A Class of Non-Hopf Bi-Frobenius Algebras Generated by n Elements

Abstract

Bi-Frobenius algebras are a class of Frobenius algebras and Frobenius coalgebras with some compatible conditions. In this paper, we construct a class of bi-Frobenius algebras generated by n elements on graded algebra $A.$ The comultiplication and counit are defined via a permutation $\pi$ on $A,$ such that A becomes a bi-Frobenius algebra. For any n, these bi-Frobenius algebras are neither Hopf algebras nor S-type bi-Frobenius algebras.

1. Introduction

Finite-dimensional group algebras and finite-dimensional Hopf algebras are Frobenius algebras. As a generalization of finite-dimensional Hopf algebras, the concept of bi-Frobenius algebras was introduced by Doi and Takeuchi in [1]. Roughly speaking, this is a Frobenius algebra as well as a Frobenius coalgebra together with an antipode. Therefore, finite-dimensional Hopf algebras are bi-Frobenius algebras, but bi-Frobenius algebras are not necessarily finite-dimensional Hopf algebras. In [2], Haim showed that a bi-Frobenius algebra is a Hopf algebra if and only if it is a bialgebra. Consequently, constructing examples of bi-Frobenius algebras that are not Hopf algebras provides a meaningful approach to study bi-Frobenius algebras. The theory of bi-Frobenius algebras was further developed in [2,3,4,5,6,7,8,9]. So far, examples of bi-Frobenius algebras can be found in [1,2,4,6,7,8,9]. In [1], Doi and Takeuchi presented a class of bi-Frobenius algebras whose underlying algebras are given by $\mathbb{K}\left[x\right]/\left(x^n\right)$. In [2], Haim constructed a class of bi-Frobenius algebras on group-like algebras. Zhu and Tang presented a class of bi-Frobenius algebras by the extension of bialgebras in [9]. Additionally, by using quiver methods, a class of bi-Frobenius algebras was constructed in [6,7]. Jin and Zhang constructed a class of bi-Frobenius algebras on quantum complete intersections in [4,10]. In [8], a class of bi-Frobenius algebras generated by $2n$ elements was constructed.

In [11], an algebra A is Koszul AS regular if and only if the Koszul dual of A is Koszul and Frobenius. Based on the results in [12,13,14], the purpose of this paper is to construct a class bi-Frobenius algebras generated by n elements, which is a class of Yoneda algebras of Artin–Schelter regular algebras. Moreover, bi-Frobenius algebras in this article have a different Frobenius coalgebra structure from [8].

Let $\mathrm{\Lambda }={\left({\lambda }_{ij}\right)}_{n\times n}$ be a non-degenerate matrix and A be a $\mathbb{Z}$-graded algebra with $\mathbb{K}$-space $A=A_0\oplus A_1\oplus A_2$, where $A_0=\mathbb{K}{1}_A$, $A_1=\mathbb{K}{x}_1\oplus \cdots \oplus \mathbb{K}{x}_n$, $A_2=\mathbb{K}y$. The algebra structure of A is given by $x_i{x}_j={\lambda }_{ij}y,x_{i}y=y{x}_i=y^2=0,$ for $i=1,2,\cdots ,n$. We endow a comultiplication on A by a permutation $\pi$ ( see Lemma 2), thus A is a pre-bi-Frobenius algebra. Then, we provide the necessary and sufficient condition fir A to become a bi-Frobenius algebra (See Theorem 1). Moreover, these bi-Frobenius algebras are not S-type bi-Frobenius algebras (see Remark 2). These bi-Frobenius algebras are different from bi-Frobenius algebras in [6,7,8] (see Remark 3). Furthermore, we show that the Nakayama and co-Nakayama automorphisms of the bi-Frobenius algebra A are the identity, and the antipode $\psi$ satisfies ${\psi }^2=\mathrm{id}$ (see Proposition 2). Ultimately, we establish the necessary conditions for the automorphism of A (see Proposition 3).

Throughout, $\mathbb{K}$ will be a field with $\mathrm{char}\mathbb{K}=0$. All algebras, coalgebras, and tensors are over $\mathbb{K}$. We use Sweedler’s notation for the comultiplication of coalgebras.

2. Preliminaries

In this section, we review the definition and some basic properties of bi-Frobenius algebras.

Assume that A is a finite-dimensional algebra. Then, the dual space $A^{*}={\mathrm{Hom}}_{\mathbb{K}}(A,\mathbb{K})$ has a natural A-A-bimodules structure given by $\left(af\right)\left(b\right)=f\left(ba\right),$ $\left(fa\right)\left(b\right)=f\left(ab\right)$, for any $f\in A^{*}$ and $a,b\in A$. We say that $(A,\varphi )$ with $\varphi \in A^{*}$ is a Frobenius algebra if $A^{*}=A\varphi$ or $A^{*}=\varphi A$. The homomorphism $\varphi$ is called a Frobenius basis. For more about Frobenius algebras, see [1,14,15,16].

Dually, let $(C,\mathrm{\Delta },ϵ)$ be a finite-dimensional coalgebra with comultiplication $\mathrm{\Delta }$ and counit $ϵ$. Then, C is a $C^{*}$-$C^{*}$-bimodule via $fc=\sum c_{1}f\left(c_2\right)$, $cf=\sum f\left(c_1\right)c_2$, for any $f\in C^{*}$ and $c\in C$. A pair $(C,t)$ with $t\in C$ is called a Frobenius coalgebra if $C=t{C}^{*}$ or equivalently $C=C^{*}t$. The element t is called a coFrobenius basis of C. We refer to [1] for the notion of Frobenius coalgebras.

Let V and W be vector spaces and $a\in V\otimes W$. By the length of $a\in V\otimes W$, we mean the smallest number l of all possible expression $a={\displaystyle \sum _{i=1}^l}{v}_i\otimes w_i$. Then, $(C,t)$ is a Frobenius coalgebra if and only if the dimension of C is equal to the length of $\mathrm{\Delta }\left(t\right)$ (see 1.3 in [1]).

Definition 1.

Let A be a finite-dimensional algebra and coalgebra with $\varphi \in A^{*}$ and $t\in A$. We call the triple $(A,\varphi ,t)$ a pre-bi-Frobenius algebra if $(A,\varphi )$ is a Frobenius algebra and $(A,t)$ is a Frobenius coalgebra.

If $(A,\varphi ,t)$ is a pre-bi-Frobenius algebra, then $\varphi$ and t can be chosen, such that $\varphi \left(t\right)=1$. Define a linear map

$$\begin{array}{c}\hfill \psi :A⟶A,\psi \left(a\right)=\sum \varphi \left(t_{1}a\right)t_2,a\in A.\end{array}$$

(1)

Following Doi and Takeuchi, we recall the definition of bi-Frobenius algebras (see Definition 1.1 in [3] or 2.2 in [1]).

Definition 2.

Let $(A,\varphi ,t)$ be a pre-bi-Frobenius algebra. The quadruple $(A,\varphi ,t,\psi )$ is called a bi-Frobenius algebra if the following two conditions hold:

(B1) 

$1_A$ is a group-like element and ϵ is an algebra map.

(B2) 

$\psi :A⟶A$ is an anti-algebra and anti-coalgebra map.

The linear map ψ is called the antipode of bi-Frobenius algebra $(A,\varphi ,t,\psi )$. If in addition, the antipode ψ is the convolution inverse of the identity in ${\mathrm{Hom}}_{\mathbb{K}}(A,A)$, i.e., $\psi \ast id=uϵ=id\ast \psi$, then we say the bi-Frobenius algebra A is S-type.

If A is a bi-Frobenius algebra, then there exists the following linear isommorphisms:

$$\theta :A\cong A^{*},a\mapsto \varphi \leftharpoonup a,\mathrm{and}\kappa :A^{*}\cong A,f\mapsto t\leftharpoonup f.$$

$${\theta }^{\prime }:A\cong A^{*},a\mapsto a\rightharpoonup \varphi ,\mathrm{and}{\kappa }^{\prime }:A^{*}\cong A,f\mapsto f\rightharpoonup t.$$

Then we can obtain

$$\psi =\kappa \circ {\theta }^{\prime },\mathcal{N}={{\theta }^{\prime }}^{-1}\circ \theta ,\mathrm{and}{}^c\mathcal{N}={\kappa }^{\prime }\circ {\kappa }^{-1}.$$

Here, $\mathcal{N}$ is the Nakayama automorphism of $(A,\varphi )$, satisfying $\varphi \left(xy\right)=\varphi \left(y\mathcal{N}\right(x\left)\right)$. Meanwhile, ${}^c\mathcal{N}$ is the coNakayama automorphism of $(A,t)$, satisfying ${\displaystyle \sum _c}\mathcal{N}\left(t_2\right)\otimes t_1=\sum t_1\otimes t_2.$

3. The Construction

Let n be a positive integer and $A=A_0\oplus A_1\oplus A_2=\mathbb{K}{1}_A\oplus (\mathbb{K}{x}_1\oplus \mathbb{K}{x}_2\oplus \cdots \oplus \mathbb{K}{x}_n)\oplus \mathbb{K}y$ a $\mathbb{Z}$-graded vector space. Let $\mathrm{\Lambda }={\left({\lambda }_{ij}\right)}_{n\times n}$ be a non-degenerate matrix and let ${\mathrm{\Lambda }}^T$ denote its transpose.

Define a multiplication on A as follows:

$$\begin{array}{c}\hfill x_i{x}_j={\lambda }_{ij}y,x_{i}y=y{x}_i=y^2=0,\end{array}$$

(2)

for $i,j=1,\cdots ,n$. It is obvious that the multiplication of A is associative. We call $\mathrm{\Lambda }$ the structure matrix of A.

Lemma 1.

The algebra A is a Frobenius algebra.

Proof. 

As $\mathrm{\Lambda }={\left({\lambda }_{ij}\right)}_{n\times n}$ is a non-degenerate matrix, one has a unique non-degenerate associative bilinear form $\chi :A\times A⟶\mathbb{K}$, such that $\chi (x_i,x_j)={\lambda }_{ij}$, $\chi (1,y)=\chi (y,1)=1$ and $\chi (1,x_i)=\varphi (x_i,1)=\varphi (y,x_i)=\varphi (x_i,y)=0$. Let $\varphi$ be $\chi (1,-)=\chi (-,1)\in A^{*}$. Then, $(A,\varphi )$ is a Frobenius algebra with a Frobenius basis $\varphi$. Clearly, $\varphi \left(y\right)=1$ and $\varphi \left(1\right)=\varphi \left(x_i\right)=0$ for $i\in \{1,2,\cdots ,n\}$. □

Let ${\tau }_k=(12\cdots k),$ and $P_{{\tau }_k}=\left(\begin{array}{ccc}& 1& \\ & & 1\\ & & & \ddots \\ & & & & 1\\ 1\end{array}\right)$ be the permutation matrix of ${\tau }_k.$ Generally, let ${\pi }_k=(i_1{i}_2\cdots i_k),$ and $P_{{\pi }_k}$ be the permutation matrix of ${\pi }_k$. Then, $P_{{\tau }_k}+P_{{{\tau }_k}^{-1}}$ is similiar to $P_{{\pi }_k}+P_{{{\pi }_k}^{-1}}.$

Let $\pi \in S_n,$$i_j\in \{1,2,\cdots ,n\}$ and ${\pi }_{k_j}=(i_j\pi \left(i_j\right)\cdots {\pi }^{k_j-1}\left(i_j\right)),$ then

$$\pi ={\pi }_{k_1}{\pi }_{k_2}\cdots {\pi }_{k_s},$$

(3)

for any $1\le s\le n$ and $k_1+k_2+\cdots +k_s=n.$ It is obvious that ${\pi }_{k_1},{\pi }_{k_2},\cdots ,{\pi }_{k_s}$ are disjoint permutations.

In the following, denote $[·]$ as the floor function.

Lemma 2.

The matrix P is similiar to the block matrix

$$\left(\begin{array}{c}{P}_{{\tau }_{k_1}}+P_{{\tau }_{k_1}^{-1}}\\ & P_{{\tau }_{k_2}}+P_{{\tau }_{k_2}^{-1}}\\ & & \ddots \\ & & & P_{{\tau }_{k_s}}+P_{{\tau }_{k_s}^{-1}}\end{array}\right).$$

By applying the algorithm proposed in [17], we compute that when $k_j$ is a multiple of 4, $|P_{{\tau }_{k_j}}+P_{{\tau }_{k_j}^{-1}}|=0,$ where $|·|$ evaluates the determinant. In what follows, we provide a proof of this fact.

Lemma 3.

Let π be a permutation in $S_n$. Then, $\left|P\right|=0$ if and only if there exists some j, such that $k_j=4m$ for any positive integer $m\le \left[{\displaystyle \frac{n}{4}}\right]$.

Proof. 

Let $m\le n$ be a positive integer. Note that P is similiar to the block matrix that contains a $4m$-th block

$$P_{{\tau }_{4m}}+P_{{\tau }_{4m}^{-1}}=\left(\begin{array}{cccccc}0& 1& & & & 1\\ 1& 0& 1& & & \\ & 1& 0& 1& & \\ & & \ddots & \ddots & \ddots & \\ & & & 1& 0& 1\\ 1& & & & 1& 0\end{array}\right),$$

then, the eigenvalues of $P_{{\tau }_{4m}}+P_{{\tau }_{4m}^{-1}}$ are ${\lambda }_j={\omega }_j+{\omega }_j^{4m-1}$, where ${\omega }_j$ is a primitive $4m$-th root of unity with $0\le j\le 4m-1.$ Particularly,

$${\lambda }_m={\omega }_m+{\omega }_m^{4m-1}=e^{{\displaystyle \frac{2\pi im}{4m}}}+e^{{\displaystyle \frac{2\pi i(4m-1)}{4m}}}=e^{{\displaystyle \frac{\pi i}{2}}}+e^{-{\displaystyle \frac{\pi i}{2}}}=0,$$

then $|P_{{\tau }_{4m}}+P_{{\tau }_{4m}^{-1}}|=0.$

On the contrary, if $\left|P\right|=0$, there exists an eigenvalue $\lambda$ of P, such that $\lambda =0$. Then, P is similiar to the block matrix that contains a $4m$-th block. □

In the framework of $\pi$, we assume that $4\nmid k_j$ for any index $j.$

Lemma 4.

Let π be a permutation. Then, $(A,y)$ is a Frobenius coalgebra with comultiplication and counit as follows:

$$\begin{array}{ccc}& & \mathrm{\Delta }\left(1\right)=1\otimes 1,\mathrm{\Delta }\left(x_i\right)=1\otimes x_i+x_i\otimes 1,\hfill \\ & & \mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+\sum _{i=1}^n{x}_i\otimes x_{\pi \left(i\right)}+\sum _{i=1}^n{x}_{\pi \left(i\right)}\otimes x_i,\hfill \\ & & =1\otimes y+y\otimes 1+\sum _{i=1}^n{x}_i\otimes (x_{\pi \left(i\right)}+x_{{\pi }^{-1}\left(i\right)}),\hfill \\ & & ϵ\left(1\right)=1,ϵ\left(x_i\right)=ϵ\left(y\right)=0.\hfill \end{array}$$

Proof. 

First, we prove that $(A,\mathrm{\Delta })$ is a coalgebra. It is obvious that coassociativity and counit axiom hold for 1 and $x_i$. Moreover, we have

$$\begin{array}{ccc}\hfill (\mathrm{\Delta }\otimes 1)\mathrm{\Delta }\left(y\right)& =& 1\otimes 1\otimes y+1\otimes y\otimes 1+y\otimes 1\otimes 1+\sum _{k=1}^n{x}_k\otimes (x_{\pi \left(k\right)}+x_{{\pi }^{-1}\left(k\right)})\otimes 1\hfill \\ & & +\sum _{k=1}^{n}1\otimes x_k\otimes (x_{\pi \left(k\right)}+x_{{\pi }^{-1}\left(k\right)})+\sum _{k=1}^n{x}_k\otimes 1\otimes (x_{\pi \left(k\right)}+x_{{\pi }^{-1}\left(k\right)})\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill (1\otimes \mathrm{\Delta })\mathrm{\Delta }\left(y\right)& =& 1\otimes 1\otimes y+1\otimes y\otimes 1+\sum _{k=1}^{n}1\otimes x_k\otimes (x_{\pi \left(k\right)}+x_{{\pi }^{-1}\left(k\right)})+y\otimes 1\otimes 1\hfill \\ & & +\sum _{k=1}^n{x}_k\otimes 1\otimes (x_{\pi \left(k\right)}+x_{{\pi }^{-1}\left(k\right)})+\sum _{k=1}^n{x}_k\otimes (x_{\pi \left(k\right)}+x_{{\pi }^{-1}\left(k\right)})\otimes 1.\hfill \end{array}$$

Therefore, coassociativity also holds for y. A direct calculation also shows that $(ϵ\otimes id)\mathrm{\Delta }\left(y\right)=y=(id\otimes ϵ)\mathrm{\Delta }\left(y\right)$. Thus, $(A,\mathrm{\Delta },ϵ)$ is a coalgebra.

Note that

$$\left|\begin{array}{cc}0& 1\\ 1& 0\\ & & P\end{array}\right|\ne 0,$$

then $A=A^{*}y.$ □

Note that Frobenius coalgebra $(A,y)$ is different from Frobenius coalgebra in [8] for n is even. For example, $A=\mathbb{K}{1}_A\oplus \mathbb{K}{x}_1\oplus \mathbb{K}{x}_2\oplus \mathbb{K}y$, the comultiplication of element y in [8] is $\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+x_1\otimes x_2+x_2\otimes x_1$, but the comultiplication of y in the previous conclusion is $\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+2x_1\otimes x_2+2x_2\otimes x_1$.

In what follows, we refer to the Frobenius algebra $(A,\varphi )$ and Frobenius coalgebra $(A,y)$ defined above as the triple $(A,\varphi ,y)$. Combining Lemma 1 and Lemma 4, we obtain the following corollary.

Proposition 1.

Let π be a permutation. Then, $(A,\varphi ,y)$ is a pre-bi-Frobenius algebra.

Let $(A,\varphi ,y)$ be a pre-bi-Frobenius algebra. We define a linear map $\psi :A\to A$ as in (1) by

$$\begin{array}{c}\hfill \psi \left(a\right)=\sum \varphi \left(y_{1}a\right)y_2,\forall a\in A,\end{array}$$

where $\mathrm{\Delta }\left(y\right)=\sum y_1\otimes y_2.$ Then, we have

$$\begin{array}{ccc}& & \psi \left(1_A\right)=1_A,\hfill \\ & & \psi \left(x_i\right)=\sum _{k=1}^n({\lambda }_{{\pi }^{-1}\left(k\right)i}+{\lambda }_{\pi \left(k\right)i})x_k,\hfill \\ & & \psi \left(y\right)=y,\hfill \end{array}$$

for $i=1,2,\cdots ,n$.

Now we will give the main theorem of this paper.

Theorem 1.

Let π be a permutation, $P_{\pi }$ be the permutation matrix of $\pi ,$ and P is defined as $P_{\pi }+P_{{\pi }^{-1}}.$ Then, the pre-bi-Frobenius algebra $(A,\varphi ,y)$ is a bi-Frobenius algebra with antipode ψ if and only if Λ and π satisfy Equations (4) and (5):

$$\begin{array}{ccc}& & {\left(P\mathrm{\Lambda }\right)}^2=I_n\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & \mathrm{\Lambda }={\mathrm{\Lambda }}^T\hfill \end{array}$$

(5)

Proof. 

We first prove the “if” part. If $(A,\varphi ,y)$ is a bi-Frobenius algebra with antipode $\psi$, then $\psi$ is an anti-algebra map. This means that $\psi \left(x_i{x}_j\right)=\psi \left(x_j\right)\psi \left(x_i\right).$ Now, we have

$$\begin{array}{c}\hfill \psi \left(x_i{x}_j\right)=\psi \left({\lambda }_{ij}y\right)={\lambda }_{ij}y\end{array}$$

and

$$\begin{array}{ccc}& & \psi \left(x_j\right)\psi \left(x_i\right)\hfill \\ & =& \sum _{k=1}^n({\lambda }_{{\pi }^{-1}\left(k\right)j}+{\lambda }_{\pi \left(k\right)j})x_k\sum _{s=1}^n({\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{\pi \left(s\right)i})x_s\hfill \\ & =& \sum _{k=1}^n\sum _{s=1}^n({\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{\pi \left(s\right)i}+{\lambda }_{\pi \left(k\right)j}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{\pi \left(k\right)j}{\lambda }_{\pi \left(s\right)i})x_k{x}_s\hfill \\ & =& \sum _{k=1}^n\sum _{s=1}^n({\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{\pi \left(s\right)i}+{\lambda }_{\pi \left(k\right)j}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{\pi \left(k\right)j}{\lambda }_{\pi \left(s\right)i}){\lambda }_{ks}y\hfill \\ & =& \sum _{k=1}^n\sum _{s=1}^n({\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{ks}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{ks}{\lambda }_{\pi \left(s\right)i}\hfill \\ & & +{\lambda }_{\pi \left(k\right)j}{\lambda }_{ks}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{\pi \left(k\right)j}{\lambda }_{ks}{\lambda }_{\pi \left(s\right)i})y.\hfill \end{array}$$

Comparing the coefficient of y in $\psi \left(x_i{x}_j\right)$ and $\psi \left(x_j\right)\psi \left(x_i\right)$, we have

$$\begin{array}{ccc}\hfill {\lambda }_{ij}& =& \sum _{k=1}^n\sum _{s=1}^n({\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{ks}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)j}{\lambda }_{ks}{\lambda }_{\pi \left(s\right)i}\hfill \\ & & +{\lambda }_{\pi \left(k\right)j}{\lambda }_{ks}{\lambda }_{{\pi }^{-1}\left(s\right)i}+{\lambda }_{\pi \left(k\right)j}{\lambda }_{ks}{\lambda }_{\pi \left(s\right)i})\hfill \\ & =& \sum _{k=1}^n\sum _{s=1}^n({\lambda }_{{\pi }^{-1}\left(s\right)i}{\lambda }_{ks}{\lambda }_{{\pi }^{-1}\left(k\right)j}+{\lambda }_{\pi \left(s\right)i}{\lambda }_{ks}{\lambda }_{{\pi }^{-1}\left(k\right)j}\hfill \\ & & +{\lambda }_{{\pi }^{-1}\left(s\right)i}{\lambda }_{ks}{\lambda }_{\pi \left(k\right)j}+{\lambda }_{\pi \left(s\right)i}{\lambda }_{ks}{\lambda }_{\pi \left(k\right)j}).\hfill \end{array}$$

This means

$$\begin{array}{ccc}\hfill \mathrm{\Lambda }& =& {\mathrm{\Lambda }}^T{P}_{{\pi }^{-1}}{\mathrm{\Lambda }}^T{P}_{{\pi }^{-1}}\mathrm{\Lambda }+{\mathrm{\Lambda }}^T{P}_{\pi }{\mathrm{\Lambda }}^T{P}_{{\pi }^{-1}}\mathrm{\Lambda }+{\mathrm{\Lambda }}^T{P}_{{\pi }^{-1}}{\mathrm{\Lambda }}^T{P}_{\pi }\mathrm{\Lambda }+{\mathrm{\Lambda }}^T{P}_{\pi }{\mathrm{\Lambda }}^T{P}_{\pi }\mathrm{\Lambda }\hfill \\ & =& {\mathrm{\Lambda }}^T(P_{\pi }+P_{{\pi }^{-1}}){\mathrm{\Lambda }}^T(P_{\pi }+P_{{\pi }^{-1}})\mathrm{\Lambda }\hfill \\ & =& {\mathrm{\Lambda }}^{T}P{\mathrm{\Lambda }}^{T}P\mathrm{\Lambda }.\hfill \end{array}$$

As $\mathrm{\Lambda }$ is non-degenerate, the above equality yields

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}^{T}P{\mathrm{\Lambda }}^{T}P=I_n,\end{array}$$

(6)

where $I_n$ the identity matrix of $n\times n$. Recall that $P_{\pi }^{-1}=P_{{\pi }^{-1}}=P_{\pi }^T$, so ${(P_{\pi }+P_{{\pi }^{-1}})}^T=P_{\pi }+P_{\pi }^{-1}$, i.e., $P^T=P$. Take the transposition of (6),

$$\begin{array}{c}\hfill P\mathrm{\Lambda }P\mathrm{\Lambda }=I_n.\end{array}$$

As a consequence,

$${\left(P\mathrm{\Lambda }\right)}^2=I_n.$$

In addition, $\psi$ is also an anti-coalgebra map, i.e., $(\psi \otimes \psi ){\mathrm{\Delta }}^{cop}=\mathrm{\Delta }\psi$. We have

$$\begin{array}{ccc}\hfill (\psi \otimes \psi ){\mathrm{\Delta }}^{cop}\left(x_i\right)& =& (\psi \otimes \psi )(1\otimes x_i+x_i\otimes 1)\hfill \\ & =& 1\otimes \sum _{k=1}^n({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})x_k+\sum _{k=1}^n({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})x_k\otimes 1\hfill \\ & =& \sum _{k=1}^n({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})(1\otimes x_k+x_k\otimes 1)\hfill \\ & =& \sum _{k=1}^n({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})\mathrm{\Delta }\left(x_k\right)\hfill \\ & =& \mathrm{\Delta }\psi \left(x_i\right).\hfill \end{array}$$

This means that $(\psi \otimes \psi ){\mathrm{\Delta }}^{cop}\left(x_i\right)=\mathrm{\Delta }\psi \left(x_i\right)$ is automatically true for $i=1,2,\cdots n$.

Moreover,

$$\begin{array}{ccc}\hfill (\psi \otimes \psi ){\mathrm{\Delta }}^{cop}\left(y\right)& =& (\psi \otimes \psi )(1\otimes y+y\otimes 1+\sum _{i=1}^n{x}_i\otimes (x_{\pi \left(i\right)}+x_{{\pi }^{-1}\left(i\right)}))\hfill \\ & =& 1\otimes y+y\otimes 1+\sum _{i=1}^n\sum _{k=1}^n({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})x_k\hfill \\ & & \otimes \sum _{s=1}^n({\lambda }_{\pi \left(s\right)\pi \left(i\right)}+{\lambda }_{{\pi }^{-1}\left(s\right)\pi \left(i\right)}+{\lambda }_{\pi \left(s\right){\pi }^{-1}\left(i\right)}+{\lambda }_{{\pi }^{-1}\left(s\right){\pi }^{-1}\left(i\right)})x_s,\hfill \\ \hfill \mathrm{\Delta }\psi \left(y\right)& =& 1\otimes y+y\otimes 1+\sum _{i=1}^n(x_{\pi \left(i\right)}+x_{{\pi }^{-1}\left(i\right)})\otimes x_i.\hfill \end{array}$$

Comparing the coefficient of $x_k\otimes x_s$ in $(\psi \otimes \psi ){\mathrm{\Delta }}^{cop}\left(y\right)$ and $\mathrm{\Delta }\psi \left(y\right)$, we have

$$P\mathrm{\Lambda }P{\mathrm{\Lambda }}^{T}P=P.$$

As A is a Frobenius coalgebra, then

$$P\mathrm{\Lambda }P{\mathrm{\Lambda }}^T=I_n.$$

By Lemma 3 and Equation (4)

$$\mathrm{\Lambda }={\mathrm{\Lambda }}^T.$$

Next, we prove the “only if” part. If $(A,\varphi ,y)$ is a pre-bi-Frobenius algebra, it is easy to show that the linear map $\psi :A⟶A$ is an anti-algebra map if (4) holds. By the proof of “if” part, $\psi$ is also an anti-coalgebra map if (5) holds. Moreover, one observes that 1 is a group-like element and $ϵ$ is an algebra map. Thus, $(A,\varphi ,y,\psi )$ is a bi-Frobenius algebra. □

Remark 1.

The bi-Frobenius algebras $(A,\varphi ,y,\psi )$ are not a Hopf algebra. In fact, we have

$$\begin{array}{c}\hfill \mathrm{\Delta }\left(x_i{x}_j\right)=\mathrm{\Delta }\left({\lambda }_{ij}y\right)={\lambda }_{ij}(1\otimes y+y\otimes 1+\sum _{i=1}^n{x}_i\otimes (x_{\pi \left(i\right)}+x_{{\pi }^{-1}\left(i\right)})),\end{array}$$

but

$$\begin{array}{ccc}\hfill \mathrm{\Delta }\left(x_i\right)\mathrm{\Delta }\left(x_j\right)& =& (1\otimes x_i+x_i\otimes 1)(1\otimes x_j+x_j\otimes 1)\hfill \\ & =& {\lambda }_{ij}(1\otimes y+y\otimes 1)+x_i\otimes x_j+x_j\otimes x_i.\hfill \end{array}$$

Thus, $\mathrm{\Delta }\left(x_i{x}_j\right)\ne \mathrm{\Delta }\left(x_i\right)\mathrm{\Delta }\left(x_j\right)$ for $i=1\cdots n$, which says that the comultiplication Δ is not an algebra map. So the bi-Frobenius algebra $(A,\varphi ,y,\psi )$ is not a Hopf algebra.

Remark 2.

The bi-Frobenius algebra $(A,\varphi ,y,\psi )$ is not S-type as the antipode ψ is not the inverse of identity.

Remark 3.

The bi-Frobenius algebras are different from bi-Frobenius algebras constructed from quivers in [6,7]. As Frobenius coalgebras in this paper are different from Frobenius coalgebras in [8], so bi-Frobenius algebras in this paper are not a generalization of bi-Frobenius algebras in [8] when n is even.

Remark 4.

The bi-Frobenius algebra $(A,\varphi ,y,\psi )$ is commutative.

Next, two corollaries will be further described for Equations (4) and (5).

Corollary 1.

If $\pi =Id$, then $(A,\varphi ,y)$ is a bi-Frobenius algebra if and only if

$$\begin{array}{ccc}& & {\mathrm{\Lambda }}^2={\displaystyle \frac{1}{4}}I,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & \mathrm{\Lambda }={\mathrm{\Lambda }}^T.\hfill \end{array}$$

(8)

Corollary 2.

Let π is a permutation of order 2. Then, $(A,\varphi ,y)$ is a bi-Frobenius algebra if and only if

$$\begin{array}{ccc}& & {\left(P_{\pi }\mathrm{\Lambda }\right)}^2={\displaystyle \frac{1}{4}}I,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & \sum _{i=1}^n{\lambda }_{ji}{\lambda }_{\pi \left(j\right)\pi \left(i\right)}={\displaystyle \frac{1}{4}}.\hfill \end{array}$$

(10)

Example 1.

Let $n=1,$ then $A=\mathbb{K}{1}_A+\mathbb{K}x+\mathbb{K}y$ is a bi-Frobenius algebra with multiplication

$$x^2=y,x^3=y^2=0,$$

and comultiplication

$$\mathrm{\Delta }\left(x\right)=1\otimes x+x\otimes 1,\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+x\otimes x,$$

and the antipode $\psi =id.$

Example 2.

Let $n=2$, then $A=\mathbb{K}{1}_A+(\mathbb{K}{x}_1+\mathbb{K}{x}_2)+\mathbb{K}y$ is a bi-Frobenius algebra with structure matrix $\mathrm{\Lambda }=\left(\begin{array}{cc}{\lambda }_{11}& {\lambda }_{12}\\ {\lambda }_{21}& {\lambda }_{22}\end{array}\right).$ Next, we will discuss the structure matrix Λ when $\pi =Id$ and $\pi =\left(12\right)$.

Case 1: If $\pi =Id$, we have $\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+2x_1\otimes x_1+2x_2\otimes x_2.$ Then, we have

$$\left\{\begin{array}{c}{\lambda }_{11}^2+{\lambda }_{12}^2={\displaystyle \frac{1}{4}},\hfill \\ {\lambda }_{21}^2+{\lambda }_{22}^2={\displaystyle \frac{1}{4}},\hfill \\ {\lambda }_{11}^2+{\lambda }_{12}{\lambda }_{21}={\displaystyle \frac{1}{4}},\hfill \\ {\lambda }_{11}{\lambda }_{21}+{\lambda }_{12}{\lambda }_{22}=0,\hfill \\ {\lambda }_{11}{\lambda }_{12}+{\lambda }_{12}{\lambda }_{22}=0,\hfill \\ {\lambda }_{22}^2+{\lambda }_{12}{\lambda }_{21}={\displaystyle \frac{1}{4}}.\hfill \end{array}\right.$$

(11)

By Equation (11), we have

(1) $\mathrm{\Lambda }=\left(\begin{array}{cc}a& \pm {\displaystyle \frac{1}{2}}\sqrt{1-4a^2}\\ \pm {\displaystyle \frac{1}{2}}\sqrt{1-4a^2}& -a\end{array}\right),$ where $a\in \mathbb{K}.$ The multiplication can be found that

$$x_1^2=ay,x_2^2=-ay,x_1{x}_2=x_2{x}_1=\pm {\displaystyle \frac{1}{2}}\sqrt{1-4a^2}y.$$

(2) $\mathrm{\Lambda }=\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& 0\\ 0& {\displaystyle \frac{1}{2}}\end{array}\right)or\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}& 0\\ 0& -{\displaystyle \frac{1}{2}}\end{array}\right)or\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}& 0\\ 0& {\displaystyle \frac{1}{2}}\end{array}\right)or\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& 0\\ 0& -{\displaystyle \frac{1}{2}}\end{array}\right).$

Especially, when $\mathrm{\Lambda }=\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& 0\\ 0& {\displaystyle \frac{1}{2}}\end{array}\right),$ the multiplication can be found that

$$x_1^2=x_2^2={\displaystyle \frac{1}{2}}y,x_1{x}_2=x_2{x}_1=0.$$

Case 2: If $\pi =\left(12\right)$, we have $\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+2x_1\otimes x_2+2x_2\otimes x_1.$ Then, we have

$$\left\{\begin{array}{c}{\lambda }_{21}^2+{\lambda }_{11}{\lambda }_{22}-{\displaystyle \frac{1}{4}}=0,\hfill \\ {\lambda }_{11}{\lambda }_{12}+{\lambda }_{11}{\lambda }_{21}=0,\hfill \\ {\lambda }_{21}-{\lambda }_{12}=0,\hfill \\ {\lambda }_{12}{\lambda }_{22}+{\lambda }_{21}{\lambda }_{22}=0.\hfill \end{array}\right.$$

(12)

By Equation (12), we have

(1) $\mathrm{\Lambda }=\left(\begin{array}{cc}a& 0\\ 0& {\displaystyle \frac{1}{4a}}\end{array}\right),$ where $a\in \mathbb{K}\setminus \left\{0\right\}.$ The multiplication can be founded that

$$x_1^2=ay,x_2^2={\displaystyle \frac{1}{4a}}y,x_1{x}_2=x_2{x}_1=0.$$

(2) $\mathrm{\Lambda }=\pm {\displaystyle \frac{1}{2}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$ The multiplication can be founded that

$$x_1^2=x_2^2=0,x_1{x}_2=x_2{x}_1=\pm {\displaystyle \frac{1}{2}}y.$$

Example 3.

Let $n=3$, then $A=\mathbb{K}{1}_A+(\mathbb{K}{x}_1+\mathbb{K}{x}_2+\mathbb{K}{x}_3)+\mathbb{K}y$ has the structure matrix $\mathrm{\Lambda }=\left(\begin{array}{ccc}{\lambda }_{11}& {\lambda }_{12}& {\lambda }_{13}\\ {\lambda }_{12}& {\lambda }_{22}& {\lambda }_{23}\\ {\lambda }_{13}& {\lambda }_{23}& {\lambda }_{33}\end{array}\right).$ Let A be a bi-Frobenius algebra, then the permutations are given by $\pi =Id,$ $\pi =\left(12\right),$ $\pi =\left(23\right),$ $\pi =\left(13\right),$ $\pi =\left(123\right),$ $\pi =\left(132\right)$. We now present the structure matrix Λ for the cases $\pi =Id,$$\pi =\left(12\right),$ and $\pi =\left(123\right).$

Case 1: When $\pi =Id,$ we have $\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+2x_1\otimes x_1+2x_2\otimes x_2+2x_3\otimes x_3.$ Then

$$P=\left(\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right),$$

such that

$${\left(PA\right)}^2=4A^2=I_3.$$

It implies that

$$A={\displaystyle \frac{1}{2}}{Q}^T\mathrm{\Omega }Q,$$

where Q is the orthogonal matrix and Ω is the diagonal matrix whose diagonal elements are either 1 or $-1$.

Case 2: When $\pi =\left(12\right),$ we have $\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+2x_1\otimes x_2+2x_2\otimes x_1+2x_3\otimes x_3.$ Then

$$P=\left(\begin{array}{ccc}0& 2& 0\\ 2& 0& 0\\ 0& 0& 2\end{array}\right),$$

and

$$\left\{\begin{array}{c}-4{\lambda }_{12}^2-4{\lambda }_{11}{\lambda }_{22}-4{\lambda }_{13}{\lambda }_{23}+1=0,\hfill \\ -4{\lambda }_{13}^2-8{\lambda }_{11}{\lambda }_{12}=0,\hfill \\ -4{\lambda }_{12}{\lambda }_{13}-4{\lambda }_{11}{\lambda }_{23}-4{\lambda }_{13}{\lambda }_{33}=0,\hfill \\ -4{\lambda }_{23}^2-8{\lambda }_{12}{\lambda }_{22}=0,\hfill \\ -4{\lambda }_{12}^2-4{\lambda }_{11}{\lambda }_{22}-4{\lambda }_{13}{\lambda }_{23}+1=0,\hfill \\ -4{\lambda }_{12}{\lambda }_{23}-4{\lambda }_{13}{\lambda }_{22}-4{\lambda }_{23}{\lambda }_{33}=0,\hfill \\ -4{\lambda }_{12}{\lambda }_{23}-4{\lambda }_{13}{\lambda }_{22}-4{\lambda }_{23}{\lambda }_{33}=0,\hfill \\ -4{\lambda }_{12}{\lambda }_{13}-4{\lambda }_{11}{\lambda }_{23}-4{\lambda }_{13}{\lambda }_{33}=0,\hfill \\ -4{\lambda }_{33}^2-8{\lambda }_{13}{\lambda }_{23}+1=0.\hfill \end{array}\right.$$

Let $q=\sqrt{1-8{\lambda }_{13}{\lambda }_{23}},$ then there are 16 solutions, as listed in

Table 1. Matrix Coefficients ${\mathit{\lambda }}_{\mathit{ij}}$ under Permutation $\mathit{\pi }=\left(\mathit{12}\right)$.

	${\mathit{\lambda }}_{11}$	${\mathit{\lambda }}_{12}$	${\mathit{\lambda }}_{13}$	${\mathit{\lambda }}_{22}$	${\mathit{\lambda }}_{23}$	${\mathit{\lambda }}_{33}$
1	${\displaystyle \frac{2{\lambda }_{13}^2}{1+q}}$	${\displaystyle \frac{1}{4}}(-1-q)$	*	${\displaystyle \frac{{\lambda }_{13}{\lambda }_{23}-{\lambda }_{13}{\lambda }_{23}q}{4{\lambda }_{13}^2}}$	*	${\displaystyle \frac{4{\lambda }_{13}{\lambda }_{23}(q^2+3q)}{24{\lambda }_{13}{\lambda }_{23}+q-q^3}}$
2	${\displaystyle \frac{2{\lambda }_{13}^2}{-1+q}}$	${\displaystyle \frac{1}{4}}(1-q)$	*	${\displaystyle \frac{-{\lambda }_{13}{\lambda }_{23}-{\lambda }_{13}{\lambda }_{23}q}{4{\lambda }_{13}^2}}$	*	${\displaystyle \frac{-4{\lambda }_{13}{\lambda }_{23}(q^3-3q)}{24{\lambda }_{13}{\lambda }_{23}-q+q^3}}$
3	${\displaystyle \frac{2{\lambda }_{13}^2}{1-q}}$	${\displaystyle \frac{1}{4}}(-1+q)$	*	${\displaystyle \frac{{\lambda }_{13}{\lambda }_{23}+{\lambda }_{13}{\lambda }_{23}q}{4{\lambda }_{13}^2}}$	*	${\displaystyle \frac{4{\lambda }_{13}{\lambda }_{23}(q^2-3q)}{24{\lambda }_{13}{\lambda }_{23}-q+q^3}}$
4	${\displaystyle \frac{2{\lambda }_{13}^2}{-1-q}}$	${\displaystyle \frac{1}{4}}(1+q)$	*	${\displaystyle \frac{-{\lambda }_{13}{\lambda }_{23}+{\lambda }_{13}{\lambda }_{23}q}{4{\lambda }_{13}^2}}$	*	${\displaystyle \frac{-4{\lambda }_{13}{\lambda }_{23}(q^2+3q)}{24{\lambda }_{13}{\lambda }_{23}+q-q^3}}$
5	0	$-{\displaystyle \frac{1}{2}}$	0	${\lambda }_{23}^2$	*	${\displaystyle \frac{1}{2}}$
6	0	${\displaystyle \frac{1}{2}}$	0	$-{\lambda }_{23}^2$	*	$-{\displaystyle \frac{1}{2}}$
7	*	0	0	${\displaystyle \frac{1}{4{\lambda }_{11}}}$	0	$-{\displaystyle \frac{1}{2}}$
8	*	0	0	${\displaystyle \frac{1}{4{\lambda }_{11}}}$	0	${\displaystyle \frac{1}{2}}$
9	${\lambda }_{13}^2$	$-{\displaystyle \frac{1}{2}}$	*	0	0	${\displaystyle \frac{1}{2}}$
10	$-{\lambda }_{13}^2$	${\displaystyle \frac{1}{2}}$	*	0	0	$-{\displaystyle \frac{1}{2}}$
11	${\lambda }_{13}^2$	$-{\displaystyle \frac{1}{2}}$	*	${\displaystyle \frac{1}{{\lambda }_{13}^2}}$	$-{\displaystyle \frac{1}{{\lambda }_{13}}}$	${\displaystyle \frac{3}{2}}$
12	${\lambda }_{13}^2$	${\displaystyle \frac{1}{2}}$	*	$-{\displaystyle \frac{1}{{\lambda }_{13}^2}}$	$-{\displaystyle \frac{1}{{\lambda }_{13}}}$	$-{\displaystyle \frac{3}{2}}$
13	0	$-{\displaystyle \frac{1}{2}}$	0	0	0	$-{\displaystyle \frac{1}{2}}$
14	0	$-{\displaystyle \frac{1}{2}}$	0	0	0	${\displaystyle \frac{1}{2}}$
15	0	${\displaystyle \frac{1}{2}}$	0	0	0	$-{\displaystyle \frac{1}{2}}$
16	0	${\displaystyle \frac{1}{2}}$	0	0	0	${\displaystyle \frac{1}{2}}$

Here, the symbol ∗ is used to indicate free variables.

.

Case 3: $\pi =\left(123\right),$ we have $\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+x_1\otimes (x_2+x_3)+x_2\otimes (x_3+x_1)+x_3\otimes (x_1+x_2).$ Then,

$$P=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right),$$

and

$$\left\{\begin{array}{c}1-({\lambda }_{11}+{\lambda }_{13})({\lambda }_{22}+{\lambda }_{23})-({\lambda }_{11}+{\lambda }_{12})({\lambda }_{23}+{\lambda }_{33})-{({\lambda }_{12}+{\lambda }_{13})}^2=0,\hfill \\ -({\lambda }_{11}+{\lambda }_{13})({\lambda }_{12}+{\lambda }_{13})-({\lambda }_{11}+{\lambda }_{13})({\lambda }_{12}+{\lambda }_{23})-({\lambda }_{11}+{\lambda }_{12})({\lambda }_{13}+{\lambda }_{33})=0,\hfill \\ -({\lambda }_{11}+{\lambda }_{12})({\lambda }_{12}+{\lambda }_{13})-({\lambda }_{11}+{\lambda }_{13})({\lambda }_{12}+{\lambda }_{22})-({\lambda }_{11}+{\lambda }_{12})({\lambda }_{13}+{\lambda }_{23})=0,\hfill \\ -({\lambda }_{12}+{\lambda }_{13})({\lambda }_{22}+{\lambda }_{23})-({\lambda }_{12}+{\lambda }_{23})({\lambda }_{22}+{\lambda }_{23})-({\lambda }_{12}+{\lambda }_{22})({\lambda }_{23}+{\lambda }_{33})=0,\hfill \\ 1-({\lambda }_{11}+{\lambda }_{13})({\lambda }_{22}+{\lambda }_{23})-({\lambda }_{12}+{\lambda }_{22})({\lambda }_{13}+{\lambda }_{33})-{({\lambda }_{12}+{\lambda }_{23})}^2=0,\hfill \\ -({\lambda }_{11}+{\lambda }_{12})({\lambda }_{22}+{\lambda }_{23})-({\lambda }_{12}+{\lambda }_{22})({\lambda }_{12}+{\lambda }_{23})-({\lambda }_{12}+{\lambda }_{22})({\lambda }_{13}+{\lambda }_{23})=0,\hfill \\ -({\lambda }_{12}+{\lambda }_{13})({\lambda }_{23}+{\lambda }_{33})-({\lambda }_{22}+{\lambda }_{23})({\lambda }_{13}+{\lambda }_{33})-({\lambda }_{13}+{\lambda }_{23})({\lambda }_{23}+{\lambda }_{33})=0,\hfill \\ -({\lambda }_{11}+{\lambda }_{13})({\lambda }_{23}+{\lambda }_{33})-({\lambda }_{12}+{\lambda }_{23})({\lambda }_{13}+{\lambda }_{33})-({\lambda }_{13}+{\lambda }_{23})({\lambda }_{13}+{\lambda }_{33})=0,\hfill \\ 1-({\lambda }_{11}+{\lambda }_{12})({\lambda }_{23}+{\lambda }_{33})-({\lambda }_{12}+{\lambda }_{22})({\lambda }_{13}+{\lambda }_{33})-{({\lambda }_{13}+{\lambda }_{23})}^2=0.\hfill \end{array}\right.$$

Let $p={\lambda }_{12}+{\lambda }_{13},$ then there are 20 solutions, as listed in

Table 2. Matrix Coefficients ${\mathit{\lambda }}_{\mathit{ij}}$ under Permutation $\mathit{\pi }=\left(\mathbf{123}\right)$.

	${\mathit{\lambda }}_{11}$	${\mathit{\lambda }}_{12}$	${\mathit{\lambda }}_{13}$	${\mathit{\lambda }}_{22}$	${\mathit{\lambda }}_{23}$	${\mathit{\lambda }}_{33}$
1	${\displaystyle \frac{-1-4{\lambda }_{12}{\lambda }_{13}}{4p}}$	*	*	${\displaystyle \frac{1-2{\lambda }_{12}(1+2p)}{2(1+2{\lambda }_{13})}}$	${\displaystyle \frac{-1-2p}{2}}$	${\displaystyle \frac{1-2{\lambda }_{13}-4{\lambda }_{12}{\lambda }_{13}-4{\lambda }_{13}^2}{2(1+2{\lambda }_{12})}}$
2	${\displaystyle \frac{-1-4{\lambda }_{12}{\lambda }_{13}}{4p}}$	*	*	${\displaystyle \frac{1+2{\lambda }_{12}(1-2p)}{2(-1+2{\lambda }_{13})}}$	${\displaystyle \frac{1-2p}{2}}$	${\displaystyle \frac{1+2{\lambda }_{13}-4{\lambda }_{12}{\lambda }_{13}-4{\lambda }_{13}^2}{2(-1+2{\lambda }_{12})}}$
3	*	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3+2{\lambda }_{11}}{2(-1+2{\lambda }_{11})}}$
4	*	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3-2{\lambda }_{11}}{2(1+2{\lambda }_{11})}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$
5	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	*	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3+2{\lambda }_{22}}{-2+4{\lambda }_{22}}}$
6	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	*	${\displaystyle \frac{1}{2}}$	$1-{\lambda }_{13}$	${\displaystyle \frac{-1-4{\lambda }_{13}+4{\lambda }_{13}^2}{4}}$
7	*	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3+2{\lambda }_{11}}{2(-1+2{\lambda }_{11})}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$
8	*	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3-2{\lambda }_{11}}{2(1+2{\lambda }_{11})}}$
9	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	*	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3-2{\lambda }_{22}}{2(1+2{\lambda }_{22})}}$
10	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	*	$-{\displaystyle \frac{1}{2}}$	$-1-{\lambda }_{13}$	${\displaystyle \frac{1-4{\lambda }_{13}-4{\lambda }_{13}^2}{4}}$
11	${\displaystyle \frac{1}{2}}$	*	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{-1-4{\lambda }_{12}+4{\lambda }_{12}^2}{4}}$	$1-{\lambda }_{12}$	${\displaystyle \frac{1}{2}}$
12	$-{\displaystyle \frac{1}{2}}$	*	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1-4{\lambda }_{12}-4{\lambda }_{12}^2}{4}}$	$-1-{\lambda }_{12}$	$-{\displaystyle \frac{1}{2}}$
13	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$
14	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3}{2}}$	${\displaystyle \frac{1}{2}}$
15	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$
16	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$
17	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$
18	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$
19	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{3}{2}}$	$-{\displaystyle \frac{1}{2}}$
20	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{2}}$

Here, the symbol ∗ is used to indicate free variables.

.

Based on the commutative and cocommutative of A, we can naturally obtain the properties of its Nakayama automorphism, Nakayama co-automorphism, and the formula of ${\psi }^2$.

Proposition 2.

If $(A,\varphi ,y,\psi )$ is bi-Frobenius algebra, $\mathcal{N}$ is the Nakayama automorphism of $(A,\varphi ),$ and ${}^c\mathcal{N}$ is the coNakayama automorphism of $(A,y),$ then $\mathcal{N}=id,$ ${}^c\mathcal{N}=id$ and ${\psi }^2=id.$

Proposition 3.

Let $(A,\varphi ,y,\psi )$ be a bi-Frobenius algebra. Define the linear map $f:A⟶A$ by

$$f\left(\begin{array}{c}1\\ x_1\\ x_2\\ ⋮\\ x_n\\ y\end{array}\right)=\left(\begin{array}{cccccc}1& 0& 0& \cdots & 0& 0\\ k_1& d_{11}& d_{12}& \cdots & d_{1n}& d_{1,n+1}\\ k_2& d_{21}& d_{22}& \cdots & d_{2n}& d_{2,n+1}\\ ⋮& ⋮& & ⋮& ⋮\\ k_n& d_{n1}& d_{n2}& \cdots & d_{nn}& d_{n,n+1}\\ k_{n+1}& d_{n+1,1}& d_{n+1,2}& \cdots & d_{n+1,n}& d_{n+1,n+1}\end{array}\right)\left(\begin{array}{c}1\\ x_1\\ x_2\\ ⋮\\ x_n\\ y\end{array}\right),$$

where $k_i,d_{ij}\in \mathbb{K},$ for $i,j\in \{1,2,\cdots ,n+1\}.$ Denote the coefficient matrix as $\tilde{D}$ and the block matrix $D={\left(d_{ij}\right)}_{n\times n}.$ If f is an automorphism of $A,$ then the following conditions hold:

(i) 

$\left|D\right|\ne 0.$

(ii) 

$k_i=0,$ for all $i\in \{1,2,\cdots ,n+1\}.$

(iii) 

$d_{i,n+1}=d_{n+1,i}=0,$$i\in \{1,2,\cdots ,n\}.$

(iv) 

$d_{n+1,n+1}P=D^{T}PD.$

(v) 

$d_{n+1,n+1}\mathrm{\Lambda }=D\mathrm{\Lambda }{D}^T.$

(vi) 

$\mathrm{\Lambda }PD=D\mathrm{\Lambda }P.$

Proof. 

It is evident that (i) holds. Suppose f is a morphism of algebra, then for all $i\in \{1,2,\cdots ,n\},$

$$\begin{array}{cc}\hfill & f\left(x_i\right)f\left(y\right)\hfill \\ \hfill =& (k_i+d_{i1}{x}_1+\cdots +d_{in}{x}_n+d_{i,n+1}y)(k_{n+1}+d_{n+1,1}{x}_1+\cdots +d_{n+1,n+1}y)\hfill \\ \hfill =& k_i{k}_{n+1}+(k_i{d}_{n+1,1}+d_{i1}{k}_{n+1})x_1+\cdots +(k_i{d}_{n+1,n}+k_{n+1}{d}_{i,n})x_n\hfill \\ \hfill & +(k_i{d}_{n+1,n+1}+k_{n+1}{d}_{i,n+1})y\hfill \\ \hfill =& 0.\hfill \end{array}$$

Thus, it follows that $k_i=0$ and $k_{n+1}=0.$

Furthermore,

$$\begin{array}{cc}\hfill & f\left(x_i\right)f\left(x_j\right)\hfill \\ \hfill =& (d_{i1}{x}_1+\cdots +d_{in}{x}_n+d_{i,n+1}y)(d_{j1}{x}_1+\cdots +d_{jn+1}y)\hfill \\ \hfill =& {\displaystyle \sum _{s,t=1}^n}{\lambda }_{st}{d}_{is}{d}_{jt}y,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & f\left(x_i{x}_j\right)\hfill \\ \hfill =& {\lambda }_{ij}f\left(y\right)\hfill \\ \hfill =& {\lambda }_{ij}(d_{n+1,1}{x}_1+\cdots +d_{n+1,n}{x}_n+d_{n+1,n+1}y),\hfill \end{array}$$

(14)

As Equations (13) and (14) are equal, then it follows that

$$d_{n+1,i}=0,$$

and

$${\lambda }_{ij}{d}_{n+1,n+1}=\sum _{s,t=1}^n{\lambda }_{st}{d}_{is}{d}_{jt},$$

which means

$$d_{n+1,n+1}\mathrm{\Lambda }=D\mathrm{\Lambda }{D}^T.$$

Next, suppose f is a morphism of coalgebra, then

$$\begin{array}{cc}\hfill & (f\otimes f)\mathrm{\Delta }\left(x_i\right)\hfill \\ \hfill =& (f\otimes f)(1\otimes x_i+x_i\otimes 1)\hfill \\ \hfill =& 2k_i(1\otimes 1)+d_{i1}(1\otimes x_1+x_1\otimes 1)+\cdots +d_{i,n+1}(1\otimes y+y\otimes 1)\hfill \\ \hfill =& 2k_i\mathrm{\Delta }\left(1\right)+d_{i1}\mathrm{\Delta }\left(x_1\right)+\cdots +d_{in}\mathrm{\Delta }\left(x_n\right)+d_{i,n+1}(1\otimes y+y\otimes 1).\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \mathrm{\Delta }f\left(x_i\right)\hfill \\ \hfill =& k_i\mathrm{\Delta }\left(1\right)+d_{i1}\mathrm{\Delta }\left(x_1\right)+\cdots +d_{in}\mathrm{\Delta }\left(x_n\right)+d_{i,n+1}\mathrm{\Delta }\left(y\right)\hfill \\ \hfill =& k_i\mathrm{\Delta }\left(1\right)+d_{i1}\mathrm{\Delta }\left(x_1\right)+\cdots +d_{in}\mathrm{\Delta }\left(x_n\right)+d_{i,n+1}(1\otimes y+y\otimes 1\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}{x}_i\otimes (x_{\pi \left(i\right)}+x_{{\pi }^{-1}\left(i\right)})).\hfill \end{array}$$

(16)

As Equations (15) and (16) are euqal, then it follows that

$$k_i=0,d_{i,n+1}=0.$$

Similarly, using $(f\otimes f)\mathrm{\Delta }\left(y\right)=\mathrm{\Delta }f\left(y\right),$ we can obtain

$$k_{n+1}=0,$$

and

$$d_{n+1,n+1}P=D^{T}PD.$$

Suppose f is a morphism of a bi-Frobenius algebra, then the matrix $\tilde{D}$ can be denoted as

$$\tilde{D}=\left(\begin{array}{ccc}1& & \\ & D& \\ & & d_{n+1,n+1}\end{array}\right).$$

Moreover, it satisfies $\psi \circ f=f\circ \psi .$ As

$$\psi \circ f\left(1\right)=f\circ \psi \left(1\right)=1,\psi \circ f\left(y\right)=f\circ \psi \left(y\right)=d_{n+1,n+1}y,$$

it suffices to consider $\psi \circ f\left(x_i\right)$ and $f\circ \psi \left(x_i\right)$.

$$\begin{array}{cc}\hfill \psi \circ f\left(x_i\right)=& \psi (d_{i1}{x}_1+d_{i2}{x}_2+\cdots +d_{in}{x}_n)\hfill \\ \hfill =& {\displaystyle \sum _{j=1}^n}{d}_{ij}\psi \left(x_j\right)\hfill \\ \hfill =& {\displaystyle \sum _{j,k=1}^n}{d}_{ij}({\lambda }_{\pi \left(k\right)j}+{\lambda }_{{\pi }^{-1}\left(k\right)j})x_k,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill f\circ \psi \left(x_i\right)=& f({\displaystyle \sum _{k=1}^n}({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})x_k)\hfill \\ \hfill =& {\displaystyle \sum _{k=1}^n}({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})(d_{k1}{x}_1+d_{k2}{x}_2+\cdots +d_{kn}{x}_n)\hfill \\ \hfill =& {\displaystyle \sum _{k,j=1}^n}({\lambda }_{\pi \left(k\right)i}+{\lambda }_{{\pi }^{-1}\left(k\right)i})d_{kj}{x}_j.\hfill \end{array}$$

(18)

As Equations (17) and (18) are euqal, then

$$\mathrm{\Lambda }PD=D\mathrm{\Lambda }P.$$

(19)

□

Example 4.

Recall Example 2 Case 2. Let $n=2$ and $\pi =\left(12\right),$ then $A=\mathbb{K}{1}_A+(\mathbb{K}{x}_1+\mathbb{K}{x}_2)+\mathbb{K}y$ is a bi-Frobenius algebra with structure matrix $\mathrm{\Lambda }=\left(\begin{array}{cc}1& 0\\ 0& {\displaystyle \frac{1}{4}}\end{array}\right).$ The multiplication is

$$x_1^2=y,x_2^2={\displaystyle \frac{1}{4}}y,x_1{x}_2=x_2{x}_1=0,$$

and the comultiplication is

$$\mathrm{\Delta }\left(y\right)=1\otimes y+y\otimes 1+2x_1\otimes x_2+2x_2\otimes x_1.$$

Define $\tilde{D}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& d_{11}& d_{12}& 0\\ 0& d_{21}& d_{22}& 0\\ 0& 0& 0& d_{33}\end{array}\right),$ where $d_{ij}\in \mathbb{K}.$ By Proposition 3, there exists

$$\left\{\begin{array}{c}{d}_{11}{d}_{21}=0,\hfill \\ d_{33}-d_{11}{d}_{22}-d_{12}{d}_{21}=0,\hfill \\ 4d_{11}^2+d_{12}^2-4d_{33}=0,\hfill \\ 4d_{11}{d}_{21}+d_{12}{d}_{22}=0,\hfill \\ 4d_{21}^2+d_{22}^2-d_{33}=0,\hfill \\ d_{11}{d}_{22}-d_{21}{d}_{12}\ne 0.\hfill \end{array}\right.$$

(20)

By Equation (20), we have $\tilde{D}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& k& 0& 0\\ 0& 0& k& 0\\ 0& 0& 0& k^2\end{array}\right)$ or $\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 4k& 0\\ 0& k& 0& 0\\ 0& 0& 0& 4k^2\end{array}\right),$ where $k\in \mathbb{K}\setminus \left\{0\right\}.$ Then, the automorphism f of A is

$$f\left(\begin{array}{c}1\\ x_1\\ x_2\\ y\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& k& 0& 0\\ 0& 0& k& 0\\ 0& 0& 0& k^2\end{array}\right)\left(\begin{array}{c}1\\ x_1\\ x_2\\ y\end{array}\right),$$

or

$$f\left(\begin{array}{c}1\\ x_1\\ x_2\\ y\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 4k& 0\\ 0& k& 0& 0\\ 0& 0& 0& 4k^2\end{array}\right)\left(\begin{array}{c}1\\ x_1\\ x_2\\ y\end{array}\right).$$