |\ 0\leq i\leq(kn-1),0\leq
j\leq(n-1)\rangle$ $\displaystyle=\langle\delta_{N}(g^{rk+s})g^{rk+s}x^{j}\ |\
0\leq s\leq(k-1),0\leq r,j\leq(n-1)\rangle$
$\displaystyle=\langle\delta_{N}(g^{rk})g^{rk}x^{j}\ |\ 0\leq
r,j\leq(n-1)\rangle$ $\displaystyle=\langle g^{rk}x^{j}\ |\ 0\leq
r,j\leq(n-1)\rangle$ $\displaystyle=\varphi(T_{n}(\omega^{k}))$
$\displaystyle\simeq T_{n}(\omega^{k}),$

where $\varphi$ is the injective homomorphism of Hopf algebras given in Remark
4.14.

We observe that the choice of the parameter $k$ was arbitrary. Thus, the
dimension of $T_{n}^{k}(\omega)$ can be chosen as large as desired. This
situation is the same as for group algebras (Example 4.1) and also in Theorem
4.13.

######  Example 4.15.

Consider the Sweedler’s Hopf algebra $\mathbb{H}_{4}=T_{2}(-1)$. Note that
$T_{2}^{2}(q)=\mathcal{A}^{{}^{\prime\prime\prime}}_{4,q}$ and
$T_{2}^{4}(\tilde{q})=\mathbf{H_{19}}$, where $q$ and $\tilde{q}$ are
primitive $4^{th},8^{th}$ roots of unity, respectively (see Subsection 3.3).
Then, the partial actions $\lambda_{2}=\lambda_{\\{1,g^{2}\\}}$ of
$\mathcal{A}^{{}^{\prime\prime\prime}}_{4,q}$ on $\Bbbk$ and
$\lambda_{4}=\lambda_{\\{1,g^{4}\\}}$ of $\mathbf{H_{19}}$ on $\Bbbk$, are
such that
$(\mathcal{A}^{{}^{\prime\prime\prime}}_{4,q})_{\lambda_{2}}\simeq\mathbb{H}_{4}\simeq(\mathbf{H_{19}})_{\lambda_{4}}$.

###  4.3. Examples of $\lambda$-Hopf algebras

In this subsection we show, schematically, the examples of $\lambda$-Hopf
algebras obtained with the partial actions of each pointed non-semisimple Hopf
algebra $H$ with $dim_{\Bbbk}(H)=8,16$ (see Subsection 3.3).

######  Theorem 4.16.

Let $H$ be one of the following pointed non-semisimple Hopf algebras:

  * •

Dimension 8: $\mathcal{A}_{2},\mathcal{A}^{{}^{\prime}}_{4}$ or
$\mathcal{A}^{{}^{\prime\prime}}_{4}$;

  * •

Dimension 16: $\mathbf{H_{1}}$, $\mathbf{H_{4}},$ $\mathbf{H_{5}},$
$\mathbf{H_{6}},$ $\mathbf{H_{7}},$ $\mathbf{H_{8}},$ $\mathbf{H_{9}},$
$\mathbf{H_{10}},$ $\mathbf{H_{11}},$ $\mathbf{H_{14}},$ $\mathbf{H_{15}},$
$\mathbf{H_{16}},$ $\mathbf{H_{18}}$ or $\mathbf{H_{22}}$.

Then, $H$ has not a partial action $\lambda$, $\lambda\neq\varepsilon$, such
that $H_{\lambda}$ is a Hopf subalgebra of $H$.

###### Proof.

Note that for these Hopf algebras, there exists a $(1,g)$-primitive element
$x$. Almost all the partial actions $\lambda,\lambda\neq\varepsilon,$ of these
Hopf algebras are such that $\lambda(g)=0$. The only exceptions are
$\lambda_{\\{1,g\\}}$ of $\mathbf{H_{i}}$, $i\in\\{14,15\\}$. For these two
specific cases, there exists $y\in\mathbf{H_{i}}$ such that $y$ is a
$(1,h)$-primitive element and $\lambda_{\\{1,g\\}}(h)=0$. Thus, by Corollary
4.9, it follows that $H_{\lambda}$ is not a Hopf algebra of $H$. ∎

In the following diagram, an arrow from $H$ to $B$,
$H\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}B$, means that $\lambda$
is a partial action of $H$ on $\Bbbk$ such that $B\simeq H_{\lambda}$, as Hopf
algebras. At this point, Corollary 4.9 is useful to discard a lot of partial
actions $\lambda$ of $H$ on $\Bbbk$, since the existence of a
$(g,h)$-primitive element such that $\lambda(g)\neq\lambda(h)$ implies that
$H_{\lambda}$ is not a Hopf subalgebra of $H$. To verify whether $H_{\lambda}$
is or not a Hopf subalgebra of $H$, one uses Theorem 4.6 or, which is usually
easier, Proposition 4.11.

|
---|---
|
$\textstyle{\underline{\textrm{dimension}\,\,16}}$$\textstyle{\underline{\textrm{dimension}\,\,8}}$$\textstyle{\mathbf{H_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2}\\}}}$$\textstyle{\mathbf{H_{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2}\\}}}$$\textstyle{\mathbf{H_{13}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g\\}}}$$\textstyle{\mathcal{A}_{2}}$$\textstyle{\mathbf{H_{12}}=\mathcal{A}_{2}\otimes\Bbbk
C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\,\,\,\quad\lambda_{\\{1,g\\}}=\varepsilon_{\mathcal{A}_{2}}\otimes\lambda_{\\{1\\}}^{\Bbbk
C_{2}}}$$\textstyle{\mathbf{H_{28}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g,g^{2},g^{3}\\}}}$$\textstyle{\mathcal{A}_{4}^{\prime\prime}}$$\textstyle{\mathbf{H_{29}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2},gh,g^{3}h\\}}}$

$\textstyle{\underline{\textrm{dimension}\,\,16}}$$\textstyle{\underline{\textrm{dimension}\,\,8}}$$\textstyle{\underline{\textrm{dimension}\,\,4}}$$\textstyle{\mathbf{H_{20}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}}$$\textstyle{\mathbf{H_{21}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}}$$\textstyle{\mathcal{A}_{4}^{\prime}}$$\textstyle{\mathbf{H_{23}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g,g^{2},g^{3}\\}}}$$\textstyle{\mathbf{H_{24}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\lambda_{\\{1,g^{2},gh,g^{3}h\\}}}$$\textstyle{\mathbf{H_{27}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\lambda_{\\{1,g^{2},h,g^{2}h\\}}}$$\scriptstyle{\lambda_{\\{1,g^{2}h\\}}}$$\textstyle{\mathbf{H_{17}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g,a,ga\\}}}$$\scriptstyle{\lambda_{\\{1,g,ab,gab\\}}}$$\scriptstyle{\lambda_{\\{1,g,b,gb\\}}}$$\scriptstyle{\lambda_{\\{1,g\\}}}$$\textstyle{\mathcal{A}_{2,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g\\}}}$$\textstyle{\mathbf{H_{26}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2},h,g^{2}h\\}}}$$\scriptstyle{\lambda_{\\{1,h\\}}}$$\textstyle{\mathbb{H}_{4}}$$\textstyle{\mathbf{H_{25}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2},h,g^{2}h\\}}}$$\scriptstyle{\lambda_{\\{1,g,g^{2},g^{3}\\}}}$$\scriptstyle{\lambda_{\\{1,g^{2}\\}}}$$\scriptstyle{\lambda_{\\{1,g^{2},gh,g^{3}h\\}}}$$\textstyle{\mathcal{A}_{4,q}^{\prime\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2}\\}}}$$\textstyle{\mathbf{H_{19}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\\{1,g^{2},g^{4},g^{6}\\}}}$$\scriptstyle{\lambda_{\\{1,g^{4}\\}}}$

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