# Higher Frobenius-Schur indicators for semisimple Hopf algebras in positive
characteristic

Zhihua Wang  Z. Wang
Department of Mathematics, Taizhou University, Taizhou 225300, China
<EMAIL_ADDRESS>,  Gongxiang Liu  G. Liu
Department of Mathematics, Nanjing University, Nanjing 210093, China
<EMAIL_ADDRESS>and  Libin Li  L. Li
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
<EMAIL_ADDRESS>
###### Abstract.

Let $H$ be a semisimple Hopf algebra over an algebraically closed field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. We show that
the antipode $S$ of $H$ satisfies the equality
$S^{2}(h)=\mathbf{u}h\mathbf{u}^{-1}$, where $h\in H$,
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ and $\Lambda$ is a nonzero integral
of $H$. The formula of $S^{2}$ enables us to define higher Frobenius-Schur
indicators for the Hopf algebra $H$. This generalizes the notions of higher
Frobenius-Schur indicators from the case of characteristic 0 to the case of
characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. These indicators defined here
share some properties with the ones defined over a field of characteristic 0.
Especially, all these indicators are gauge invariants for the tensor category
Rep$(H)$ of finite dimensional representations of $H$.

###### Key words and phrases:

semisimple Hopf algebra, $S^{2}$-formula, Frobenius-Schur indicator, gauge
invariant

###### 2010 Mathematics Subject Classification:

16T05

##  1\. Introduction

Linchenko-Montgomery [9] generalized the classical Frobenius-Schur (FS)
indicators from group-theoretic result to the setting of a semisimple
involutory Hopf algebra $H$. They also defined higher FS indicators
$\nu_{n}(V)$ by using idempotent integral $\Lambda$ of $H$, namely,

(1.1) $\nu_{n}(V)=\chi_{V}(\Lambda_{(1)}\cdots\Lambda_{(n)})\ \text{for}\
n\geq 1,$

where $\chi_{V}$ is the character afforded by finite dimensional
representation $V$ of $H$. The higher FS indicators were later extensively
studied by Kashina-Sommerh$\ddot{a}$user-Zhu for semisimple Hopf algebras over
an algebraically closed field of characteristic zero [6], and by Ng-
Schauenburg for semisimple quasi-Hopf algebras over the field of complex
numbers [11]. The notions of higher FS indicators have been generalized to
objects of a pivotal category [12, 13].

However, the notions of higher FS indicators for semisimple Hopf algebras over
a field of positive characteristic seem not to be considered (except for those
semisimple involutory Hopf algebras). In this paper, we consider higher FS
indicators for a finite dimensional semisimple Hopf algebra $H$ over an
algebraically closed field $\mathbbm{k}$ of characteristic
$p>\dim_{\mathbbm{k}}(H)^{1/2}$. We need to point out that the Hopf algebra
$H$ here is not necessarily involutory unless the characteristic $p$ is larger
than a certain number (see [16, 3]).

For the antipode $S$ of $H$, we first obtain a formula for $S^{2}$ as follows:

$S^{2}(h)=\mathbf{u}h\mathbf{u}^{-1},$

where $h\in H$, $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ and $\Lambda$ is a
nonzero integral of $H$. According to the formula of $S^{2}$, we have an
isomorphism of $H$-modules

$j_{\mathbf{u},V}:V\rightarrow V^{**},\ \
j_{\mathbf{u},V}(v)(f)=f(\mathbf{u}\cdot v)\ \text{for}\ v\in V,f\in V^{*},$

which is functorial in $V$. As the element
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ is not necessarily a group-like
element, the functorial isomorphism $j_{\mathbf{u}}:id\rightarrow(-)^{**}$ is
not necessarily a tensor isomorphism. In other words, the category Rep$(H)$ of
finite dimensional representations of $H$ is not necessarily pivotal with
respect to the structure $j_{\mathbf{u}}$. Even though, using the functorial
isomorphism $j_{\mathbf{u}}$ we may still define the $n$-th FS indicator
$\nu_{n}(V)$ of $V$ to be the trace of a certain $\mathbbm{k}$-linear operator
as Ng-Schauenburg did in [12]. It is similar to the case of characteristic 0
that the $n$-th FS indicator $\nu_{n}(V)$ defined here can also be entirely
described in terms of the integral $\Lambda$ of $H$ and the character
$\chi_{V}$ of $H$-module $V$:

(1.2) $\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})\
\text{for}\ n\geq 1.$

Moreover, the formula (1.2) does not depend on the choice of the nonzero
integral $\Lambda$ and it recovers the original formula (1.1) when the
characteristic of $\mathbbm{k}$ is zero and $\Lambda$ is idempotent.

Note that the formula (1.2) can be written as
$\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}P_{n}(\Lambda))$ for $n\geq 1$, where
$P_{n}$ is the $n$-th Sweedler power map of $H$. Clearly, the $n$-th Sweedler
power map $P_{n}$ is valid for all $n\in\mathbb{Z}$, this motivates us to
extend the $n$-th FS indicator from $n\geq 1$ to $n\in\mathbb{Z}$. That is, by
definition, $\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}P_{n}(\Lambda))$ for all
$n\in\mathbb{Z}$. We find that the higher FS indicators defined over a field
of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and the ones defined over a
field of characteristic 0 share some common properties. For instance, it is
similar to the case of characteristic 0 (see [5, 6]) that by replacing $V$
with the regular representation $H$, we reconstruct the $n$-th indicator of
$H$, a notion defined by the trace of the map $S\circ P_{n-1}$. Also, it is
similar to characteristic 0 case that $V$ and its dual $V^{*}$ have the same
higher FS indicators. Especially, similar to the case of characteristic 0 that
the $n$-th FS indicator $\nu_{n}(V)$ defined here is an invariant of the
tensor category Rep$(H)$ for any $n\in\mathbb{Z}$ and any finite dimensional
representation $V$ of $H$.

The paper is organized as follows: In Section 2, we present some basic results
on semisimple Hopf algebras. In Section 3, we deduce the formula of $S^{2}$ by
comparing two different forms of the character $\chi_{H}$ of the regular
representation $H$. We investigate some properties of the element
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ and show that the integral
$\Lambda$ of $H$ is cocommutative if and only if $S^{2}=id$. In Section 4, we
generalize the notions of higher FS indicators from characteristic 0 case to
characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ case and find that the
indicators defined here share some common properties with the ones defined
over a field of characteristic 0. In Section 5, we show that the $n$-th FS
indicator $\nu_{n}(V)$ is a gauge invariant for any integer $n$ and any finite
dimensional representation $V$ of $H$.

##  2\. Preliminaries

Throughout this paper, $H$ is a finite dimensional semisimple Hopf algebra
over an algebraically closed field $\mathbbm{k}$ of characteristic
$p>\dim_{\mathbbm{k}}(H)^{1/2}$. We need to stress that all results presented
here are also valid for the case of characteristic 0, although we only deal
with the case of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.

As a Hopf algebra, $H$ has a counit $\varepsilon$, antipode $S$,
multiplication $m$ and comultiplication $\Delta$. The comultiplication
$\Delta(a)$ will be written as $\Delta(a)=a_{(1)}\otimes a_{(2)}$ for $a\in
H$, where we omit the summation sign. We denote by $\Lambda$ and $\lambda$ the
left and right integrals of $H$ and $H^{*}$ respectively so that
$\lambda(\Lambda)=1$. Since the semisimple Hopf algebra $H$ is unimodular, the
left and right integrals of $H$ are the same. We refer to [10] for basic
theory of Hopf algebras.

If $V$ is a finite dimensional $H$-module, then $V$ is also called a
representation of $H$ via the algebra homomorphism
$\rho_{V}:H\rightarrow\text{End}_{\mathbbm{k}}(V)$ given by
$\rho_{V}(h)(v)=h\cdot v$ for $h\in H$ and $v\in V$. We will make no
distinction between the two notions. The character of $V$ is the map
$\chi_{V}:H\rightarrow\mathbbm{k}$ given by
$\chi_{V}(h)=\text{tr}(\rho_{V}(h))$ for $h\in H$. The $\mathbbm{k}$-linear
dual space $V^{*}$ is also an $H$-module via $(h\cdot f)(v):=f(S(h)\cdot v)$
for $h\in H$, $f\in V^{*}$ and $v\in V$. In particular, the dual module
$V^{*}$ has the character $\chi_{V^{*}}=\chi_{V}\circ S$. The category
Rep$(H)$ of finite dimensional representations of $H$ is a semisimple tensor
category, where the monoidal structure stems from the comultiplication
$\Delta$.

Recall that the dual Hopf algebra $H^{*}$ has an $H$-bimodule structure given
by

$(a\rightharpoonup f)(b)=f(ba),\ (f\leftharpoonup a)(b)=f(ab)\ \text{for}\
a,b\in H,\ f\in H^{*}.$

Moreover, $(H^{*},\leftharpoonup)$ and $(\rightharpoonup,H^{*})$ are free
$H$-modules generated by $\lambda$, i.e., $H^{*}=\lambda\leftharpoonup H$ and
$H^{*}=H\rightharpoonup\lambda$ (see [15, Corollary 2(b)]). This provides an
associative and non-degenerate bilinear form $H\times H\rightarrow\mathbbm{k}$
by $a\times b\mapsto\lambda(ab)$ for $a,b\in H$. Moreover, the pair
$(H,\lambda)$ is a Frobenius algebra with the Frobenius homomorphism $\lambda$
satisfying the equality (see [15, Eq.(1)]):

(2.1)
$a=\lambda(a\Lambda_{(1)})S(\Lambda_{(2)})=\lambda(S(\Lambda_{(2)})a)\Lambda_{(1)}\
\text{for}\ a\in H.$

The pair $\Lambda_{(1)}\otimes S(\Lambda_{(2)})$ satisfying (2.1) is called
the dual basis of $H$ with respect to the Frobenius homomorphism $\lambda$.

Since the right integral $\lambda$ of $H^{*}$ satisfies
$\lambda(ab)=\lambda(S^{2}(b)a)$ for all $a,b\in H$ (see [15, Theorem 3(a)]),
the Hopf algebra $H$ is a symmetric algebra with a symmetric bilinear form
given by

$H\times H\rightarrow\mathbbm{k},\ a\times
b\mapsto\lambda(uab)=(\lambda\leftharpoonup
u)(ab)=(u\rightharpoonup\lambda)(ab),$

where $u$ is a unit of $H$ satisfying $S^{2}(h)=uhu^{-1}$ for all $h\in H$ and
the Frobenius homomorphism $\lambda\leftharpoonup u=u\rightharpoonup\lambda$
holds because $\lambda(au)=\lambda(S^{2}(u)a)=\lambda(ua)$ for all $a\in H$.
Using (2.1) we may see that the pair $\Lambda_{(1)}\otimes
u^{-1}S(\Lambda_{(2)})$ is a dual basis of $H$ with respect to
$\lambda\leftharpoonup u\ (=u\rightharpoonup\lambda)$ (see also [2, Lemma
1.4(2)]). The symmetry of the Frobenius homomorphism $\lambda\leftharpoonup u\
(=u\rightharpoonup\lambda)$ means that

(2.2) $\Lambda_{(1)}\otimes
u^{-1}S(\Lambda_{(2)})=u^{-1}S(\Lambda_{(2)})\otimes\Lambda_{(1)}.$

By Wedderburn’s theorem, the semisimple Hopf algebra $H$ is isomorphic to a
direct sum of full matrix algebras over $\mathbbm{k}$, namely,

$H\cong\bigoplus_{i\in I}M_{d_{i}}(\mathbbm{k}).$

Let $e_{i}$ be the idempotent of $H$ satisfying that $He_{i}\cong
M_{d_{i}}(\mathbbm{k})$. Then $\\{e_{i}\\}_{i\in I}$ forms a complete set of
central primitive idempotents of $H$. Let $V_{i}$ be a simple left module
(unique up to isomorphism) over the matrix algebra $M_{d_{i}}(\mathbbm{k})$.
Then $\dim_{\mathbbm{k}}(V_{i})=d_{i}$ and $\\{V_{i}\\}_{i\in I}$ forms a
complete set of simple left $H$-modules up to isomorphism. The left regular
representation $H$ has the decomposition $H\cong\bigoplus_{i\in
I}V_{i}^{\oplus d_{i}}$ as $H$-modules, so the character $\chi_{H}$ of the
left regular representation $H$ is equal to $\sum_{i\in I}d_{i}\chi_{i}$,
where each $\chi_{i}$ is the character of $V_{i}$.

For any simple $H$-module $V_{i}$ and any
$\varphi\in\text{End}_{\mathbbm{k}}(V_{i})$, we use the dual basis
$\Lambda_{(1)}\otimes u^{-1}S(\Lambda_{(2)})$ with respect to the Frobenius
homomorphism $\lambda\leftharpoonup u$ to define the map
$\mathcal{I}(\varphi)\in\text{End}_{\mathbbm{k}}(V_{i})$ by

$\mathcal{I}(\varphi)(v)=\Lambda_{(1)}\varphi(u^{-1}S(\Lambda_{(2)})v)\
\text{for}\ v\in V_{i}.$

Note that $\mathcal{I}(\varphi)$ lies in
$\text{End}_{H}(V_{i})\cong\mathbbm{k}$. There exists a unique element
$c_{i}\in\mathbbm{k}$ such that

(2.3) $\mathcal{I}(\varphi)=c_{i}\text{tr}(\varphi)id_{V_{i}}\ \text{for\
all}\ \varphi\in\text{End}_{k}(V_{i}).$

Such an element $c_{i}$, depending only on the isomorphism class of $V_{i}$,
is called the Schur element associated to $V_{i}$ (see [4, Theorem 7.2.1]).
Since $H$ is semisimple, it follows from [4, Theorem 7.2.6] that the Schur
element $c_{i}\neq 0$ in $\mathbbm{k}$ and the Frobenius homomorphism
$\lambda\leftharpoonup u$ can be written explicitly as follows:

(2.4) $\lambda\leftharpoonup u=u\rightharpoonup\lambda=\sum_{i\in
I}\frac{1}{c_{i}}\chi_{i}.$

##  3\. A formula for the square of antipodes

In this section, we will provide a formula for $S^{2}$ by virtue of a nonzero
integral $\Lambda$ of $H$. Then we study some properties of the element
$\mathbf{u}:=S(\Lambda_{(2)})\Lambda_{(1)}$. Especially, we will give a
sufficient and necessary condition for $S^{2}=id$ via the integral $\Lambda$.

Let $u$ be a unit of $H$ satisfying $S^{2}(a)=uau^{-1}$ for all $a\in H$. We
fix a left integral $\Lambda$ of $H$ and a right integral $\lambda$ of $H^{*}$
such that $\lambda(\Lambda)=1$. We denote $\\{V_{i}\\}_{i\in I}$ the set of
all simple left $H$-modules up to isomorphism. For each $V_{i}$ we denote
$c_{i}$ the Schur element of $V_{i}$ associated to the dual basis
$\Lambda_{(1)}\otimes u^{-1}S(\Lambda_{(2)})$ of $H$ with respect to the
Frobenius homomorphism $\lambda\leftharpoonup u$. We denote $\\{e_{i}\\}_{i\in
I}$ the set of all central primitive idempotents of $H$. We first establish a
relationship between the elements $u$ and
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$.

######  Proposition 3.1.

With the notions above, we have $\mathbf{u}=u\sum_{i\in
I}\dim_{\mathbbm{k}}(V_{i})c_{i}e_{i}$, which is a unit of $H$.

###### Proof.

Note that each central primitive idempotent $e_{i}$ acts as the identity on
$V_{i}$ and annihilates $V_{j}$ for $j\neq i$. It follows that
$\chi_{j}(e_{i})=\dim_{\mathbbm{k}}(V_{i})$ if $i=j$ and 0 otherwise. By (2.4)
we have

$\chi_{i}(a)=\chi_{i}(ae_{i})=\sum_{j\in
I}\frac{1}{c_{j}}\chi_{j}(c_{i}ae_{i})=(u\rightharpoonup\lambda)(c_{i}ae_{i})=(uc_{i}e_{i}\rightharpoonup\lambda)(a).$

Thus, $\chi_{i}=uc_{i}e_{i}\rightharpoonup\lambda$ and hence

(3.1) $\chi_{H}=\sum_{i\in I}\dim_{\mathbbm{k}}(V_{i})\chi_{i}=u\sum_{i\in
I}\dim_{\mathbbm{k}}(V_{i})c_{i}e_{i}\rightharpoonup\lambda.$

For any map $\varphi\in\text{End}_{\mathbbm{k}}(H)$, the trace of $\varphi$ is
$\text{tr}(\varphi)=\lambda(\varphi(S(\Lambda_{(2)}))\Lambda_{(1)})$ (see [15,
Theorem 2]). Taking into account that $\varphi=L_{a}$, where $L_{a}$ is the
left multiplication operator of $H$ by $a$, we have

$\chi_{H}(a)=\text{tr}(L_{a})=\lambda(aS(\Lambda_{(2)})\Lambda_{(1)})=(S(\Lambda_{(2)})\Lambda_{(1)}\rightharpoonup\lambda)(a).$

This implies that
$\chi_{H}=S(\Lambda_{(2)})\Lambda_{(1)}\rightharpoonup\lambda.$ Comparing it
with (3.1) and using the non-degeneracy of the Frobenius homomorphism
$\lambda$, we have

$S(\Lambda_{(2)})\Lambda_{(1)}=u\sum_{i\in
I}\dim_{\mathbbm{k}}(V_{i})c_{i}e_{i}.$

Since $p>\dim_{\mathbbm{k}}(H)^{1/2}$, it follows that
$p^{2}>\dim_{\mathbbm{k}}(H)=\sum_{i\in
I}\dim_{\mathbbm{k}}(V_{i})^{2}\geq\dim_{\mathbbm{k}}(V_{i})^{2}$. Hence
$p>\dim_{\mathbbm{k}}(V_{i})$ and $\dim_{\mathbbm{k}}(V_{i})\neq 0$ in
$\mathbbm{k}$ for any $i\in I$. Thus, the element $u$ is the same as
$S(\Lambda_{(2)})\Lambda_{(1)}$ up to a central unit $\sum_{i\in
I}\dim_{\mathbbm{k}}(V_{i})c_{i}e_{i}$. ∎

######  Remark 3.2.

Proposition 3.1 also holds if the field $\mathbbm{k}$ has characteristic 0. In
this case, $S^{2}=id$ (see [7] or [8]) implying that
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}=S(\Lambda_{(2)})S^{2}(\Lambda_{(1)})=S(S(\Lambda_{(1)})\Lambda_{(2)})=\varepsilon(\Lambda)$.

Proposition 3.1 gives a formula for $S^{2}$, namely,

$S^{2}(a)=\mathbf{u}a\mathbf{u}^{-1}\ \text{for}\ a\in H,$

where $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}.$ In the sequel, we will
replace $u$ with $\mathbf{u}$. In this case, the equality (2.2) turns out to
be

(3.2)
$\Lambda_{(1)}\otimes\mathbf{u}^{-1}S(\Lambda_{(2)})=\mathbf{u}^{-1}S(\Lambda_{(2)})\otimes\Lambda_{(1)},$

which is the dual basis of $H$ with respect to the Frobenius homomorphism
$\lambda\leftharpoonup\mathbf{u}$. The Schur element associated to the simple
$H$-module $V_{i}$ under the new dual basis
$\Lambda_{(1)}\otimes\mathbf{u}^{-1}S(\Lambda_{(2)})$ with respect to the
Frobenius homomorphism $\lambda\leftharpoonup\mathbf{u}$ is
$\frac{1}{\dim_{\mathbbm{k}}(V_{i})}$. Therefore, the equality (2.4) turns out
to be

(3.3)
$\lambda\leftharpoonup\mathbf{u}=\mathbf{u}\rightharpoonup\lambda=\sum_{i\in
I}\dim_{\mathbbm{k}}(V_{i})\chi_{i}=\chi_{H}.$

By applying [2, Theorem 1.5] and (3.2), we obtain the expression of each
central primitive idempotent $e_{i}$ of $H$ as follows:

(3.4)
$e_{i}=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\Lambda_{(1)})\mathbf{u}^{-1}S(\Lambda_{(2)})=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\mathbf{u}^{-1}S(\Lambda_{(2)}))\Lambda_{(1)}.$

Let $g\in G(H)$ and $\alpha\in\text{Alg}(H,k)$ be the modular elements of $H$
and $H^{*}$ respectively. Recall that the Radford’s formula of $S^{4}$ has the
form (see [14, Proposition 6]):

$S^{4}(a)=\alpha^{-1}\rightharpoonup(gag^{-1})\leftharpoonup\alpha.$

Since $H$ is unimodular, i.e., $\alpha=\varepsilon$, the Radford’s formula of
$S^{4}$ now becomes

$S^{4}(a)=gag^{-1}.$

The distinguished group-like element $g$ and the integral $\Lambda$ of $H$
satisfy the following useful equality (see [15, Theorem 3(d)]):

(3.5) $\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes
S^{2}(\Lambda_{(2)})g.$

After these preparations, we give some properties of the element $\mathbf{u}$
as follows:

######  Proposition 3.3.

The element $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$ satisfies the the
following properties:

  1. (1)

$\mathbf{u}=\chi_{H}(\Lambda_{(1)})S(\Lambda_{(2)})$.

  2. (2)

$\Lambda_{(1)}\mathbf{u}^{-1}S(\Lambda_{(2)})=1.$

  3. (3)

$\lambda(e_{i})=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\mathbf{u}^{-1})$.

  4. (4)

$\mathbf{u}S(\mathbf{u})=S(\mathbf{u})\mathbf{u}=\varepsilon(\Lambda)\sum_{i\in
I}\frac{\dim_{\mathbbm{k}}(V_{i})^{2}}{\lambda(e_{i})}e_{i}$.

  5. (5)

$S(\mathbf{u}^{-1})\mathbf{u}=\mathbf{u}S(\mathbf{u}^{-1})$, which is the
distinguished group-like element $g$ of $H$.

###### Proof.

(1) It follows from (3.4) that
$e_{i}\mathbf{u}=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\Lambda_{(1)})S(\Lambda_{(2)})$.
Thus,

$\mathbf{u}=\sum_{i\in I}e_{i}\mathbf{u}=\sum_{i\in
I}\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\Lambda_{(1)})S(\Lambda_{(2)})=\chi_{H}(\Lambda_{(1)})S(\Lambda_{(2)}).$

(2) Since
$\Lambda_{(1)}\otimes\mathbf{u}^{-1}S(\Lambda_{(2)})=\mathbf{u}^{-1}S(\Lambda_{(2)})\otimes\Lambda_{(1)}$
by (3.2), we obtain the desired result by multiplying the tensor factors
together.

(3) Since
$e_{i}=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\Lambda_{(1)})\mathbf{u}^{-1}S(\Lambda_{(2)})$,
it follows that

$e_{i}=\mathbf{u}e_{i}\mathbf{u}^{-1}=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\Lambda_{(1)})S(\Lambda_{(2)})\mathbf{u}^{-1}.$

Hence

$\lambda(e_{i})=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\Lambda_{(1)})\lambda(S(\Lambda_{(2)})\mathbf{u}^{-1})=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(\mathbf{u}^{-1}),$

where the last equality follows from (2.1).

(4) For any $a\in H$, we have
$S^{3}(a)=S(S^{2}(a))=S(\mathbf{u}a\mathbf{u}^{-1})=S(\mathbf{u}^{-1})S(a)S(\mathbf{u})$,
we also have $S^{3}(a)=S^{2}(S(a))=\mathbf{u}S(a)\mathbf{u}^{-1}$. It follows
that $S(\mathbf{u})\mathbf{u}$ is a central unit of $H$. The equality
$\mathbf{u}S(\mathbf{u})=S(\mathbf{u})\mathbf{u}$ holds because
$S(\mathbf{u})=S(S^{2}(\mathbf{u}))=S^{2}(S(\mathbf{u}))=\mathbf{u}S(\mathbf{u})\mathbf{u}^{-1}$.
For the central unit $\mathbf{u}S(\mathbf{u})$, we suppose that
$\mathbf{u}S(\mathbf{u})=\sum_{i\in I}k_{i}e_{i}$, where each scalar
$k_{i}\neq 0$ in $\mathbbm{k}$. Then
$e_{i}\mathbf{u}^{-1}=\frac{1}{k_{i}}e_{i}S(\mathbf{u})$. We have

$\displaystyle\lambda(e_{i})$
$\displaystyle=(\mathbf{u}^{-1}\rightharpoonup\chi_{H})(e_{i})=\chi_{H}(e_{i}\mathbf{u}^{-1})=\frac{1}{k_{i}}\chi_{H}(e_{i}S(\mathbf{u}))$
$\displaystyle=\frac{\dim_{\mathbbm{k}}(V_{i})}{k_{i}}\chi_{i}(e_{i}S(\mathbf{u}))=\frac{\dim_{\mathbbm{k}}(V_{i})}{k_{i}}\chi_{i}(S(\mathbf{u}))$
$\displaystyle=\frac{\dim_{\mathbbm{k}}(V_{i})}{k_{i}}(\chi_{i}\circ
S)(\mathbf{u})=\frac{\dim_{\mathbbm{k}}(V_{i})}{k_{i}}(\chi_{i}\circ
S)(S(\Lambda_{(2)})\Lambda_{(1)})$
$\displaystyle=\frac{\dim_{\mathbbm{k}}(V_{i})}{k_{i}}(\chi_{i}\circ
S)(\Lambda_{(1)}S(\Lambda_{(2)}))=\frac{\dim_{\mathbbm{k}}(V_{i})^{2}\varepsilon(\Lambda)}{k_{i}}\neq
0.$

It follows that
$k_{i}=\frac{\dim_{\mathbbm{k}}(V_{i})^{2}\varepsilon(\Lambda)}{\lambda(e_{i})}$
and $\mathbf{u}S(\mathbf{u})=\sum_{i\in
I}k_{i}e_{i}=\varepsilon(\Lambda)\sum_{i\in
I}\frac{\dim_{\mathbbm{k}}(V_{i})^{2}}{\lambda(e_{i})}e_{i}$.

(5) Note that $\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes
S^{2}(\Lambda_{(2)})g$ by (3.5). Applying $S\otimes id$ to both sides of this
equality and multiplying the tensor factors together, we have
$\mathbf{u}=S(\mathbf{u})g$ or $g=S(\mathbf{u}^{-1})\mathbf{u}$. ∎

As a consequence, we obtain the following result:

######  Corollary 3.4.

For any central primitive idempotent $e_{i}$ of $H$, we have
$\lambda(e_{i})=\lambda(S(e_{i}))$.

###### Proof.

We denote $S(e_{i})=e_{i^{*}}$ for some $i^{*}\in I$, then $V_{i}^{*}\cong
V_{i^{*}}$, or equivalently, $\chi_{i}\circ S=\chi_{i^{*}}$ (see [2, Lemma
1.8]). By Proposition 3.3 (3) we have

$\lambda(S(e_{i}))=\lambda(e_{i^{*}})=\dim_{\mathbbm{k}}(V_{i^{*}})\chi_{i^{*}}(\mathbf{u}^{-1})=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(S(\mathbf{u}^{-1})).$

Since $\mathbf{u}S(\mathbf{u})=\varepsilon(\Lambda)\sum_{i\in
I}\frac{\dim_{\mathbbm{k}}(V_{i})^{2}}{\lambda(e_{i})}e_{i}$, it follows that
$S(\mathbf{u}^{-1})=\mathbf{u}\frac{1}{\varepsilon(\Lambda)}\sum_{i\in
I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}e_{i}$. Thus,

$\displaystyle\lambda(S(e_{i}))$
$\displaystyle=\dim_{\mathbbm{k}}(V_{i})\chi_{i}(S(\mathbf{u}^{-1}))=\frac{\lambda(e_{i})}{\varepsilon(\Lambda)\dim_{\mathbbm{k}}(V_{i})}\chi_{i}(\mathbf{u})$
$\displaystyle=\frac{\lambda(e_{i})}{\varepsilon(\Lambda)\dim_{\mathbbm{k}}(V_{i})}\chi_{i}(\Lambda_{(1)}S(\Lambda_{(2)}))=\lambda(e_{i}).$

We complete the proof. ∎

If the field $\mathbbm{k}$ has characteristic 0, then the antipode $S$ of $H$
satisfies $S^{2}=id$ (see [7] or [8]). This further implies that the integral
$\Lambda$ of $H$ is cocommutative (see [7, Proposition 2(b)]). The following
result shows that $\Lambda$ being cocommutative is equivalent to $S^{2}=id$
when the characteristic of the field $\mathbbm{k}$ is larger then
$\dim_{\mathbbm{k}}(H)^{1/2}$.

######  Proposition 3.5.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. The following
statements are equivalent:

  1. (1)

The nonzero integral $\Lambda$ of $H$ is cocommutative.

  2. (2)

The nonzero integral $\lambda$ of $H^{*}$ is cocommutative.

  3. (3)

$S^{2}=id$.

###### Proof.

It can be seen from [15, Corollary 5] that Part (2) and Part (3) are
equivalent. We next show that Part (1) and Part (3) are equivalent. If
$\Lambda$ is cocommutative, then
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}=S(\Lambda_{(1)})\Lambda_{(2)}=\varepsilon(\Lambda)$.
It follows from $S^{2}(a)=\mathbf{u}a\mathbf{u}^{-1}$ that $S^{2}=id$.
Conversely, if $S^{2}=id$, then
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}=S(\Lambda_{(2)})S^{2}(\Lambda_{(1)})=S(S(\Lambda_{(1)})\Lambda_{(2)})=\varepsilon(\Lambda)$.
By Proposition 3.3, we have $g=S(\mathbf{u}^{-1})\mathbf{u}=1$. Since
$\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes S^{2}(\Lambda_{(2)})g$
by (3.5), it follows that
$\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes\Lambda_{(2)}$. We
complete the proof. ∎

##  4\. Higher FS indicators

If the field $\mathbbm{k}$ has characteristic 0, the $n$-th FS indicators of
finite dimensional representations of semisimple Hopf algebras have been
studied in [6]. In this section, we will generalize these indicators from
characteristic 0 to the case of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$
and describe them via a nonzero integral $\Lambda$ of $H$. We begin with the
following preparations.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ with a nonzero
integral $\Lambda$ and $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$. Applying
$\Delta_{n-1}\otimes id$ to both sides of the equality:
$\Lambda_{(2)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes S^{2}(\Lambda_{(2)})g$
(see (3.5)), we have

$\Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}\otimes\Lambda_{(1)}=\Lambda_{(1)}\otimes\cdots\otimes\Lambda_{(n-1)}\otimes
S^{2}(\Lambda_{(n)})g.$

Since $g=\mathbf{u}S(\mathbf{u}^{-1})$ and
$S^{2}(\Lambda_{(n)})=\mathbf{u}\Lambda_{(n)}\mathbf{u}^{-1}$, the above
equality induces the following equality:

(4.1)
$\Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}\otimes\mathbf{u}^{-1}\Lambda_{(1)}=\Lambda_{(1)}\otimes\cdots\otimes\Lambda_{(n-1)}\otimes\Lambda_{(n)}S(\mathbf{u}^{-1}).$

Note that the category Rep$(H)$ of finite dimensional representations of $H$
is a semisimple tensor category. Let $j_{\mathbf{u}}:id\rightarrow(-)^{**}$ be
a natural isomorphism between the identity functor and the functor of taking
the second dual. It is completely determined by a collection of $H$-module
isomorphisms

$j_{\mathbf{u},V}:V\rightarrow V^{**},\ \
j_{\mathbf{u},V}(v)(f)=f(\mathbf{u}v)\ \text{for}\ v\in V,f\in V^{*}.$

The inverse of $j_{\mathbf{u},V}$ is

$j_{\mathbf{u},V}^{-1}:V^{**}\rightarrow V,\ \ \alpha\mapsto
j_{\mathbf{u},V}^{-1}(\alpha),$

where $j_{\mathbf{u},V}^{-1}(\alpha)\in V$ satisfies the equality
$f(j_{\mathbf{u},V}^{-1}(\alpha))=\alpha(S^{-1}(\mathbf{u}^{-1})f)$ for $f\in
V^{*}$. Since $S^{2}(h)=\mathbf{u}h\mathbf{u}^{-1}$ and $\mathbf{u}$ is not
known to be a group-like element, the natural isomorphism $j_{\mathbf{u}}$ is
not necessarily a tensor isomorphism. Although the representation category
Rep$(H)$ with respect to the structure $j_{\mathbf{u}}$ is not necessarily
pivotal, we may still define higher FS indicators for any finite dimensional
representation of $H$ using the structure $j_{\mathbf{u}}$ of Rep$(H)$.

We denote $V^{\otimes n}$ the $n$-th tensor power of $V$ where $V^{\otimes 0}$
is the trivial $H$-module $\mathbbm{k}$. For any natural number $n\geq 1$, we
define the following $\mathbbm{k}$-linear map

$E_{V}^{n}:\text{Hom}_{H}(\mathbbm{k},V^{\otimes
n})\rightarrow\text{Hom}_{H}(\mathbbm{k},V^{\otimes n}),\ \ f\mapsto
E_{V}^{n}(f),$

where $E_{V}^{n}(f)$ is an $H$-module morphism from $\mathbbm{k}$ to
$V^{\otimes n}$ given by

$E_{V}^{n}(f):\mathbbm{k}\xrightarrow{\text{coev}_{V^{*}}}V^{*}\otimes
V^{**}=V^{*}\otimes\mathbbm{k}\otimes V^{**}\xrightarrow{id\otimes f\otimes
id}V^{*}\otimes V^{\otimes n}\otimes V^{**}$
$\xrightarrow{\text{ev}_{V}\otimes id}V^{\otimes(n-1)}\otimes
V^{**}\xrightarrow{id\otimes j_{\mathbf{u},V}^{-1}}V^{\otimes n}.$

Here the maps $\text{coev}_{V^{*}}$ and $\text{ev}_{V}$ are the usual
coevaluation morphism of $V^{*}$ and evaluation morphism of $V$ respectively.
If we set $f(1)=\sum v_{1}\otimes\cdots\otimes v_{n}\in V^{\otimes n}$, the
above definition of $E_{V}^{n}(f)$ shows that

(4.2) $E_{V}^{n}(f)(1)=\sum v_{2}\otimes\cdots\otimes
v_{n}\otimes\mathbf{u}^{-1}v_{1}.$

Similar to [12], we give the definition of the $n$-th FS indicator of $V$ to
be the trace of the linear operator $E_{V}^{n}$ as follows:

######  Definition 4.1.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. For any
finite dimensional representation $V$ of $H$, the $n$-th FS indicator of $V$
is defined by

$\nu_{n}(V)=\text{tr}(E_{V}^{n})\ \text{for}\ n\geq 1.$

Similar to the characteristic 0 case, the $n$-th FS indicator of $V$ defined
above can also be described by a nonzero integral $\Lambda$ of $H$:

######  Theorem 4.2.

Let $\Lambda$ be a nonzero integral of $H$ and
$\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$. Suppose $\chi_{V}$ is the
character of a finite dimensional representation $V$ of $H$. We have

$\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})\
\text{for}\ n\geq 1.$

###### Proof.

We first show that the equality
$\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$ holds
for an idempotent integral $\Lambda$. Suppose that $\alpha$ is the following
$\mathbbm{k}$-linear map

$\alpha:V^{\otimes n}\rightarrow V^{\otimes n},\ \ v_{1}\otimes
v_{2}\otimes\cdots\otimes v_{n}\mapsto v_{2}\otimes\cdots\otimes v_{n}\otimes
v_{1}$

and
$\delta=\alpha\circ(\mathbf{u}^{-1}\Lambda_{(1)}\otimes\Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}).$
We have

$\displaystyle\delta(v_{1}\otimes v_{2}\otimes\cdots\otimes v_{n})$
$\displaystyle=\alpha(\mathbf{u}^{-1}\Lambda_{(1)}v_{1}\otimes\Lambda_{(2)}v_{2}\otimes\cdots\otimes\Lambda_{(n)}v_{n})$
$\displaystyle=\Lambda_{(2)}v_{2}\otimes\cdots\otimes\Lambda_{(n)}v_{n}\otimes\mathbf{u}^{-1}\Lambda_{(1)}v_{1}$
(4.3)
$\displaystyle=\Lambda_{(1)}v_{2}\otimes\cdots\otimes\Lambda_{(n-1)}v_{n}\otimes\Lambda_{(n)}S(\mathbf{u}^{-1})v_{1}\
\text{by}\ (\ref{equ00})$
$\displaystyle=\Lambda\cdot(v_{2}\otimes\cdots\otimes v_{n}\otimes
S(\mathbf{u}^{-1})v_{1}).$

This shows that $\delta(V^{\otimes n})\subseteq\Lambda\cdot V^{\otimes
n}=(V^{\otimes n})^{H}.$ Note that the map

$\Phi:\textrm{Hom}_{H}(\mathbbm{k},V^{\otimes n})\rightarrow(V^{\otimes
n})^{H},\ \ f\mapsto f(1)$

is an $H$-module isomorphism. We claim that the following diagram is
commutative:

$\textstyle{\textrm{Hom}_{H}(\mathbbm{k},V^{\otimes
n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{E_{V}^{n}}$$\textstyle{\textrm{Hom}_{H}(\mathbbm{k},V^{\otimes
n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{(V^{\otimes
n})^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{(V^{\otimes
n})^{H}.}$

Indeed, for any $f\in\text{Hom}_{H}(\mathbbm{k},V^{\otimes n})$, we suppose
that $f(1)=\sum v_{1}\otimes\cdots\otimes v_{n}\in V^{\otimes n}$. It follows
from $f(1)=f(\Lambda\cdot 1)=\Lambda\cdot f(1)$ that

(4.4) $\sum v_{1}\otimes\cdots\otimes
v_{n}=\sum\Lambda_{(1)}v_{1}\otimes\cdots\otimes\Lambda_{(n)}v_{n}.$

On the one hand, we have

$\displaystyle(\delta\circ\Phi)(f)$ $\displaystyle=\delta(f(1))=\delta(\sum
v_{1}\otimes\cdots\otimes v_{n})$ $\displaystyle=\Lambda\cdot(\sum
v_{2}\otimes\cdots\otimes v_{n}\otimes S(\mathbf{u}^{-1})v_{1})\ \ \text{by}\
(\ref{equ29})$

On the other hand, we have

$\displaystyle(\Phi\circ E_{V}^{n})(f)$ $\displaystyle=E_{V}^{n}(f)(1)=\sum
v_{2}\otimes\cdots\otimes v_{n}\otimes\mathbf{u}^{-1}v_{1}\ \ \ \text{by}\
(\ref{equ09})$
$\displaystyle=\sum\Lambda_{(2)}v_{2}\otimes\cdots\otimes\Lambda_{(n)}v_{n}\otimes\mathbf{u}^{-1}\Lambda_{(1)}v_{1}\
\ \ \text{by}\ (\ref{equ19})$
$\displaystyle=\sum\Lambda_{(1)}v_{2}\otimes\cdots\otimes\Lambda_{(n-1)}v_{n}\otimes\Lambda_{(n)}S(\mathbf{u}^{-1})v_{1}\
\ \ \text{by}\ (\ref{equ00})$ $\displaystyle=\Lambda\cdot(\sum
v_{2}\otimes\cdots\otimes v_{n}\otimes S(\mathbf{u}^{-1})v_{1}).$

We obtain that $\delta\circ\Phi=\Phi\circ E_{V}^{n}$, or equivalently,
$E_{V}^{n}=\Phi^{-1}\circ\delta\circ\Phi$. It follows that

$\displaystyle\nu_{n}(V)$
$\displaystyle=\text{tr}(E_{V}^{n})=\text{tr}_{V^{\otimes n}}(\delta)$
$\displaystyle=\text{tr}_{V^{\otimes
n}}(\alpha\circ(\mathbf{u}^{-1}\Lambda_{(1)}\otimes\Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}))$
$\displaystyle=\text{tr}_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)}),$

where the equality $\text{tr}_{V^{\otimes
n}}(\alpha\circ(\mathbf{u}^{-1}\Lambda_{(1)}\otimes\Lambda_{(2)}\otimes\cdots\otimes\Lambda_{(n)}))=\text{tr}_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$
follows from [6, Lemma 2.3]. We have shown that
$\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$ where
$\Lambda$ is idempotent. Since
$\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)}$ does not depend on the
choice of the nonzero integral $\Lambda$, the equality
$\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)})$ holds
for any nonzero integral $\Lambda$ of $H$. ∎

######  Remark 4.3.

If the field $\mathbbm{k}$ has characteristic 0 and $\Lambda$ is idempotent,
then $\mathbf{u}=\varepsilon(\Lambda)=1$. In this case, the $n$-th FS
indicator of $V$ is $\chi_{V}(\Lambda_{(1)}\cdots\Lambda_{(n)})$, which is the
one defined in [6, Definition 2.3].

In the rest of this section, we will extend the $n$-th FS indicator
$\nu_{n}(V)$ of $V$ from $n\geq 1$ to the case $n\in\mathbb{Z}$. Recall that
the $n$-th Sweedler power map $P_{n}:H\rightarrow H$ is defined by

$P_{n}(a)=\left\\{\begin{array}[]{ll}a_{(1)}\cdots a_{(n)},&n\geq 1;\\\
\varepsilon(a),&n=0;\\\ S(a_{(1)})\cdots
S(a_{(-n)}),&n\leq-1.\end{array}\right.$

From the $n$-th Sweedler power map $P_{n}$ of $H$, we may see that

$\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}P_{n}(\Lambda))\ \text{for}\ n\geq 1.$

However, this expression is well-defined for any integer $n$. Thus, we may
extend this formula from $n\geq 1$ to any integer $n$ stated as follows:

######  Definition 4.4.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. For any
finite dimensional representation $V$ of $H$ and any $n\in\mathbb{Z}$, the
$n$-th FS indicator of $V$ is defined by

$\nu_{n}(V)=\chi_{V}(\mathbf{u}^{-1}P_{n}(\Lambda)),$

where $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}$.

######  Remark 4.5.

  1. (1)

Note that $S(\Lambda)=\Lambda$. The $n$-th FS indicator of $V$ can be written
as

$\nu_{n}(V)=\left\\{\begin{array}[]{ll}\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\cdots\Lambda_{(n)}),&n\geq
1;\\\ \chi_{V}(\mathbf{u}^{-1}\varepsilon(\Lambda)),&n=0;\\\
\chi_{V}(\mathbf{u}^{-1}\Lambda_{(-n)}\cdots\Lambda_{(1)}),&n\leq-1.\end{array}\right.$

  2. (2)

By Proposition 3.3 (4), we have

$\mathbf{u}^{-1}S(\mathbf{u}^{-1})=\sum_{i\in
I}\frac{\lambda(e_{i})}{\varepsilon(\Lambda)\dim_{\mathbbm{k}}(V_{i})^{2}}e_{i}\in
Z(H).$

It follows that

$\displaystyle\nu_{0}(V)$
$\displaystyle=\varepsilon(\Lambda)\chi_{V}(\mathbf{u}^{-1})=\varepsilon(\Lambda)\chi_{V}(\mathbf{u}^{-1}S(\mathbf{u}^{-1})S(\mathbf{u}))$
$\displaystyle=\sum_{i\in
I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}\chi_{V}(e_{i}S(\mathbf{u}))=\sum_{i\in
I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}\chi_{V}(e_{i}S(\Lambda_{(1)})S^{2}(\Lambda_{(2)}))$
$\displaystyle=\sum_{i\in
I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}\chi_{V}(e_{i}S^{2}(\Lambda_{(2)})S(\Lambda_{(1)}))=\varepsilon(\Lambda)\sum_{i\in
I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}\chi_{V}(e_{i}).$

  3. (3)

$\nu_{-1}(V)=\nu_{1}(V)=\chi_{V}(\mathbf{u}^{-1}\Lambda)=\chi_{V}(\frac{\Lambda}{\varepsilon(\Lambda)})$.

  4. (4)

By [17, Proposition 3.1], $\Lambda_{(1)}\Lambda_{(2)}$ and
$\Lambda_{(2)}\Lambda_{(1)}$ are both central elements of $H$, they are
determined by the values that the characters $\chi_{i}$ for all $i\in I$ take
on them. It follows from
$\chi_{i}(\Lambda_{(1)}\Lambda_{(2)})=\chi_{i}(\Lambda_{(2)}\Lambda_{(1)})$
that $\Lambda_{(1)}\Lambda_{(2)}=\Lambda_{(2)}\Lambda_{(1)}$. Therefore,
$\nu_{-2}(V)=\nu_{2}(V)$.

The higher FS indicators of any simple module $V_{i}$ can be described as
follows:

######  Proposition 4.6.

For any $n\in\mathbb{Z}$ and any simple module $V_{i}$ with the character
$\chi_{i}$, we have

$\nu_{n}(V_{i})=\frac{\chi_{i}(P_{n}(\Lambda))\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}.$

###### Proof.

Since $P_{n}(\Lambda)\in Z(H)$ for any $n\in\mathbb{Z}$ (see [17, Proposition
3.1]), it follows that $P_{n}(\Lambda)=\sum_{i\in
I}\frac{\chi_{i}(P_{n}(\Lambda))}{\dim_{\mathbbm{k}}(V_{i})}e_{i}$. The $n$-th
FS indicator of $V_{i}$ is

$\nu_{n}(V_{i})=\chi_{i}(\mathbf{u}^{-1}P_{n}(\Lambda))=\frac{\chi_{i}(P_{n}(\Lambda))}{\dim_{\mathbbm{k}}(V_{i})}\chi_{i}(\mathbf{u}^{-1})=\frac{\chi_{i}(P_{n}(\Lambda))\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}},$

where the last equality follows from Proposition 3.3 (3). ∎

For any semisimple Hopf algebra over a field $\mathbbm{k}$ of characteristic
0, the finite dimensional representation $V$ and its dual $V^{*}$ have the
same $n$-th FS indicators for all $n\geq 1$ (see [6, Section 2.3]). The
following result shows that this result also holds for the $n$-th FS
indicators defined for the Hopf algebra $H$ over the field $\mathbbm{k}$ of
characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.

######  Proposition 4.7.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. Let $V$ be a
finite dimensional representation of $H$ with the dual $V^{*}$. We have
$\nu_{n}(V)=\nu_{n}(V^{*})$ for all $n\in\mathbb{Z}$.

###### Proof.

Since $S(\Lambda)=\Lambda$, we have $S(P_{n}(\Lambda))=P_{n}(\Lambda)$ for any
$n\in\mathbb{Z}$. For the case $n\geq 1$, the $n$-th FS indicator of $V^{*}$
is

$\displaystyle\nu_{n}(V^{*})$
$\displaystyle=(\chi_{V^{*}})(\mathbf{u}^{-1}P_{n}(\Lambda))=(\chi_{V}\circ
S)(\mathbf{u}^{-1}P_{n}(\Lambda))$
$\displaystyle=\chi_{V}(\Lambda_{(1)}\cdots\Lambda_{(n)}S(\mathbf{u}^{-1}))=\chi_{V}(\Lambda_{(2)}\cdots\Lambda_{(n)}\mathbf{u}^{-1}\Lambda_{(1)})\
\ \text{by}\ (\ref{equ00})$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(1)}\Lambda_{(2)}\cdots\Lambda_{(n)})=\nu_{n}(V).$

For the case $n\leq-1$, the $n$-th FS indicator of $V^{*}$ is

$\displaystyle\nu_{n}(V^{*})$
$\displaystyle=(\chi_{V^{*}})(\mathbf{u}^{-1}P_{n}(\Lambda))=(\chi_{V}\circ
S)(\mathbf{u}^{-1}P_{n}(\Lambda))$
$\displaystyle=\chi_{V}(\Lambda_{(-n)}\cdots\Lambda_{(1)}S(\mathbf{u}^{-1}))=\chi_{V}(S(\mathbf{u}^{-1})\Lambda_{(-n)}\cdots\Lambda_{(1)})$
$\displaystyle=\chi_{V}(S(\mathbf{u}^{-1})\mathbf{u}^{-1}\Lambda_{(1)}S(\mathbf{u})\Lambda_{(-n)}\cdots\Lambda_{(2)})\
\ \text{by}\ (\ref{equ00})$
$\displaystyle=\chi_{V}(\Lambda_{(1)}\mathbf{u}^{-1}\Lambda_{(-n)}\cdots\Lambda_{(2)})=\chi_{V}(\mathbf{u}^{-1}\Lambda_{(-n)}\cdots\Lambda_{(1)})$
$\displaystyle=\nu_{n}(V).$

For the case $n=0$, we denote $S(e_{i})=e_{i^{*}}$ for any $i\in I$, then $*$
is a permutation of $I$, $V_{i^{*}}\cong V_{i}^{*}$ and
$\lambda(e_{i^{*}})=\lambda(e_{i})$ by Corollary 3.4. We have

$\displaystyle\nu_{0}(V^{*})$ $\displaystyle=\varepsilon(\Lambda)\sum_{i\in
I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}\chi_{V}(S(e_{i}))\ \
\text{by\ Remark}\ \ref{rem1}(2)$
$\displaystyle=\varepsilon(\Lambda)\sum_{i\in
I}\frac{\lambda(e_{i^{*}})}{\dim_{\mathbbm{k}}(V_{i^{*}})^{2}}\chi_{V}(e_{i^{*}})=\varepsilon(\Lambda)\sum_{i\in
I}\frac{\lambda(e_{i})}{\dim_{\mathbbm{k}}(V_{i})^{2}}\chi_{V}(e_{i})$
$\displaystyle=\nu_{0}(V).$

We complete the proof. ∎

Kashina-Sommerh$\ddot{a}$user-Zhu has shown in [6, Proposition 2.5] that the
$n$-th FS indictor of the regular representation of a semisimple Hopf algebra
over a field of characteristic 0 can be described as $\text{tr}(S\circ
P_{n-1})$ for $n\geq 1.$ The following result shows that this formula also
holds for the $n$-th FS indicators defined for the Hopf algebra $H$ over the
field $\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$.

######  Proposition 4.8.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$. For any
$n\in\mathbb{Z}$, the $n$-th FS indictor of the regular representation of $H$
can be written as $\nu_{n}(H)=\text{tr}(S\circ P_{n-1}),$ where $P_{n-1}$ is
the $(n-1)$-th Sweedler power map of $H$.

###### Proof.

We choose a left integral $\Lambda$ of $H$ and a right integral $\lambda$ of
$H^{*}$ such that $\lambda(\Lambda)=1$. For any $n\in\mathbb{Z}$, by Radford’s
trace formula [15, Theorem 2], we have

$\displaystyle\text{tr}(S\circ P_{n-1})$ $\displaystyle=\text{tr}(P_{n-1}\circ
S)=\lambda(S(\Lambda_{(2)})(P_{n-1}\circ S)(\Lambda_{(1)}))$
$\displaystyle=\lambda(S(\Lambda_{(2)})P_{n-1}(S(\Lambda_{(1)})))=\lambda(\Lambda_{(1)}P_{n-1}(\Lambda_{(2)}))$
$\displaystyle=\lambda(P_{n}(\Lambda))=\chi_{H}(\mathbf{u}^{-1}P_{n}(\Lambda))\
\ \text{by}\ (\ref{equ0077})$ $\displaystyle=\nu_{n}(H).$

We complete the proof. ∎

##  5\. Gauge invariants

In this section, we will show that the $n$-th FS indicator $\nu_{n}(V)$
defined in Section 4 is a gauge invariant of the tensor category Rep$(H)$ for
any $n\in\mathbb{Z}$ and any finite dimensional representation $V$ of the
semisimple Hopf algebra $H$ .

Recall from [1] that a (normalized) twist for semisimple Hopf algebra $H$ is
an invertible element $J\in H\otimes H$ that satisfies $(\varepsilon\otimes
id)(J)=(id\otimes\varepsilon)(J)=1$ and

$(\Delta\otimes id)(J)(J\otimes 1)=(id\otimes\Delta)(J)(1\otimes J).$

We write $J=J^{(1)}\otimes J^{(2)}$ and $J^{-1}=J^{-(1)}\otimes J^{-(2)}$,
where the summation is understood.

Given a twist $J$ for $H$ one can define a new Hopf algebra $H^{J}$ with the
same algebra structure and counit as $H$, for which the comultiplication
$\Delta^{J}$ and antipode $S^{J}$ are given respectively by

$\Delta^{J}(a)=J^{-1}\Delta(a)J,$ $S^{J}(a)=Q^{-1}_{J}S(a)Q_{J},\ \
\text{for}\ a\in H,$

where $Q_{J}=S(J^{(1)})J^{(2)}$, which is an invertible element of $H$ with
the inverse $Q_{J}^{-1}=J^{-(1)}S(J^{-(2)})$. With the notions above, we have
the following result:

######  Proposition 5.1.

Let $H$ be a finite dimensional semisimple Hopf algebra over the field
$\mathbbm{k}$ of characteristic $p>\dim_{\mathbbm{k}}(H)^{1/2}$ and $V$ a
finite dimensional representation of $H$. The $n$-th FS indicator $\nu_{n}(V)$
of $V$ is invariant under twisting for any $n\in\mathbb{Z}$.

###### Proof.

Let $\Lambda$ be a nonzero integral of $H$ and $J$ a normalized twist for $H$.
It follows from [17, Theorem 3.4] that $P^{J}_{n}(\Lambda)=P_{n}(\Lambda)$,
where $P^{J}_{n}$ and $P_{n}$ are the $n$-th Sweedler power maps of $H^{J}$
and $H$ respectively. Moreover, $P_{n}(\Lambda)$ is a central element of $H$
(see [17, Proposition 3.1]). Since
$\Delta^{J}(\Lambda)=Q_{J}^{-1}\Lambda_{(1)}\otimes\Lambda_{(2)}Q_{J},$ it
follows that

(5.1)
$\mathbf{u}^{J}:=S^{J}(\Lambda_{(2)}Q_{J})Q_{J}^{-1}\Lambda_{(1)}=Q_{J}^{-1}S(Q_{J})\mathbf{u},$

where $\mathbf{u}=S(\Lambda_{(2)})\Lambda_{(1)}.$ For $H$-module $V$ with the
character $\chi_{V}$, we denote $V^{J}$ the same as $V$ as
$\mathbbm{k}$-linear space but thought of as an $H^{J}$-module. Then the
character of $V^{J}$ is also $\chi_{V}$. For any $n\in\mathbb{Z}$, we have

$\displaystyle\nu_{n}(V^{J})$
$\displaystyle=\chi_{V}((\mathbf{u}^{J})^{-1}P_{n}^{J}(\Lambda))$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}S(Q_{J}^{-1})Q_{J}P_{n}^{J}(\Lambda))\
\ \text{by}\ (\ref{equ006})$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}S(Q_{J}^{-1})Q_{J}P_{n}(\Lambda))$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}S^{2}(J^{-(2)})S(J^{-(1)})S(J^{(1)})J^{(2)}P_{n}(\Lambda))$
$\displaystyle=\chi_{V}(J^{-(2)}\mathbf{u}^{-1}S(J^{-(1)})S(J^{(1)})J^{(2)}P_{n}(\Lambda))$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}S(J^{-(1)})S(J^{(1)})J^{(2)}P_{n}(\Lambda)J^{-(2)})$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}S(J^{-(1)})S(J^{(1)})J^{(2)}J^{-(2)}P_{n}(\Lambda))$
$\displaystyle=\chi_{V}(\mathbf{u}^{-1}P_{n}(\Lambda))$
$\displaystyle=\nu_{n}(V).$

We complete the proof. ∎

We are now ready to state the main result which says that higher FS indicators
are gauge invariants of the tensor category Rep$(H)$.

######  Theorem 5.2.

Let $H$ and $H^{\prime}$ be two finite dimensional semisimple Hopf algebras
over the field $\mathbbm{k}$ of characteristic
$p>\dim_{\mathbbm{k}}(H)^{1/2}$. If
$\mathcal{F}:\text{Rep}(H)\rightarrow\text{Rep}(H^{\prime})$ is an equivalence
of tensor categories, then $\nu_{n}(V)=\nu_{n}(\mathcal{F}(V))$ for any
$n\in\mathbb{Z}$ and any finite dimensional representation $V$ of $H$.

###### Proof.

Since the $\mathbbm{k}$-linear equivalence
$\mathcal{F}:\text{Rep}(H)\rightarrow\text{Rep}(H^{\prime})$ is a tensor
equivalence, it follows from [11, Theorem 2.2] that $H$ and $H^{\prime}$ are
gauge equivalent in the sense that there exist a twist $J$ of $H$ such that
$H^{\prime}$ is isomorphic to $H^{J}$ as bialgebras. Let
$\sigma:H^{\prime}\rightarrow H^{J}$ be such an isomorphism. Then $\sigma$ is
automatically a Hopf algebra isomorphism. The isomorphism $\sigma$ induces a
$\mathbbm{k}$-linear equivalence
$(-)^{\sigma}:\text{Rep}(H)\rightarrow\text{Rep}(H^{\prime})$ as follows: for
any finite dimensional $H$-module $V$, $V^{\sigma}=V$ as $\mathbbm{k}$-linear
space with the $H^{\prime}$-module action given by
$h^{\prime}v=\sigma(h^{\prime})v$ for $h^{\prime}\in H^{\prime}$, $v\in V$,
and $f^{\sigma}=f$ for any morphism $f$ in Rep$(H)$. Moreover, the equivalence
$\mathcal{F}$ is naturally isomorphic to the $\mathbbm{k}$-linear equivalence
$(-)^{\sigma}$ (see [5, Theorem 1.1]). Therefore,

$\nu_{n}(\mathcal{F}(V))=\nu_{n}(V^{\sigma}).$

Let $\Lambda^{\prime}$ be a nonzero integral of $H^{\prime}$ and $S^{\prime}$
the antipode of $H^{\prime}$. Note that the map $\sigma:H^{\prime}\rightarrow
H^{J}$ is a Hopf algebra isomorphism. It follows that
$\sigma(\Lambda^{\prime})=\Lambda$, which is a nonzero integral of $H^{J}$ and
$\sigma(P^{\prime}_{n}(\Lambda^{\prime}))=P^{J}_{n}(\Lambda)$, where
$P^{\prime}_{n}$ and $P^{J}_{n}$ are the $n$-th Sweedler power maps of
$H^{\prime}$ and $H^{J}$ respectively. In particular,

$\sigma((\mathbf{u}^{\prime})^{-1}P^{\prime}_{n}(\Lambda^{\prime}))=(\mathbf{u}^{J})^{-1}P^{J}_{n}(\Lambda),$

where
$\mathbf{u}^{\prime}=S^{\prime}(\Lambda^{\prime}_{(2)})\Lambda^{\prime}_{(1)}$
and $\mathbf{u}^{J}=S^{J}(\Lambda_{\langle 2\rangle})\Lambda_{\langle
1\rangle}$. We have

$\displaystyle\nu_{n}(V^{\sigma})$
$\displaystyle=\chi_{V^{\sigma}}((\mathbf{u}^{\prime})^{-1}P^{\prime}_{n}(\Lambda^{\prime}))$
$\displaystyle=\chi_{V^{J}}(\sigma((\mathbf{u}^{\prime})^{-1}P^{\prime}_{n}(\Lambda^{\prime})))$
$\displaystyle=\chi_{V^{J}}((\mathbf{u}^{J})^{-1}P^{J}_{n}(\Lambda))$
$\displaystyle=\nu_{n}(V^{J})$ $\displaystyle=\nu_{n}(V),$

where the last equality follows from Proposition 5.1. We conclude that
$\nu_{n}(\mathcal{F}(V))=\nu_{n}(V)$ for any $n\in\mathbb{Z}$ and any finite
dimensional representation $V$ of $H$. ∎

## Acknowledgement

The authors would like to thank Professor Juan Cuadra for pointing out a
mistake in a previous version of the manuscript. The first author was
supported by Qing Lan Project. The second author was supported by National
Natural Science Foundation of China (Grant No. 11722016). The third author was
supported by National Natural Science Foundation of China (Grant No.
11871063).

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