Cat弱Hopf代數

陳 笑 緣

(浙江商業職業技術學院， 浙江 杭州 310053 )

Cat弱Hopf代數

陳 笑 緣

(浙江商業職業技術學院， 浙江 杭州 310053 )

首先引入pre-cat弱Hopf代數和cat弱Hopf代數來刻畫具有投射的弱Hopf代數的性質，并建立pre-cat弱Hopf代數的張量范疇，證明了pre-cat弱Hopf代數是cat弱Hopf代數的充要條件，從而推廣了LODAY引入的cat-群和cat Hopf代數的相應結論.

弱Hopf代數；投射；cat 弱Hopf代數

定義1 范疇C的一個fork是指圖

(1)

其中，f°i=g°i.fork可分指在范疇C中存在態射h:B→A和p:A→I滿足h°f=idA,h°g=i°p及p°i=idI.

例1 下式是可分fork，

(2)

∑af(11)SA(fg(f(12)))=

∑af(11)SA(f(12))=a;

∑a1f(SH(g(a2))g(a3)l)=

證明 定義I的代數、余代數和對極分別為μI(x?y)=x·y=p(i(x)i(y)),ηI=p(1A),ΔI(x)=∑p(i(x)1)?p(i(x)2),εI(x)=ε(i(x)),SI(x)=pSA(i(x)).

首先，驗證I是一個代數且i是代數同態.事實上，對任意x,y,z∈I，有

(x·y)·z=p(h(gi(x)gi(y))i(z))=

p(h(fi(x)fi(y))i(z))=p((i(x)i(y))i(z))=

p(i(x)(i(y)i(z)))=x·(y·z);

x·p(1)=p(i(x)ip(1))=p(i(x)hg(1))=

p(i(x))=x;

i(x·y)=h(gi(x)gi(y))=h(fi(x)fi(y))=

hf(i(x)i(y))=i(x)i(y);

i(1I)=ip(1A)=hg(1A)=hf(1A)=1A.

其次，ΔI顯然是余結合的且εI(1I)=ε(i(1A))=1.并且i是余代數同態，因為對任意x∈I有

(i?i)ΔI(x)=∑ip(i(x)1)?ip(i(x)2)=

∑h(gi(x)1)?h(gi(x)2)=∑hf(i(x)1)?

hf(i(x)2)=∑i(x)1?i(x)2.

所以只需驗證I是一個弱雙代數.因為對任意x,y,z∈I，有

ΔI(x)·ΔI(y)=∑p(h(gi(x)1gi(y)1))?p(h(gi(x)2gi(y)2))=∑p(hf(i(x)1)hf(i(y)1))?p(hf(i(x)2)hf(i(y)2))=∑p(i(x)1i(y)1)?p(i(x)2i(y)2)=ΔI(x·y);

εI(x·y1)εI(y2·z)=∑εA(h(g(i(x))gip(i(y)1)))εA(h(gip(i(y)2))gi(z))=∑εA(hf(i(x)ip(i(y)1)))εA(hf(ip(i(y)2)i(z)))=

∑εA(i(x)i(y)1)εA(i(y)2i(z))=

εA(i(x)i(y)i(z))=εI(x·y·z).

同理可證εI(x·y2)εI(y1·z)=εI(x·y·z).

(ΔI(p(1))?p(1))·(p(1)?ΔI(p(1)))=

(p(1)?ΔI(p(1)))·(ΔI(p(1))?p(1)).

最后，證明I是弱Hopf代數，i是弱Hopf代數同態.實際上，對任意x∈I，有

∑SI(x1)·x2·SI(x3)=

∑p(ip(SA(i(x1))i(x2))ip(SA(i(x3))))=

∑p(h(SB(gi(x1))gi(x2))hSB(gi(x3)))=

∑p(hf(SA(i(x1))i(x2))hf(SA(i(x3))))=

∑p(SA(i(x1))i(x2)SA(i(x3)))=

p(SA(i(x)))=SI(x);

i(SI(x))=ip(SA(i(x)))=hSA(gi(x))=hf(SA(i(x)))=SA(i(x));

∑x1·SI(x2)=

∑p(ip(i(x)1)ip(SA(ip(i(x)2))))=

∑p(i(x1)hSB(gi(x2)))=

∑p(i(x1)hf(SA(i(x2)))=

∑p(i(x1)SA(i(x2))=p(i(x)l);

∑εI(p(1)1·x)p(1)2=

∑εA(ip(11)i(x))p(12)=

∑εA(11i(x))p(12)=p(i(x)l),

因此，x1·SI(x2)=∑εI(p(1)1·x)p(1)2.同理可證SI(x1)·x2=∑εI(x·p(1)2)p(1)1.證畢.

α°γ=β°γ=idH;

(3)

∑α(a2)?a1=∑α(a1)?a2,a∈A;

(4)

∑β(a2)?a1=∑β(a1)?a2,a∈A.

(5)

(6)

γA(βB(b1)lβB(b2))?b3=

∑a1γA(SH(αA(a4)))1γA(βB(b1))?αA(a2γA(SH(αA(a4)))2)βB(b2)?b3=

∑a1γA(SH(12αA(a2)))γA(βB(b1))?SH(11)βB(b2)?b3=

hA,B°(idA?((βB?idB)°ΔB))(a?b)=

hA,B°(((idA?αA)°ΔA)?idB)(a?b)=

∑a11γA(SH(αA(12)))?b=

∑aγA(11)γA(SH(αA(γA(12))))?b=a?b.

顯然，映射((idA?αA)°ΔA)?idB和idA?((βB?idB)°ΔB)均為弱Hopf代數同態，由定理1可知A?HB是弱Hopf代數.

最后，驗證A?HB是pre-cat弱Hopf代數.定義2中的條件(4)和(5)顯然成立，只需證明條件(3)成立.因為對任意h∈H，有

(βA?εB)°iA,B°γA?HB(h)=

∑βA(γA(h1))βAγA(SH(αA(γA(h2))))h3=

∑h1SH(h2)h3=h.

定理2證畢.

范疇CH的對象是H上的pre-cat弱Hopf代數，態射是pre-cat弱Hopf代數同態.則有以下結論.

定理3 范疇CH是monoidal范疇.

φX=(idX?αX)°ΔX,ψX=(βX?idX)°ΔX):

A?H?B?H?C,

(7)

A?H?B?H?C.

(8)

(9)

(10)

∑a1γA(SH(12αA(a2)))γA(x1)?SH(11)x2=

(2)μA°(βi?iα)=μA°τA,A°(βi?iα).

證明 (1)?(2)的證明.因為m是代數同態，所以有m°μA?HA((1?a)?(b?1))=μA°(m?m)((1?a)?(b?1)).方便起見，將iA,A(a?Hb)記為∑a*?b*，進一步有，

μA°(m?m)((1?a)?(b?1))=

故

μA°(βi?iα)=μA°τA,A°(βi?iα).

(2)?(1)的證明.首先證明m是代數同態.事實上，對任意a,b,x,y∈A,有

μA°(m?m)((a?Hb)?(x?Hy));

m°ηA?HA=

∑11γA(S(αA(12)))=γA(1)=1.

αA°m(a?Hb)=

∑γA(h1)γA(S(αA(γA(h2))))γA(h3)=

∑γA(h1)γA(S(h2))γA(h3)=γA(h).

再者，必須證明m°SA?HA=S°m.事實上，對任意a,b∈A和h∈H，有

γA(S(βA(S(b2)1)))S(b2)2=

∑S(γA(βA(b1)))γA(S(βA(S(b2)1)))S(b2)2×

∑a1γA(S(αA(a2))βA(b1))γA(S(βA(b1)))b3=

因而有

∑a1γA(S(αA(a4)))b1γA(S(αA(b2)))×

γA(S(αA(a2γA(αA(a3)))))c=∑a1γA(S(αA(a3)))b1γA(S(αA(b2)))γA(S(αA(a2))l)c=

∑a1γA(S(αA(a2)))b1γA(S(αA(b2)))c=

定理4證畢.

[1]LODAYLJ.Spaceswithfinitelymanynontrivialhomotopygroups[J]. J Pure Appl Algebra,1982,24(2):179-202.

[4] BOHM G, NILL F, SZLACHANYI K. Weak Hopf Algebras (I): Integral theory andC*-structure[J].Journal of Algebra,1999,221(2):385-438.

[5] BESPALOV Y. Crossed modules and quantum groups in Braided Categories[J]. Appl Categ Structure,1997,5(2):155-204.

CHEN Xiaoyuan

(ZhejiangBusinessCollege,Hangzhou310053,China)

In this paper, we first introduce the notions of pre-cat weak Hopf algebras and cat weak Hopf algebras to characterize the structures of weak Hopf algebras with projections. Then, we give the monoidal category of these objects which generalize the results of cat Hopf algebras and cat-groups introduced by LODAY.

weak Hopf algebra; projection; cat weak Hopf algebra

2015-01-21.

陳笑緣(1963-)，ORCID:http://orcid.org/0000-0003-2898-9976,女，教授，主要從事代數學研究,E-mail:cxy5988@sina.com.

10.3785/j.issn.1008-9497.2017.02.010

O 153.3

A

1008-9497(2017)02-181-05

Cat weak Hopf algebras. Journal of Zhejiang University(Science Edition), 2017,44(2):181-185