陳全國(guó)

(伊犁師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,新疆伊寧835000)

1 引言及預(yù)備知識(shí)

為了構(gòu)造 Knizhnik-Zamolodchikov的解,V.G.Drinfeld[1]引入了擬Hopf代數(shù).擬Hopf代數(shù)與通常的Hopf代數(shù)在定義準(zhǔn)則方面存在很大區(qū)別,主要表現(xiàn)在余結(jié)合律不再滿(mǎn)足,其余結(jié)合性通過(guò)一個(gè)可逆元素φ∈H?H?H連接起來(lái),還有擬Hopf代數(shù)的定義不是自對(duì)偶的,導(dǎo)致無(wú)法定義擬Hopf代數(shù)上的余模.類(lèi)似于Hopf代數(shù),擬Hopf代數(shù)的表示范疇是辮子張量范疇.

在Hopf代數(shù)中,研究Doi-Hopf模是非常有意義的,因?yàn)樗y(tǒng)一了各種Hopf模,例如:Sweedler Hopf模、Doi相對(duì) Hopf模[2]、Takeuchi相對(duì) Hopf模[3]以及Yetter-Drinfeld模.作為推廣,D.Bulacu等[4]在擬Hopf代數(shù)情形中引入了Doi-Hopf模.因此,本文以此類(lèi)模為研究對(duì)象.

可分函子的概念由 C.Nǎstǎsescu 等[5]引入,并在不同的框架內(nèi)得到充分的研究[5-11].但是,在擬Hopf代數(shù)中,有關(guān)刻畫(huà)Doi-Hopf模的半單性的結(jié)果很少,其原因歸結(jié)為擬Hopf代數(shù)的復(fù)雜結(jié)構(gòu),導(dǎo)致構(gòu)造上的困難.目前,有關(guān)擬Hopf代數(shù)的結(jié)果主要集中于文獻(xiàn)[12-14]中.本文在已有工作基礎(chǔ)之上,進(jìn)一步研究擬Hopf代數(shù)中的Doi-Hopf模.本文的出發(fā)點(diǎn)就是證明Doi-Hopf模的Maschke型定理,其問(wèn)題歸結(jié)為刻畫(huà)忘卻函子(忽略余作用)的可分性.

本文中假設(shè)k為一個(gè)域.所有代數(shù)、余代數(shù)的張量積均在k上.下面簡(jiǎn)單介紹一些本文中涉及到的概念和符號(hào),更多的有關(guān)擬Hopf代數(shù)中概念和結(jié)果,參見(jiàn)文獻(xiàn)[4,12-14].

2 主要結(jié)果

[1]Drinfeld V G.Quasi-Hopf algebras[J].Leningrad Math J,1990,1:1419-1457.

[2]Doi Y.On the structure of relative Hopf algebras[J].Commun Algebra,1981,11:31-50.

[3]Takeuchi M.A correspondence between Hopf ideals and sub-Hopf algebras[J].Manuscripta Mathematica,1972,7:251-270.

[4]Bulacu D,Caenepeel S.Two-sided two-cosided Hopf modules and Doi-Hopf modules for Quasi-Hopf algebras[J].J Algebra,2003,270:55-95.

[5]Nǎstǎsescu C,Van den Bergh M,Van Oystaeyen F.Separable functors applied to graded rings[J].J Algebra,1989,123:397-413.

[6]Caenepeel S,Ion B,Militaru G,et al.Separable functors for the category of Doi-Hopf modules,applications[J].Adv Math,1999,145:239-290.

[7]Casta~no Iglesias F,Gòmez-Torrecillas J,Nǎstǎsescu C.Separable functors in coalgbras applications[J].Tsukuba J Math,1997,21:329-344.

[8]Casta~no Iglesias F,Gòmez-Torrecillas J,Nǎstǎsescu C.Separable functors in graded rings[J].J Pure Appl Algebra,1998,127:219-230.

[9]Brzezi'nski T.The structure of corings:induction functors,Maschke-type theorem,and Frobenius and Galois properties[J].Algebra Represent Theory,2002,5:389-410.

[10]Chen Q G,Wang S H.Separable functors for the category of Doi-Hopf group modules[J].Abh aus dem Math Seminar der Univ Hamburg,2011,81:261-272.

[11]Chen Q G,Wang D G.The category of partial Doi-Hopf modules and functors[J].Rendiconti di Padova,2013,129:189-204.

[12]Bulacu D,Nauwelaerts E.Relative Hopf modules for(dual)quasi-Hopf algebras[J].J Algebra,2000,229:632-659.

[13]Bulacu D,Caenepeel S,Torrecillas B.Involutory Quasi-Hopf algebras[J].Algebra Represent Theory,2009,12:257-285.

[14]Hausser F,Nill F.Diagonal crossed products by duals of quasi-quantum groups[J].Rev Math Phys,1999,11:553-629.

[15]陳全國(guó),王栓宏.弱Doi-Hopf群模的Maschke型定理[J].四川師范大學(xué)學(xué)報(bào):自然科學(xué)版,2012,35(5):615-617.