Automorphism groups of representation rings of the weak Sweedler Hopf algebras

1.   Introduction

The study of representation rings has attracted extensive attention of mathematicians. Chen et al. ^[5] described the structure of the representation rings of the Taft algebra $H_{n}(q)$. Li and Zhang ^[12] determined the representation rings of the generalized Taft Hopf algebras $H_{n, d}(q)$, and determined all nilpotent elements in the representation ring of $H_{n, d}(q)$. In ^[14], we constructed two classes of weak Hopf algebras $\mathfrak{w}^{s}_{n, d}(s = 0, 1)$ corresponding to generalized Taft algebra $H_{n, d}$, and investigate the representation rings $r(\mathfrak{w}^{s}_{n, d})$ of $\mathfrak{w}^{s}_{n, d}$. More conclusions related to representation rings can be seen in ^[4,15].

Many significant researches focused on studying automorphisms of algebras or rings. van der Kulk ^[11], Zhao ^[18], Yu ^[17], Vesselin and Yu ^[16] did some significant contributions on the automorphisms of polynomial algebras. Alperin ^[2] investigated the homology of the group of automorphisms of $k[x, y]$ over a field $k$. Furthermore, Dicks ^[7] researched automorphisms of polynomial ring in two variables. Chen et al. ^[3,6] considered the coalgebra automorphism groups of the Hopf algebras. Han and Su ^[8] studied automorphism group of Witt algebras. Jia et al. ^[10] proved that the automorphism group of representation ring of Sweedler Hopf algebra is isomorphic to the Klein four-group. Motivated by the above works, in this paper, we investigate the automorphism groups ${{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2}))$ of representation rings $r(\mathfrak{w}^{s}_{2, 2})$ of two classes of weak Sweedler Hopf algebra $\mathfrak{w}^{s}_{2, 2} (s = 0, 1)$. It is shown that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2}))$ is isomorphic to $K_4$, where $K_4$ is the Klein four-group. Through calculation, the structure of automorphism group ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ has been constructed. We prove that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ is isomorphic to $(\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2}$. Its centre is isomorphic to $\mathbb{Z}_{2}$. In addition, ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ is a non-commutative infinite solvable group, but it is not nilpotent.

The paper is organized as follows. In Section 1, we recall some relative background and knowledge in detail. In Section 2, the structures of automorphism groups ${{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) (s = 0, 1)$ are described. We obtain that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ is a non-commutative infinite group, and the automorphism group ${{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2}))$ is isomorphic to $K_4$. In Section 3, the properties of the automorphism groups ${{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2}))$ are discussed. It is shown that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ is isomorphic to $(\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2}$, and its centre is isomorphic to $\mathbb{Z}_{2}$. Finally, we prove that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ is a solvable and non-nilpotent group. It is interesting that although both $\mathfrak{w}^{0}_{2, 2}$ and $\mathfrak{w}^{1}_{2, 2}$ are weak Hopf algebras of Sweedler Hopf algebra, the automorphism groups of their representation rings are strongly different.

2.   Preliminaries

Throughout, we work over an algebraically closed field $\mathbb{K}$ of characteristic $0$ unless otherwise stated. All algebras, Hopf algebras and weak Hopf algebras are defined over $\mathbb{K}$.

In the sequel, we fix two integers $n, d \geq 2$ such that $d\mid n$, and assume that $q\in \mathbb{K}$ is a primitive $d$-th root of unity. In ^[14], we constructed the weak Hopf algebras $\mathfrak{w}^{s}_{n, d}(s = 0, 1)$ corresponding to generalized Taft algebra^[13], and investigated the representation rings $r(\mathfrak{w}^{s}_{n, d})(s = 0, 1)$ of $\mathfrak{w}^{s}_{n, d}(s = 0, 1)$. As an algebra $\mathfrak{w}^{s}_{n, d}(s = 0, 1)$ is generated by $g, x$ subject to the relations

The comultiplication, counit and weak antipode $T$ are given by

If $r = 0$, we get the weak Hopf algebra $\mathfrak{w}^{1}_{n, d}$, where $g^{0} = 1$. If $r = n$ and $x = g^{n}x$, we get the weak Hopf algebra $\mathfrak{w}^{0}_{n, d}$. Let $E = g^n$, it is easy to see that the dimension of ${\mathfrak{w}}^s_{n, d}$ is $nd+(d-1)s+1$, and the set

forms a PBW basis for ${\mathfrak{w}}^s_{n, d}$.

In particular, when $n = d = 2$, ${\mathfrak{w}}^s_{2, 2}(s = 0, 1)$ are exactly two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra (see also ^[1]).

Let $H$ be a weak Hopf algebra, the representation ring $r(H)$ of $H$ is defined as follows. Assume that $F(H)$ is a free abelian group generated by isomorphism classes $[V]$ of finite dimensional $H$-modules $V$. Let $r(H)$ be the quotient group $F(H)$ modulo the relations $[M\oplus V] = [M]+[V]$, We equip $r(H)$ with the multiplication $[M][V] = [M\otimes V]$. It is well known that $r(H)$ is an associative ring with $\mathbb{Z}$-basis $\{[V]\mid V\in{{{\rm{ind}}}} \rm{-}H\}$.

Theorem 2.1. ^[14] The representation ring $r(\mathfrak{w}^{s}_{n, d})\cong\mathbb{Z}\langle\, {x_{1}, x_{2}, x_{3}}\, \rangle/I$ as ring isomorphisms, and the ideal $I$ is generated by the relations

where $m = \frac{n}{d}$ and $F_{t}(y, z)$ are the generalized Fibonacci polynomials defined by $F_{t+2}(y, z) = zF_{t+1}(y, z)-yF_{t}(y, z), t > 1, F_{0}(y, z) = 0, F_{1}(y, z) = 1, F_{2}(y, z) = z.$

Corollary 2.2. The representation ring $r(\mathfrak{w}^{s}_{2, 2})\cong\mathbb{Z}\langle\, {x, y, z}\, \rangle/I$ as ring isomorphisms, and the ideal $I$ is generated by the relations

Notice that

are $\mathbb{Z}-$basis of $r(\mathfrak{w}^{0}_{2, 2})$ and $r(\mathfrak{w}^{1}_{2, 2})$, respectively.

Let $\mathbf{A}_{f}$ denote the corresponding coefficient matrix of $\mathbb{Z}-$linear map

where $s = 0, 1$. And let $|\mathbf{A}_{f}|$ denote the determinant of $\mathbf{A}_{f}$.

3.   Automorphism groups of representation rings $r(\mathfrak{w}^{s}_{2, 2})$

In this section, we will discuss the automorphism groups of representation rings of weak Sweedler Hopf algebras. By Corollary 2.2, one see that $r(\mathfrak{w}^{0}_{2, 2})$ is a commutative ring, but $r(\mathfrak{w}^{1}_{2, 2})$ is a non-commutative ring. We mainly consider the automorphism group of $r(\mathfrak{w}^{1}_{2, 2})$, the automorphism group of $r(\mathfrak{w}^{0}_{2, 2})$ will be stated directly.

For any $i, j, k, z\in \mathbb{Z}$, let $\omega_{i}, \delta_{j}, \tau_{k}, \varphi_{z}$ be $\mathbb{Z}-$linear maps of $r(\mathfrak{w}^{1}_{2, 2})$. They are determined by the following maps:

It is easy to check that $\omega_{i}, \delta_{j}, \tau_{k}, \varphi_{z}$ are four classes of automorphisms of $r(\mathfrak{w}^{1}_{2, 2})$. For any $i, j, k, z\in\mathbb{ Z}\; (i = 1, 2, 3, 4),$ we have

Let

then $G$ is a group under the composition of functions. For any $i, j, k, z, i', j', k', z'\in \mathbb{Z}$ the multiplication is described as follows

It follow that $G$ is a non-commutative infinite group.

In the sequel, we shall show the automorphism group ${\rm Aut}(r(\mathfrak{w}^{1}_{2, 2}))$ is just the group $G.$

Lemma 3.1. Let $g$ be an automorphism of $r(\mathfrak{w}^{1}_{2, 2})$, then

where $a\in \mathbb{Z}$.

Proof. Indeed, we have $(g(x_{1}))^{2} = 1$ since $g$ is an automorphism of $r(\mathfrak{w}^{1}_{2, 2})$ and $x_{1}^{2} = 1$.

Assume that

Then we get

and

Thanks to $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\in \mathbb{Z}$, we obtain that the system of equations has eight distinct solutions $(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ as follows:

where $a\in \mathbb{Z}$. Therefore, we get that only the fifth and seventh solutions are unreasonable, and

The proof is finished.

Lemma 3.2. Let $g$ be an automorphism of $r(\mathfrak{w}^{1}_{2, 2})$, then

where $c\in \mathbb{Z}$.

Proof. Noting that $x_{3}^{2} = x_3$, we have

Assume that

Then we have

It is easy to get that the system of equations has four distinct solutions $(b_{0}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5})$ as follows:

where $c\in \mathbb{Z}$. Only the first and third solutions are unreasonable. Therefore

Lemma 3.3. Let $g$ be an automorphism of $r(\mathfrak{w}^{1}_{2, 2})$, we have

1. if $g(x_{3}) = (1-2c)x_{3}+cx_{3} x_{2}$, then $g(x_{1}) = x_{1} \; \mathrm{or}\; g(x_{1}) = -x_{1}+(2-4c)x_{3} +2cx_{3}x_{2} \; \mathrm{or}\; g(x_{1}) = -1+(2-4c)x_{3} +2cx_{3}x_{2};$

2. if $g(x_{3}) = 1-(1+2c) x_{3}+cx_{3} x_{2}$, then $g(x_{1}) = 1-(2+4c)x_{3} +2cx_{3}x_{2};$

where $c\in \mathbb{Z}$.

Proof. Noting that $x_1x_{3} = x_{3}x_{1} = x_3$, then we have

1. Since $g(x_{3}) = (1-2c)x_{3}+cx_{3} x_{2}$, and by the Lemma 3.1, then

(a) if $g(x_{1}) = x_{1}$, we have

(b) if $g(x_{1}) = x_{1}-(2+2a)x_{3}+ax_{2}x_{3}$, we have

(c) if $g(x_{1}) = -x_{1}$, we have

(d) if $g(x_{1}) = -x_{1}+ (2-2a)x_3+ax_{3}x_{2}$, we have

Let $a = 2c$, then $g(x_{1}) = -x_{1}+ (2-4c)x_3+2cx_{3}x_{2}$ and $g(x_{3})g(x_{1}) = g(x_{3})$;

(e) if $g(x_{1}) = 1-(2+2a)x_{3}+ax_{2}x_{3}$, we have

(f) if $g(x_{1}) = -1+ (2-2a)x_3+ax_{3}x_{2}$, we have

Let $a = 2c$, then $g(x_{1}) = -1+ (2-4c)x_3+2cx_{3}x_{2}$ and $g(x_{3})g(x_{1}) = g(x_{3})$.

2. Similar to the proof of 1.

Proposition 3.4. Let $g$ be an automorphism of $r(\mathfrak{w}^{1}_{2, 2})$, if $g(x_{1}) = x_{1}$ and $g(x_{3}) = (1-2c) x_{3}+cx_{3}x_{2}\; (c\in\mathbb{ Z})$, then $g\in G$.

Proof. Since $g$ is an automorphism of $r(\mathfrak{w}^{1}_{2, 2})$ and

Then we have

Assume

Then we have

Since $c_{0}^{2}+c_{1}^{2} = c_{0}+c_{1}$ and $2c_{0}c_{1} = c_0+c_1$, then

Therefore, we have following cases.

$\mathbf{Case \; 1}\quad c_{0} = c_{1} = 0.$

Since $c_{2}^{2}+c_{4}^{2}+2c_{2}c_{4} = c_2+c_4$, then we have

1. $c_2+c_4 = 0.$

Since $c_0 = c_1 = 0 \; \mathrm{and}\; c_2+c_4 = 0$, then $c_3 = 2-2c_{5}, c_2 = -c_4$. We let $c_4 = d, c_5 = b,$ then

And

so $\mid A_{g}\mid = 0$, and hence, $A_{g}$ is not invertible.

2. $c_2+c_4 = 1.$

Since $c_0 = c_1 = 0 \; \mathrm{and}\; c_2+c_4 = 1$, then $c_3 = -2c_{5}, c_2 = 1-c_4$. We let $c_4 = d, c_3 = b$ then

And

so $\mid A_{g}\mid = (1-2d)(1+b-2c)$. Note that $A_{g}$ is invertible if only if $\mid A_{g}\mid = \pm1$, thus, we get that

For any $c\in\mathbb{ Z}$, we have

(a) If $d = 0$ and $b = 2c$, then $g(x_{2}) = x_{2}-4cx_{3}+2cx_{3}x_{2}, g(x_{1}x_{2}) = x_{1}x_{2}-4cx_{3}+2cx_{3}x_{2}, g(x_{3}x_{2}) = -4cx_{3}+(1+2c)x_{3}x_{2},$ thus $g = \omega_{i}$.

(b) If $d = 1$ and $b = 2c-2$, then $g(x_{2}) = (4-4c)x_{3}+x_{1}x_{2}+(2c-2)x_{3}x_{2}, g(x_{1}x_{2}) = (4-4c)x_{3}+x_{2}+(2c-2)x_{3}x_{2}, g(x_{3}x_{2}) = (4-4c)x_{3}+(2c-1)x_{3}x_{2},$ thus $g = \delta_{j}$.

(c) If $d = 0$ and $b = 2c-2$, then $g(x_{2}) = x_{2}+(4-4c)x_{3}+(2c-2)x_{3}x_{2}, g(x_{1}x_{2}) = x_{1}x_{2}+(4-4c)x_{3}+(2c-2)x_{3}x_{2}, g(x_{3}x_{2}) = (4-4c)x_{3}+(2c-1)x_{3}x_{2},$ thus $g = \tau_{k}$.

(d) If $d = 1$ and $b = 2c$, then $g(x_{2}) = -4cx_{3}+x_{1}x_{2}+2cx_{3}x_{2}, g(x_{1}x_{2}) = -4cx_{3}+x_{2}+2cx_{3}x_{2}, g(x_{3}x_{2}) = -4cx_{3}+(2c+1)x_{3}x_{2},$ thus $g = \varphi_{z}$.

$\mathbf{Case \; 2}\quad c_{0} = c_{1} = 1.$

Since $2c_2+2c_4+c_{2}^{2}+c_{4}^{2}+2c_{2}c_{4} = c_2+c_4$, then we have

1. $c_2+c_4 = 0.$

Since $c_0 = c_1 = 1 \; \mathrm{and}\; c_2+c_4 = 0$, then $c_3 = c_{5} = 0, c_2 = -c_4$. We let $c_4 = e,$ then

And

so $\mid A_{g}\mid = 0$, and hence, $A_{g}$ is not invertible.

2. $c_2+c_4 = -1.$

Since $c_0 = c_1 = 1 \; \mathrm{and}\; c_2+c_4 = -1$, then $c_3 = 0, c_{5} = 1, c_2 = -1-c_4$. We let $c_4 = e,$ then

And

so $\mid A_{g}\mid = 0$, and hence, $A_{g}$ is not invertible.

In summary, if $g(x_{1}) = x_{1}$ and $g(x_{3}) = (1-2c) x_{3}+cx_{3}x_{2}(c\in\mathbb{ Z})$, then $g\in G$.

Theorem 3.5. Let ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ denote the automorphism group of $r(\mathfrak{w}^{1}_{2, 2})$. Then

Proof. Let $g$ be an automorphism of $r(\mathfrak{w}^{1}_{2, 2})$, by Lemma 3.3, we know that

● if $g(x_{3}) = (1-2c)x_{3}+cx_{3} x_{2}$, then $g(x_{1}) = x_{1} \; \mathrm{or}\; g(x_{1}) = -x_{1}+(2-4c)x_{3} +2cx_{3}x_{2} \; \mathrm{or}\; g(x_{1}) = -1+(2-4c)x_{3} +2cx_{3}x_{2};$

● if $g(x_{3}) = 1-(1+2c) x_{3}+cx_{3} x_{2}$, then $g(x_{1}) = 1-(2+4c)x_{3} +2cx_{3}x_{2}.$

By Proposition 3.4, we have that if $g(x_{1}) = x_{1}$ and $g(x_{3}) = (1-2c) x_{3}+cx_{3}x_{2}$, then $g\in G$. The similar arguments of Proposition 3.4 are applied to the remaining possibilities, show that $\mid A_{g}\mid = 0$, and hence, $A_{g}$ is not invertible, in these cases.

Thus

The proof is finished.

Let $f_{0}, f_{1}, f_{2}$ and $f_{3}$ are automorphisms of $r(\mathfrak{w}^{0}_{2, 2})$, determined by the following.

where $f_0$ is the identity map. The set $\left\{f_0, f_1, f_2, f_3\right\}$ is a group under the composition of functions. The multiplication is described as follows.

Remark 3.6. Similar to arguments of the proof of Theorem 3.5 show that

where $K_4$ is the Klein four-group.

4.   The properties of $\mathrm{ Aut}(r(\mathfrak{w}^{s}_{2, 2}))$

By Section 3, we have $\mathrm{ Aut}(r(\mathfrak{w}^{0}_{2, 2}))\cong K_4$ and ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) = G$.

The infinite group $G = \left\{\omega_{i}, \delta_{j}, \tau_{k}, \varphi_{z}\mid i, j, k, z\in \mathbb{Z}\right\}$ is not abelian. The elements $\varphi_{0}, \delta_{j}, \tau_{k}\; (j, k\in \mathbb{Z})$ of $G$ have order $2$, and other elements $\omega_{i}, \varphi_{z}\; (i, z\in \mathbb{Z}, z\neq 0)$ have infinite order. In the sequel, we will discuss some properties of $G$. The definitions of solvable group, nilpotent group, and normal subgroups, etc. can be found in ^[9], they are used in the sequel.

Proposition 4.1. Let $Z(G)$ be the centre of $G$, then $Z(G)\cong\mathbb{ Z}_{2}$.

Proof. For any $i, j, k, z\in \mathbb{Z}$, all subgroups of $G$, up to isomorphism, as follows

and we have

Hence $\langle\, {\omega_{0}}\, \rangle, \langle\, {\omega_{1}}\, \rangle, \langle\, {\varphi_{0}}\, \rangle, \langle\, {\varphi_{1}}\, \rangle, \langle\, {\delta_{0}, \delta_{1}}\, \rangle, \langle\, {\tau_{0}, \tau_{1}}\, \rangle, \langle\, {\delta_{0}, \varphi_{1}}\, \rangle$ are normal subgroups of $G$. Furthermore

for any $i, j, k, z, j', k'\in \mathbb{Z}\setminus\{0\}, j\neq j', k \neq k'$. Therefore

It is easy to show that $Z(G)\cong\mathbb{ Z}_{2},$ determined by the map $id\rightarrow 0, \varphi_{0}\rightarrow 1.$

Theorem 4.2. $G\cong (\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2}$.

Proof. We set

It is easy to know that $H$ and $K$ are subgroups, and $H\lhd G$. Since

for any $i, z\in \mathbb{Z}$, hence $G = HK,$ and $H\cap K = \{id\}$, thus we have

Let

one can get that $H_{1}\lhd H, Z(G)\lhd H,$ where $Z(G) = \langle\, {\varphi_{0}}\, \rangle = \{id, \varphi_{0}\}$ is the centre of $G$. Furthermore

and $\omega_{i}\varphi_{0} = \varphi_{i}$ for any $i \in \mathbb{Z}$. Hence $H = H_{1}Z(G)$, and

$H_{1}$ is isomorphic to $\mathbb{Z}$, determined by the map $\omega_{i}\rightarrow i,$ for any $i\in \mathbb{Z}$. $K$ is isomorphic to $\mathbb{Z}_{2}$, determined by the map

Therefore

The proof is finished.

By the Theorem 4.2, the following results are easily to get.

Corollary 4.3. $G$ is a solvable and non-nilpotent group.

Proof. By the Proposition 4.1 and Theorem 4.2, we have

Since $\mathbb{Z}$ and $\mathbb{Z}_{2}$ are solvable groups, then $\mathbb{Z}\times\mathbb{Z}_{2}$ and $(\mathbb{Z}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}\big/\mathbb{Z}\times\mathbb{Z}_{2}$ are solvable groups, hence $G\cong (\mathbb{Z}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}$ is solvable.

It is easy to see $G/Z(G) = \langle\, {\omega_{1}}\, \rangle\rtimes \langle\, {\delta_{0}}\, \rangle$ and $Z\left(G/Z(G)\right) = \{id\},$ hence, $G/Z(G)$ and $G$ are non-nilpotent groups.

5.   Conclusions

In this paper, we investigate the automorphism groups ${{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2}))$ of representation rings $r(\mathfrak{w}^{s}_{2, 2})$ of two classes of weak Sweedler Hopf algebras $\mathfrak{w}^{s}_{2, 2} (s = 0, 1)$ and discuss some properties of ${{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2}))$. We obtain that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2}))$ is isomorphic to the Klein four-group. It is shown that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ is a non-commutative infinite group, it is solvable and non-nilpotent. In addition, we prove that ${{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2}))$ is isomorphic to $(\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2}$, and its centre is isomorphic to $\mathbb{Z}_{2}$.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No.11671024) and the doctoral research start-up fund of Henan University of science and technology (Grant No. 13480069)

Conflict of interest

The authors declared that they have no conflict of interest.