通常来讲, 要构造一个Hopf代数是件很不容易的事.然而, 近年来, 很多数学家开始利用箭向来研究代数结构^[1-3], 得到很多可交换与不可交换的Hopf代数^[4-5].我们在文献[6-7]中也借助于箭向构造了群上的大量的Hopf代数.

设$G$是群, $kG$是代数闭域$k$上的一个群代数, 则Hopf双模范畴$^{kG}_{kG}{\mathcal M}^{kG}_{kG}$等价于直积范畴$\sqcap_{C\in K(G)}{\mathcal M}_{kZ_{u(C)}}$, 这里$K(G)$是群$G$的全体共轭类, 映射

$ \begin{equation}\label{e1.1} u:K(G) \to G, u(C) \in K(G), Z_{u(C)}=\{g\in G|gu(C)=u(C)g\}, \forall C \in K(G) . \end{equation} $

(1.1)

${\mathcal M}_{kZ_{u(C)}}$表示右$kZ_{u(C)}$模^{[5, 8]}. 2002年, 由于数学家Cibils和Rosso引入了Hopf箭向和群的分歧^[8], 使得利用箭向构造Hopf代数成为可能^[5]. 2008年, 张寿全教授等给出了$S_n$(此时$n \not= 6$^[9])是完全群时的分歧系统, 由此可以构造出一批Hopf代数^[10].那么对于非完全群$S_6$, 如何构造其上的Hopf代数呢？本文想在这方面做些探讨.

2011年, Andruskiewitsch, Fantino, Grana以及Vendramin研究出对称群上的有限维逐点Hopf代数都是平凡的, 且对于对称群$S_6$, 其上的一型路Hopf代数都是无限维的^[11].所以我们要构造的一型路Hopf代数都是无限维的.

本文约定在代数闭域$k$上讨论, 并且$k$的特征${\rm char}(k)\not=2.$所有代数, 余代数, Hopf代数等都在域$k$上讨论.与Hopf代数有关的概念参见文献[12].

由于非完全群$S_6$也是置换群, 故$S_6$中任意两个元素有相同的共轭类当且仅当它们有相同的循环结构.从而有以下引理.

引理2.1^[13]  设$S_6$是包含6个元素的集合的全体置换做成的群, ${\rm Aut}(S_6)$是$S_6$的自同构群, 则${\rm Aut} (S_6) ={\rm Inn}(S_6) <\delta>$, 其中$\delta$为2阶外自同构.因此${\rm Aut}(S_6)$是一个1440阶的群, 是$S_6$的内自同构群和一个2阶群的半直积.

由于群$S_6$可由$\Omega=\{(12), (13), (14), (15), (16) \}$生成, 令$\phi:S_6 \to S_6$是一个映射, 定义

$ \begin{eqnarray*} &&\phi((12))=(12)(36)(45), \phi((13))=(16)(24)(35), \phi((14))=(13)(25)(46), \\ &&\phi((15))=(15)(26)(34), \phi((16))=(14)(23)(56).\end{eqnarray*} $

可以验证$\phi$定义了$S_6$的一个外自同构.

令$\sigma=(12345)$, $i_{\sigma}$是由$\sigma$诱导的$S_6$的内自同构, 即任意$g \in S_6$, 定义$i_{\sigma}:S_6 \to S_6, $ $i_{\sigma}(g)=\sigma g \sigma^{-1}.$令$\delta =i_{\sigma}\phi$, 则$\delta$是2阶外自同构.由此得以下结论.

定理2.2 任意$h \in S_6$, 定义

$ i_g:S_6 \to S_6, i_g (h)=ghg^{-1}; \delta_g:S_6 \to S_6, \delta_g (h)=g \delta(h) g^{-1}=i_{g\sigma}\phi(h), $

则${\rm Aut} (S_6)=\{i_g|g \in S_6 \} \cup \{\delta_g | g \in S_6 \}$.

注 (ⅰ) 由于$\Omega = \{(12), (13), (14), (15), (16)\}$可以生成置换群$S_6$, 所以$i_g, \delta_g$只需要定义在$\Omega$上即可.

(ⅱ) 经过计算, 可得

$ \begin{eqnarray*}&&\delta ((12))=(23)(46)(15), \delta ((13))=(26)(35)(41), \delta ((14))=(24)(31)(56), \\ &&\delta ((15))=(21)(36)(45), \delta ((16))=(25)(34)(16).\end{eqnarray*} $

用$C_1, C_2, C_3, C_4, C_5, C_6, C_7, C_8, C_9, C_{10}, C_{11}$表示$S_6$的全体共轭类, 分别用$(1), (12), (123), $ $(1234), (12345), (123456), (12)(34), (12)(345), (12)(3456), (12)(34)(56), (123)(456)$作为这些共轭类的代表元.考虑$S_6$的共轭类与特征标的关系, 得到如表 1 ^[13-14].

定理2.3 设$S_6$是6个元素上的置换群, $i_g, \delta_g$如定理2.2中所述, 记$\widehat {S_6} =\{\chi_1, \chi_2, \cdots, \ \chi_{11} \}, $其中$\chi_1, \chi_2, \cdots, \ \chi_{11} $如表 1中所述.则对于任意$ g \in S_6, $有

$ \begin{eqnarray*} &&\chi_ji_g=\chi_j, j=1, 2, \cdots, 11, \chi_1 \delta_g=\chi_1, \chi_2 \delta_g=\chi_7, \chi_4 \delta_g=\chi_8, \\ &&\chi_5\delta_g=\chi_{10}, \chi_7\delta_g=\chi_2, \chi_8 \delta_g=\chi_4, \chi_{10}\delta_g=\chi_5.\end{eqnarray*} $

证 容易看出, 对于任意$ g \in S_6$, 有$\chi_ji_g=\chi_j, j=1, 2, \cdots, 11, \ \chi_1 \delta_g=\chi_1$.下面证明后面6个关系式.

记$\Omega = \{(12), (13), (14), (15), (16)\}$.任意$h \in \Omega$, 由于自同构保持元素的阶和共轭类的阶, 因此通过简单计算, 有$\chi_4 \delta_g (h)=\chi_4(g\delta(h)g^{-1})=\chi_4\delta(h)=-2, \ \chi_8(h)=-2$.所以$\chi_4\delta_g =\chi_8$.类似的, 得到$\chi_8\delta_g =\chi_4;\chi_2\delta_g =\chi_7;\ \chi_7\delta_g =\chi_2; \chi_5\delta_g =\chi_{10}; \ \chi_{10}\delta_g =\chi_5.$

由此, 不失一般性, 可设

$ \begin{eqnarray}\label{coz} \chi_{b_1, b_2, \cdots, b_{10}}^a =\left\{\begin{array}{rl} \chi_{11},& \quad 1 \le a \le b_1, \\ \chi_{10},&\quad b_1   a \le b_1+b_2, \\ \chi_9,&\quad b_1+b_2   a \le b_1+b_2+b_3, \\ \chi_8,&\quad \sum\limits_{i=1}^3 b_i   a \le \sum\limits_{i=1}^4 b_i, \\ \chi_7,&\quad \sum\limits_{i=1}^4 b_i   a \le \sum\limits_{i=1}^5 b_i, \\ \chi_6,&\quad \sum\limits_{i=1}^5 b_i   a \le \sum\limits_{i=1}^6 b_i, \\ \chi_5,&\quad \sum\limits_{i=1}^6 b_i   a \le \sum\limits_{i=1}^7 b_i, \\ \chi_4,&\quad \sum\limits_{i=1}^7 b_i   a \le \sum\limits_{i=1}^8 b_i, \\ \chi_3,&\quad \sum\limits_{i=1}^8 b_i   a \le \sum\limits_{i=1}^9 b_i, \\ \chi_2,&\quad \sum\limits_{i=1}^9 b_i   a \le \sum\limits_{i=1}^{10} b_i, \\ \chi_1,&\quad \sum\limits_{i=1}^{10} b_i   a, \\ \end{array}\right. \end{eqnarray} $

(2.1)

这里$a$是一个正整数, $b_1, b_2, \cdots, b_{10}$是非负整数.

设$\mathbf {N}$表示自然数集合, 得到如下结论.

定理3.1 设$G=S_6$是置换群, $m$是自然数, $\mathbf {N}$表示自然数集合, $ r$是$G$的关于$r_{_{C_i}}, i=1, 2, \cdots, 11$的分歧, $Q=(G, r)$是对应的Hopf箭向.如果$r_{_{C_1}}=m>0, r_{_{C_i}}=0, i=2, 3, \cdots, 11$, 那么路余代数$ kQ^c$有不同构的分次Hopf代数结构$kQ^c(\alpha ^{\chi_s}), s\in \mathbf {N}^{10}$, 其个数与不等式$s_1+2s_2+s_3+2s_4+2s_5+s_6+s_9 \le m$的非负整数解的个数相同.记$x_i=a_{(1), (1)}^{(i)}, y_i=a_{(12), (12)}^{(i)}, z_i=a_{(13), (13)}^{(i)}, p_i=a_{(14), (14)}^{(i)}, q_i=a_{(15), (15)}^{(i)}, v_i=a_{(16), (16)}^{(i)}, i=1, 2, \cdots, m.$则在$(kQ_1, \alpha^{\chi_s})$上的$kG$ -模作用为

$ \begin{eqnarray*} &&(12) \cdot x_i=y_i, (13) \cdot x_i=z_i, (14) \cdot x_i=p_i, (15) \cdot x_i=q_i, (16) \cdot x_i=v_i;\\ &&(12) \cdot y_i=x_i, (13) \cdot y_i=a^{(i)}_{(123), (123)}, (14) \cdot y_i=a^{(i)}_{(124), (124)}, \\ &&(15) \cdot y_i=a^{(i)}_{(125), (125)}, (16) \cdot y_i=a^{(i)}_{(126), (126)};\\ &&(12) \cdot z_i=a^{(i)}_{(132), (132)}, (13) \cdot z_i=x_i, (14) \cdot z_i=a^{(i)}_{(134), (134)}, \\ &&(15) \cdot z_i=a^{(i)}_{(135), (135)}, (16) \cdot z_i=a^{(i)}_{(136), (136)};\\ &&(12) \cdot p_i=a^{(i)}_{(142), (142)}, (13) \cdot p_i=a^{(i)}_{(143), (143)}, (14) \cdot p_i=x_i, \\ &&(15) \cdot p_i=a^{(i)}_{(145), (145)}, (16) \cdot p_i=a^{(i)}_{(146), (146)};\\&&(12) \cdot q_i=a^{(i)}_{(152), (152)}, (13) \cdot q_i=a^{(i)}_{(153), (153)}, (14) \cdot q_i=a^{(i)}_{(154), (154)}, \\ &&(15) \cdot q_i=x_i, (16) \cdot q_i=a^{(i)}_{(156), (156)};\\ &&(12) \cdot v_i=a^{(i)}_{(162), (162)}, (13) \cdot v_i=a^{(i)}_{(163), (163)}, (14) \cdot v_i=a^{(i)}_{(164), (164)}, \\ &&(15) \cdot v_i=a^{(i)}_{(165), (165)}, (16) \cdot v_i=x_i. \end{eqnarray*} $

任意$t=2, 3, 4, 5, 6, $

$ \begin{eqnarray*}x_i \cdot (1t)&=&\left\{\begin{array}{rl} -3a^{(i)}_{(1t), (1t)},&\quad s_1 <i \le \sum\limits^3_{i=1} s_i, \\ -2a^{(i)}_{(1t), (1t)},&\quad \sum\limits^3_{i=1} s_i <i \le\sum\limits^4_{i=1} s_i, \\ -a^{(i)}_{(1t), (1t)},&\quad \sum\limits^4_{i=1} s_i <i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i <i \le\sum\limits^6_{i=1} s_i, \\ a^{(i)}_{(1t), (1t)},&\quad \sum\limits^6_{i=1} s_i <i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i <i \le m, \\ 2a^{(i)}_{(1t), (1t)},&\quad \sum\limits^7_{i=1} s_i <i \le\sum\limits^8_{i=1} s_i, \\ 3a^{(i)}_{(1t), (1t)},&\quad \sum\limits^8_{i=1} s_i <i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ y_i \cdot (12)&=&\left\{\begin{array}{rl} -3a_{(1)(1)}^{(i)},&\quad s_1 <i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1)(1)}^{(i)},&\quad \sum\limits^3_{i=1} s_i <i \le\sum\limits^4_{i=1} s_i, \\ -a_{(1)(1)}^{(i)},&\quad \sum\limits^4_{i=1} s_i <i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i <i \le\sum\limits^6_{i=1} s_i, \\ a_{(1)(1)}^{(i)},&\quad \sum\limits^6_{i=1} s_i <i \le\sum\limits^7_{i=1} s_i \mbox{或} \sum\limits^{10}_{i=1}s_i <i \le m, \\ 2a_{(1)(1)}^{(i)},&\quad \sum\limits^7_{i=1} s_i <i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1)(1)}^{(i)},&\quad \sum\limits^8_{i=1} s_i <i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right.\end{eqnarray*} $

任意$t=3, 4, 5, 6, $

$ \begin{eqnarray*}y_i \cdot (1t)&=&\left\{\begin{array}{rl} -3a_{(1t2)(1t2)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1t2)(1t2)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -aa_{(1t2)(1t2)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox {或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1t2)(1t2)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1t2)(1t2)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1t2)(1t2)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ z_i \cdot (13)&=&\left\{\begin{array}{rl} -3a_{(1)(1)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1)(1)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -a_{(1)(1)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1)(1)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1)(1)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1)(1)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right.\end{eqnarray*} $

任意$t=2, 4, 5, 6, $

$ z_i \cdot (1t)=\left\{\begin{array}{rl} -3a_{(1t3)(1t3)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1t3)(1t3)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -aa_{(1t3)(1t3)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1t3)(1t3)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或} \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1t3)(1t3)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1t3)(1t3)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ p_i \cdot (14)=\left\{\begin{array}{rl} -3a_{(1)(1)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1)(1)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -a_{(1)(1)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox {或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1)(1)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1)(1)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1)(1)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right. $

任意$t=2, 3, 5, 6, $

$ \begin{eqnarray*}p_i \cdot (1t)&=&\left\{\begin{array}{rl} -3a_{(1t4)(1t4)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1t4)(1t4)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -aa_{(1t4)(1t4)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1t4)(1t4)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1t4)(1t4)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1t4)(1t4)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ q_i \cdot (15)&=&\left\{\begin{array}{rl} -3a_{(1)(1)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1)(1)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -a_{(1)(1)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1)(1)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1)(1)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1)(1)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right.\end{eqnarray*} $

任意$t=2, 3, 4, 6, $

$ \begin{eqnarray*}q_i \cdot (1t)&=&\left\{\begin{array}{rl} -3a_{(1t5)(1t5)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1t5)(1t5)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -aa_{(1t5)(1t5)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1t5)(1t5)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1t5)(1t5)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1t5)(1t5)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\v_i \cdot (16)&=&\left\{\begin{array}{rl} -3a_{(1)(1)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1)(1)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -a_{(1)(1)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1)(1)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1)(1)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1)(1)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right.\end{eqnarray*} $

任意$t=2, 3, 4, 5, $

$ v_i \cdot (1t)=\left\{\begin{array}{rl} -3a_{(1t6)(1t6)}^{(i)},&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2a_{(1t6)(1t6)}^{(i)},&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -aa_{(1t6)(1t6)}^{(i)},&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ a_{(1t6)(1t6)}^{(i)},&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2a_{(1t6)(1t6)}^{(i)},&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3a_{(1t6)(1t6)}^{(i)},&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right. $

证 设$r$是群$G$的满足$r_{_{C_1}}=m > 0, r_{_{C_i}}=0, i=2, 3, \cdots, 11$的分歧, $Q=(G, r)$是对应的Hopf箭向.则由文献[5]得$^x(Q_1)^x=\{a^{(i)}_{x, x}\mid i=1, 2, \cdots, m\}$.对任意$x, y \in G$, $x \not=y$, 有$^y(Q_1)^x$是空集.显然$Z_{u((1))}=G, \widehat {G}=\{\chi_1, \chi_2, \cdots, \chi_{11}\}$.任意$s=(s_1, s_2, \cdots, s_{10}) \in \mathbf N^{10}, $设$\chi_{_{C_1}}^{(i)}=\chi^{(i)}_{s_1s_2\cdots s_{10}}.$由于$r_{_{C_1}}=m>0, r_{_{C_i}}=0, i=2, 3, \cdots, 11$, 所以$\chi_s=\{\chi_{_{C_1}}^{(i)}\in\widehat{G}\mid 1\leq i\leq m\}$是关于$r$的带有特征标的全部分歧系统, 简记为{\rm RSC}.对于任意$s, l \in \mathbf N^{10}$, 有$\chi_s \cong \chi_l$当且仅当$s=l$或$s_1=l_2, s_2=l_7, s_3=l_3, s_4=l_8, s_5=l_{10}, s_6=l_6, s_7=l_2, s_8=l_4, $ $s_9=l_9, s_{10}=l_5.$

事实上, 如果$s=l$, 显然$\chi_s=\chi_l, $自然有$\chi_s \cong \chi_l$.对于其他情形, 由于

$ \begin{eqnarray*}\chi_{s}&=&\{\underbrace{\chi_{11}, \cdots, \chi_{11}}_{s_1}, \underbrace{\chi_{10}, \cdots, \chi_{10}}_{s_2}, \underbrace{\chi_9, \cdots, \chi_9}_{s_3}, \underbrace{\chi_8, \cdots, \chi_8}_{s_4}, \underbrace{\chi_7, \cdots, \chi_7}_{s_5}, \underbrace{\chi_6, \cdots, \chi_6}_{s_6}, \\ &&\underbrace{\chi_5, \cdots, \chi_5}_{s_7}, \underbrace{\chi_4, \cdots, \chi_4}_{s_8}, \underbrace{\chi_3, \cdots, \chi_3}_{s_9}, \underbrace{\chi_2, \cdots, \chi_2}_{s_{10}}, \underbrace{\chi_1, \cdots, \chi_1}_{m-\sum\limits_{i=1}^{10}s_i} \}, \\ \chi_{l}&=&\{\underbrace{\chi_{11}, \cdots, \chi_{11}}_{l_1}, \underbrace{\chi_{10}, \cdots, \chi_{10}}_{l_2}, \underbrace{\chi_9, \cdots, \chi_9}_{l_3}, \underbrace{\chi_8, \cdots, \chi_8}_{l_4}, \underbrace{\chi_7, \cdots, \chi_7}_{l_5}, \underbrace{\chi_6, \cdots, \chi_6}_{l_6}, \\ &&\underbrace{\chi_5, \cdots, \chi_5}_{l_7}, \underbrace{\chi_4, \cdots, \chi_4}_{l_8}, \underbrace{\chi_3, \cdots, \chi_3}_{l_9}, \underbrace{\chi_2, \cdots, \chi_2}_{l_{10}}, \underbrace{\chi_1, \cdots, \chi_1}_{m-\sum\limits_{i=1}^{10}l_i} \}.\end{eqnarray*} $

由定理2.3, 任意$g \in G, \chi_1 \delta_g=\chi_1, \chi_2 \delta_g=\chi_7, \chi_4\delta_g=\chi_8, \chi_5\delta_g=\chi_{10}, \chi_7\delta_g=\chi_2, $ $\chi_8\delta_g=\chi_4, \chi_{10}\delta_g=\chi_5, $有

(ⅰ) $\delta_g:G \to G$是一个群同构.

(ⅱ) 任意$\alpha, \beta \in G, $显然$\delta(\alpha \beta)=\delta(\alpha)\delta(\beta).$

固定映射$u:K(G) \to G, $

$ \begin{eqnarray*}&&u(C_1)=(1), u(C_2)=(12), u(C_3)=(123), u(C_4)=(1234), \\&&u(C_5)=(12345), u(C_6)=(123456), u(C_7)=(12)(34), u(C_8)=(123)(45), \\ &&u(C_9)=(1234)(56), u(C_{10})=(12)(34)(56), u(C_{11})=(123)(456).\end{eqnarray*} $

任意$C_i \in K(G), g \in G$, 存在元素$h_{_{C_i}}\in G$使得

$ \delta_g(h_{_{C_i}}u(C_i)h_{_{C_i}}^{-1})=u(\delta_g(C_i)), i=1, 2, \cdots, 11. $

事实上, 有

$ \begin{eqnarray*}&&h_{_{C_1}}=(1), h_{_{C_2}}=i^{-1}(\sigma^{-1}g^{-1}(156432)\sigma), h_{_{C_3}}= i^{-1}(\sigma^{-1}g^{-1}(2463)\sigma), \\&&h_{_{C_4}}= i^{-1}(\sigma^{-1}g^{-1}(15462)\sigma), h_{_{C_5}}= i^{-1}(\sigma^{-1}g^{-1}\sigma), h_{_{C_6}}= i^{-1}(\sigma^{-1}g^{-1}(245)(63)\sigma), \\ &&h_{_{C_7}}= i^{-1}(\sigma^{-1}g^{-1}(264)(35)\sigma), h_{_{C_8}}= i^{-1}(\sigma^{-1}g^{-1}(26354)\sigma), h_{_{C_9}}= i^{-1}(\sigma^{-1}g^{-1}(2453)\sigma), \\ &&h_{_{C_{10}}}= i^{-1}(\sigma^{-1}g^{-1}(13)\sigma), h_{_{C_{11}}}= i^{-1}(\sigma^{-1}g^{-1}(26)\sigma).\end{eqnarray*} $

(ⅲ) 任意$C\in K(G)$, 存在双射$f_{C_1} : I_{C_1}(r) \rightarrow I_{\delta_g(C_1)}(r)$使得

$ \begin{eqnarray*}&&f_{C_1}(1, \cdots, s_1, s_1+1, \cdots, s_1+s_2, (\sum\limits_{i=1}^2 s_i)+1, \cdots, \sum\limits_{i=1}^3 s_i, (\sum\limits_{i=1}^3 s_i) +1, \cdots, \sum\limits_{i=1}^4 s_i, \\&&(\sum\limits_{i=1}^4 s_i)+1, \cdots, \sum\limits_{i=1}^5 s_i, (\sum\limits_{i=1}^5 s_i)+1, \cdots, \sum\limits_{i=1}^6 s_i, (\sum\limits_{i=1}^6 s_i)+1, \cdots, \sum\limits_{i=1}^7 s_i, (\sum\limits_{i=1}^7 s_i)+1, \cdots, \\ &&\sum\limits_{i=1}^8 s_i, (\sum\limits_{i=1}^8 s_i)+1, \cdots, \sum\limits_{i=1}^9 s_i, (\sum\limits_{i=1}^9 s_i)+1, \cdots, \sum\limits_{i=1}^{10} s_i, (\sum\limits_{i=1}^{10} s_i)+1, \cdots, m)\\ &=&(1, \cdots, l_1, (\sum\limits_{i=1}^6 l_i)+1, \cdots, \sum\limits_{i=1}^7 l_i, (\sum\limits_{i=1}^2 l_i)+1, \cdots, \sum\limits_{i=1}^3 l_i, (\sum\limits_{i=1}^7 l_i)+1, \cdots, \sum\limits_{i=1}^8 l_i, \\ &&(\sum\limits_{i=1}^9 l_i)+1, \cdots, \sum\limits_{i=1}^{10} l_i, (\sum\limits_{i=1}^5 l_i)+1, \cdots, \sum\limits_{i=1}^6 l_i, l_1+1, \cdots, l_1+l_2, (\sum\limits_{i=1}^3 l_i) +1, \cdots, \\ &&\sum\limits_{i=1}^4 l_i, (\sum\limits_{i=1}^8 l_i)+1, \cdots, \sum\limits_{i=1}^9 l_i, (\sum\limits_{i=1}^4 l_i)+1, \cdots, \sum\limits_{i=1}^5 l_i, (\sum\limits_{i=1}^{10} l_i)+1, \cdots, m), \\ &&f_{C_i}:I_{C_i}(r) \rightarrow I_{\delta_g(C_i)}(r), i=2, 3, \cdots, 11\end{eqnarray*} $

是空映射.由定理2.3, 对任意$h \in Z_{u(C_1)}$,

$ i\in I_{C_1}(r), \chi_{\delta_g(C_1)}^{f_{C_1}(i)}(\delta_g(h_{_{C_1}}^{-1}hh_{_{C_1}})=\chi_{\delta_ g(C_1)}^{f_{C_1}(i)}\delta_g(h) =\chi_{_{C_1}}^{i}(h). $

故$\chi_s\cong \chi_l.$

由此, $\{\chi_s| s \in \mathbf N^{10} \}$是关于$r$的互不同构的所有的RSC, 该集合的基数恰好等于不等式$s_1+2s_2+s_3+2s_4+2s_5+s_6+s_9 \le m$的非负整数解的个数.由于域$k$的特征char$(k)\not=2$, 所有右$kG$ -模是逐点的.由文献[5]的定理2.2, 得到路余代数$kQ^c$的不同构的余路Hopf代数结构$kQ^c(\alpha ^{\chi_s}), s\in \mathbf {N}^{10}$.设$s\in \mathbf {N}^{10}$.为简便起见, 记

$ x_i=a_{(1), (1)}^{(i)}, y_i=a_{(12), (12)}^{(i)}, z_i=a_{(13), (13)}^{(i)}, p_i=a_{(14), (14)}^{(i)}, q_i=a_{(15), (15)}^{(i)}, v_i=a_{(16), (16)}^{(i)}, $

$i=1, 2, \cdots, m.$由文献[5]的等式(2.2), 得到$(kQ_1, \alpha^{\chi_s})$上的所有$kG$ -模作用.证毕.

推论3.2 设$kQ^c(\alpha ^{\chi_s}), s \in \mathbf {N}^{10}$如定理3.1中所述, 则$ kQ^c(\alpha ^{\chi_s})$的子Hopf代数$ kG[kQ_1;\alpha^{\chi_s}]$由$(12), (13), (14), (15), (16), x_i, y_i, z_i, p_i, q_i, v_i, i=1, 2, \cdots, m$生成, 生成关系为

$ x_i \cdot (12)=\left\{\begin{array}{rl} -3y_i,&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2y_i,&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -y_i,&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ y_i,&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ y_i,&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ y_i,&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ x_i \cdot (13)=\left\{\begin{array}{rl} -3z_i,&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2z_i,&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -z_i,&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ z_i,&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ z_i,&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ z_i,&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ x_i \cdot (14)=\left\{\begin{array}{rl} -3p_i,&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2p_i,&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -p_i,&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox {或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ p_i,&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2p_i,&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3p_i,&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ x_i \cdot (15)=\left\{\begin{array}{rl} -3q_i,&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2q_i,&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -q_i,&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ q_i,&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或} \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2q_i,&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3q_i,&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right. $

$ \begin{eqnarray*}x_i \cdot (16)&=&\left\{\begin{array}{rl} -3v_i,&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2v_i,&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -v_i,&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或 } 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ v_i,&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2v_i,&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3v_i,&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i, \\ \end{array}\right.\\ &&x_i \cdot x_j=x_j \cdot x_i, 1 \le i, j \le m.\\ &&z_i \cdot (13)=p_i \cdot (14)=q_i \cdot (15)=v_i \cdot (16)=y_i \cdot (12)\\ &=&\left\{\begin{array}{rl} -3x_i,&\quad s_1  i \le \sum\limits^3_{i=1} s_i, \\ -2x_i,&\quad \sum\limits^3_{i=1} s_i  i \le\sum\limits^4_{i=1} s_i, \\ -x_i,&\quad \sum\limits^4_{i=1} s_i  i \le\sum\limits^5_{i=1} s_i \mbox { 或} 1 \le i\le s_1, \\ 0,&\quad \sum\limits^5_{i=1} s_i  i \le\sum\limits^6_{i=1} s_i, \\ x_i,&\quad \sum\limits^6_{i=1} s_i  i \le\sum\limits^7_{i=1} s_i \mbox{ 或 } \sum\limits^{10}_{i=1}s_i  i \le m, \\ 2x_i,&\quad \sum\limits^7_{i=1} s_i  i \le\sum\limits^8_{i=1} s_i, \\ 3x_i,&\quad \sum\limits^8_{i=1} s_i  i \le\sum\limits^{10}_{i=1} s_i.\\ \end{array}\right.\end{eqnarray*} $

余代数结构为$\Delta ((1t))=((1t)) \otimes ((1t)), \varepsilon ((1t))=1, S((1t))=(1t), t=2, 3, 4, 5, 6, $ $\Delta(w)=(1)\otimes w+w \otimes (1), \varepsilon(w)=0, S(w)=-w, $这里$w=x_i, y_i, z_i, p_i, q_i, v_i$, 而

$ \begin{eqnarray*}&&x_i=a_{(1), (1)}^{(i)}, y_i=a_{(12), (12)}^{(i)}, z_i=a_{(13), (13)}^{(i)}, p_i=a_{(14), (14)}^{(i)}, \\ &&q_i=a_{(15), (15)}^{(i)}, v_i=a_{(16), (16)}^{(i)}, i=1, 2, \cdots, m.\end{eqnarray*} $

证 由文献[15]中一型路代数的乘法关系, 经过计算, 容易得出上述所有关系.