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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(8),2822-2833
PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.108294
Hom-Hopf“êþ
{È
ZZZ§§§½½½[[[ÂÂÂ
∗
§§§444
úô“‰ŒÆêƆOŽÅ‰ÆÆ§úô7u
ÂvFϵ2021c718F¶¹^Fϵ2021c87F¶uÙFϵ2021c819F
Á‡
•ïÄHom-{ȧÏL$^a'gŽ•{§½ÂHom-{ȧ¿ÏLOŽ‰Ñ
eZHom-{Èƒ'5Ÿ"Š•A^§Hom-{È¤Hom-{“ê¿‡
^‡§„kHom-{ÈÚHomsmashÈ/¤Hom-V“ê¿©7‡^‡"
'…c
Hom-{“ê§Hom-V“ê§HomSmashȧHom-V“ê
CrossedCoproductsoverHom-Hopf
Algebras
TianWang,JiafengLv
∗
,LingLiu
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Jul.18
th
,2021;accepted:Aug.7
th
,2021;published:Aug.19
th
,2021
∗ÏÕŠö"
©ÙÚ^:Z,½[Â,4 .Hom-Hopf“êþ{È[J].A^êÆ?Ð,2021,10(8):2822-2833.
DOI:10.12677/aam.2021.108294
Z
Abstract
InordertostudytheHom-crossed-coproduct,wedefinetheHom-crossed-coproduct
byanalogy,andgivesomepropertiesofHom-crossed-coproductbycalculation.As
anapplication,weobtainthenecessaryandsufficientconditionsforHom-crossed-
coproducttoformHom-coalgebra,andthenecessaryandsufficientconditionsfor
Hom-crossed-coproductandHom-smash-producttoformHom-bialgebra.
Keywords
Hom-Coalgebra,Hom-Bialgebra,HomSmashProduct,Hom-BimoduleAlgebra
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
Hom-.“ê•@†q-/CWitt“ê[1]Úq-/CVirasoro“ê[2]ƒ',̇A^uÔn
Æ|nØ.2008 c,MakhloufÚSilvestrov3©z[3]¥ïÄ[o“êžÚ\Hom-“êV
g.Hom-“êÚ\¢Sþ´í2“êVg,r“ê¥(Ü5{KC¤Hom-(Ü5^‡,
=α(a)(bc) = (ab)α(c).‘XHom-“êïÄ\,Hom-“ê9Ùƒ'(CƒÉ•H.˜
Æö[4–6]UYÚ\Hom-{“ê,Hom-V“êÚHom-Hopf“ê.d,Yau[7]r˜Š^Ú
{Š^•ÄùHom-(¥:Hom-,Hom-{,Hom-HopfÚHom-“ê,¿3©z[5]¥
ïÄHom-HopfÄ(½n.
{Ș†´Hopf“ê¥-‡ïÄé–,åu+Ø¥.1997c,WangS.H.,Jiao
Z.M.ÚZhaoW.Z.3©z[8]¥Õá/r+{ÈnØí2Hopf“êþ,‰Ñ
{È½Â¿ïÄÙ5Ÿ.¿…ïÄ{È´{“ê¿‡^‡.{È´
smashÈí2,Ù½ÂXe:H´Hopf“êÚC´{“ê,H3Cþk†f{Š^,
ρ: C→H⊗C,ρ(c) =
P
c
(−1)
⊗c
(0)
.ψ: C→H⊗H´˜‡‚5N.ψ(c) =
P
c
0
⊗c
00
.KC†H
{Ƚ••þ˜mC⊗
ψ
H9Ù{¦{
∆(c⊗h) =
X
c
1
⊗c
2(−1)
c
3
0
h
1
⊗c
2(0)
⊗c
3
00
h
2
DOI:10.12677/aam.2021.1082942823A^êÆ?Ð
Z
Ù¥,é?¿a∈C,h∈H.
©/ÏHom-V“ê,½ÂHom-V“êþ{È,¿?ØÙ5Ÿ,‰ÑHom{
ÈÚHomsmashÈ/¤Hom-V“ê¿©7‡^‡.
2.ý•£
©¤9•þ˜m!ÜþÈ!!‚5NÑ´3ê•kþ?1ïÄ.©¥÷^
Sweedler'u{“ê{¦Ú{P{.éu{“êC,?¿c∈C,·‚P§{¦•
∆(c) =c
1
⊗c
2
.'umC-{ M,?¿m∈M,{Š^P•ρ(m)= m
0
⊗m
1
.,,I´‚
5˜mVþðN,α,β´Œ_N.
½Â1 [7]V´‚5˜m, µ: V⊗V→V, x⊗y7→xy, α: V→VÑ´‚5N.XJé?
¿x,y,z∈V,÷vHom-(Ü^‡:
µ(α(x)⊗µ(y⊗z)) = µ(µ(x⊗y)⊗α(z)),
@o¡n|(V,µ,α)´Hom-“ê.XJk‚5Nη: k→V÷v
µ(η(1)⊗I(x)) = α(x) = µ(I(x)⊗η(1)),
@o¡V´kü Hom-“ê.
(V,µ,α)Ú(V
0
,µ
0
,α
0
)Ñ´Hom-“ê.XJ‚5Nf: V→V
0
÷v
f◦α= α
0
◦f,µ
0
◦(f⊗f) = f◦µ,
@o¡‚5Nf: V→V
0
´Hom-“êÓ.
½Â2 [7]V´‚5˜m, ∆ : V→V⊗V,β: V→VÑ´‚5N.XJ
(β⊗∆)◦∆ = (∆⊗β)◦∆,
@o¡n|(V,∆,β)´Hom-{“ê.XJk‚5Nε: V→k,÷v
(I⊗ε)◦∆ = β= (ε⊗I)◦∆,
@o¡V´k{ü Hom-{“ê.
(V,∆,β)Ú(V
0
,∆
0
,β
0
)Ñ´Hom-{“ê.XJ‚5Nf: V→V
0
÷v
f◦β= β
0
◦f,∆
0
◦f= (f⊗f)◦∆,
@o¡‚5Nf: V→V
0
´Hom-{“êÓ.
½Â3 [7]
DOI:10.12677/aam.2021.1082942824A^êÆ?Ð
Z
1)(V,µ,α,η)´kü ηHom-“ê;
2)(V,∆,α,ε)´k{ü εHom-{“ê;
3)‚5N∆ÚεÑ´Hom-“êÓ,=



























∆(e
1
)= e
1
⊗e
1
,e
1
= η(1),
∆(µ(x⊗y))= ∆(x)∆(y),
ε(e
1
)= 1,
ε(µ(x⊗y))= ε(x)ε(y),
∆(α(x))= α(x
1
)α(x
2
),
ε◦α(x)=ε(x).
K¡8|(V,µ,α,η,∆,ε)´Hom-V“ê.
½Â4[7](A,α)´Hom-“ê,M´‚5˜m,…β:M→M´‚5N.XJk‚5N
ϕ: A⊗M→M,a⊗m7→a·m,é?¿a,b∈A,m∈M,÷v
α(a)·(b·m) = (ab)·β(m),1·m= β(m),
β(a·m) = α(a)·β(m),
@o¡(M,β)´†(A,α)-Hom-.
aq/,·‚Œ±½Âm(A,α)-Hom-.(M,β
M
)Ú(N,β
N
)´ü‡†(A,α)-Hom-.X
Jé?¿a∈A,m∈M,‚5Nf: M→N÷v
f(a·m) = a·f(m),f◦β
M
= β
N
◦f,
@o¡‚5Nf: M→N´†A-Ó.
½Â5[7](H,β)´Hom-V“ê,(A,α)´Hom-“ê.XJk‚5Nρ:H⊗A→
A,h⊗a⊗7→h·a,é?¿h,g∈H,a∈A,÷v
(hg)·α(a) = β(h)·(g·a),1·a= α(a),
α(h·a) = β(h)·α(a),
β
2
(h)·(ab) = (h
1
·a)(h
2
·b),h·1 = ε
H
(h)1,
@o¡(A,α)´†(H,β)-Hom-“ê.
3.Hom-Hopf“êþ{È
̇0Hom-{È½Â,‰ÑHom-{È¤{“ê¿‡^‡,„‰Ñ
DOI:10.12677/aam.2021.1082942825A^êÆ?Ð
Z
Hom-{ÈÚHomsmashÈ/¤Hom-V“ê¿©7‡^‡.
½Â1(H,β)´Hom-V“ê,(C,α)´Hom-{“ê,XJ•3‚5Nρ:C→
H⊗C,ρ(c) =
P
c
(−1)
⊗c
(0)
,é?¿c∈C,÷v
1)
P
(c
(−1)
)c
(0)
= α(c),
2)
P
c
(−1)
(c
(0)
) = (c)1
H
,
3)
X
β
2
(c
(−1)
)⊗c
(0)1
⊗c
(0)2
=
X
c
1(−1)
c
2(−1)
⊗c
1(0)
⊗c
2(0)
,(1)
Ù¥∆(c) =
P
c
1
⊗c
2
.
@o¡•(H,β)3(C,α)þ†fHom-{Š^.
½Â2(H,β)´Hom-V“ê,(C,α)´Hom-{“ê.(H,β)3(C,α)þ†fHom-{Š^
ρ:C→H⊗C,Pρ(c)=
P
c
(−1)
⊗c
(0)
,(C,α)•†H-Hom-{“ê.ψ:C→H⊗H,´˜‡
‚5N.ψ(c) =
P
c
0
⊗c
00
.Š••þ˜mC⊗
ψ
H= C⊗H.éc∈C,h∈H,½ÂHom-{¦Ú
Hom-{ü Xe:
∆(c⊗h) =
X
α
−2
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))β
−1
(h
1
)⊗α
−2
(c
12(0)
)⊗β
−1
(c
2
00
)β
−1
(h
2
)
(c⊗h) = (c)(h)
½n1C⊗
ψ
H¤Hom-{“ê¿‡^‡´ψ÷ve^‡:
1)
P
(c
0
)c
00
=
P
c
0
(c
00
) = (c)1
H
2)
X
c
1(−1)
β(c
2
0
)⊗β
−1
(c
1(0)(−1)
)β(c
2
00
)⊗c
1(0)(0)
=
X
β(c
1
0
)β
−1
(c
2(−1)1
)⊗β(c
1
00
)β
−1
(c
2(−1)2
)⊗α(c
2(0)
)(2)
3)
X
c
1(−1)
β(c
2
0
)⊗c
1(0)
0
c
2
00
1
⊗c
1(0)
00
c
2
00
2
=
X
β(c
1
0
)c
2
0
1
⊗β(c
1
00
)c
2
0
2
⊗β
2
(c
2
00
)(3)
DOI:10.12677/aam.2021.1082942826A^êÆ?Ð
Z
y²Äk,bþªÑ¤á,ky²{ü ,é?¿c∈C,h∈H,k
(id⊗)∆(c⊗h)
=(id⊗)(
X
α
−1
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))β
−1
(h
1
)⊗α
−2
(c
12(0)
)⊗β
−1
(c
2
00
)β
−1
(h
2
))
=
X
α
−1
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))β
−1
(h
1
)⊗(c
12(0)
)⊗(c
2
00
)(h
2
)
=
X
α
−1
(c
11
)⊗(c
12
)1
H
(c
2
)β(h) =
X
c
1
⊗(c
2
)β(h) = α(c)⊗β(h).
(⊗id)∆(c⊗h)
=(⊗id)(
X
α
−1
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))β
−1
(h
1
)⊗α
−2
(c
12(0)
)⊗β
−1
(c
2
00
)β
−1
(h
2
))
=
X
(c
11
)⊗(c
12(−1)
)(c
2
0
)(h
1
)⊗α
−2
(c
12(0)
)⊗β
−2
(c
2
00
)β
−1
(h
2
)
=
X
(c
11
)⊗α
−1
(c
12
)⊗(c
2
)β(h)
=
X
c
1
⊗(c
2
)β(h)
=α(c)⊗β(h).
2y²{(Ü5
(α⊗∆)∆(c⊗h)
=(α⊗∆)(
X
α
−1
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))β
−1
(h
1
)⊗α
−2
(c
12(0)
)⊗β
−1
(c
2
00
)β
−1
(h
2
))
=
X
c
11
⊗(β
−3
(c
12(−1)
)β
−1
(c
2
0
))h
1
⊗α
−3
(c
12(0)11
)⊗(β
−6
(c
12(0)12(−1)
)β
−4
(c
12(0)2
0
))
(β
−2
(c
2
00
1
)β
−2
(h
21
))⊗α
−4
(c
12(0)12(0)
)⊗β
−3
(c
12(0)2
00
)(β
−2
(c
2
00
2
)β
−2
(h
22
))
=
X
c
11
⊗(β
−3
(c
12(−1)
)β
−1
(c
2
0
))h
1
⊗α
−2
(c
12(0)1
)⊗(β
−6
(c
12(0)21(−1)
)β
−5
(c
12(0)22
0
))
(β
−2
(c
2
00
1
)β
−2
(h
21
))⊗α
−4
(c
12(0)21(0)
)⊗β
−4
(c
12(0)22
00
)(β
−2
(c
2
00
2
)β
−2
(h
22
))
(1)
=
X
c
11
⊗((β
−5
(c
121(−1)
β
−5
(c
122(−1)
))β
−1
(c
2
0
))h
1
⊗α
−2
(c
121(0)
)⊗(β
−6
(c
122(0)1(−1)
)
β
−5
(c
122(0)2
0
))(β
−2
(c
2
00
1
)β
−2
(h
21
))⊗α
−4
(c
122(0)1(0)
)⊗β
−4
(c
122(0)2
00
)(β
−2
(c
2
00
2
)β
−2
(h
22
))
(1)
=
X
c
11
⊗((β
−5
(c
121(−1)
(β
−7
(c
1221(−1)
)β
−7
(c
1222(−1)
)))β
−1
(c
2
0
))h
1
⊗α
−2
(c
121(0)
)
⊗(β
−6
(c
1221(0)(−1)
)β
−5
(c
1222(0)
0
))(β
−2
(c
2
00
1
)β
−2
(h
21
))⊗α
−4
(c
1221(0)(0)
)⊗β
−4
(c
1222(0)
00
)
(β
−2
(c
2
00
2
)β
−2
(h
22
))
=
X
c
11
⊗((β
−5
(c
121(−1)
(β
−7
(c
1221(−1)
)β
−7
(c
1222(−1)
)))β
−1
(c
2
0
))β
−1
(h
11
)⊗α
−2
(c
121(0)
)
⊗(β
−6
(c
1221(0)(−1)
)β
−5
(c
1222(0)
0
))(β
−2
(c
2
00
1
)β
−2
(h
12
))⊗α
−4
(c
1221(0)(0)
)⊗β
−4
(c
1222(0)
00
)
(β
−2
(c
2
00
2
)β
−1
(h
2
))
=
X
c
11
⊗β
−3
(c
121(−1)
((β
−6
(c
1221(−1)
)(β
−7
(c
1222(−1)
)β
−3
(c
2
0
)))β
−2
(h
11
))⊗α
−2
(c
121(0)
)
⊗(β
−6
(c
1221(0)(−1)
)(β
−6
(c
1222(0)
0
)β
−3
(c
2
00
1
)))β
−1
(h
12
)⊗α
−4
(c
1221(0)(0)
)⊗β
−5
(c
1222(0)
00
)
(β
−2
(c
2
00
2
))h
2
)
DOI:10.12677/aam.2021.1082942827A^êÆ?Ð
Z
(∆⊗α)∆(c⊗h)
=(∆⊗α)(
X
α
−1
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))β
−1
(h
1
)⊗α
−2
(c
12(0)
)⊗β
−1
(c
2
00
)β
−1
(h
2
))
=
X
α
−2
(c
1111
)⊗(β
−5
(c
1112(−1)
)β
−3
(c
112
0
))((β
−5
(c
12(−1)1
)β
−3
c
2
0
1
))β
−2
(h
11
))
⊗α
−3
(c
1112(0)
)⊗β
−2
(c
112
00
)((β
−5
(c
12(−1)2
)β
−3
(c
2
0
2
))β
−2
(h
12
))⊗α
−1
(c
12(0)
)⊗c
2
00
h
2
=
X
c
11
⊗(β
−4
(c
121(−1)
)β
−4
(c
1221
0
))((β
−7
(c
1222(−1)1
)β
−3
c
2
0
1
))β
−2
(h
11
))
⊗α
−2
(c
121(0)
)⊗β
−3
(c
1221
00
)((β
−7
(c
1222(−1)2
)β
−3
(c
2
0
2
))β
−2
(h
12
))⊗α
−3
(c
1222(0)
)⊗c
2
00
h
2
=
X
c
11
⊗β
−3
(c
121(−1)
)(((β
−6
(c
1221
0
)β
−8
(c
1222(−1)1
))β
−3
c
2
0
1
))β
−2
(h
11
))
⊗α
−2
(c
121(0)
)⊗((β
−5
(c
1221
00
)β
−7
(c
1222(−1)2
))β
−2
(c
2
0
2
))β
−1
(h
12
)⊗α
−3
(c
1222(0)
)⊗c
2
00
h
2
=
X
c
11
⊗β
−3
(c
121(−1)
)(((β
−6
(c
1221
0
)β
−8
(c
1222(−1)1
))β
−3
(c
2
0
1
))β
−2
(h
11
))
⊗α
−2
(c
121(0)
)⊗((β
−5
(c
1221
00
)β
−7
(c
1222(−1)2
))β
−2
(c
2
0
2
))β
−1
(h
12
)⊗α
−3
(c
1222(0)
)⊗c
2
00
h
2
(2)
=
X
c
11
⊗β
−3
(c
121(−1)
)(((β
−7
(c
1221(−1)
)β
−6
(c
1222
0
))β
−3
c
2
0
1
))β
−2
(h
11
))
⊗α
−2
(c
121(0)
)⊗((β
−7
(c
1221(0)(−1)
)β
−5
(c
1222
00
))β
−2
(c
2
0
2
))β
−1
(h
12
)⊗α
−4
(c
1221(0)(0)
)⊗c
2
00
h
2
=
X
α(c
1
)⊗β
−3
(c
211(−1)
)(((β
−6
(c
212(−1)
)β
−5
(c
221
0
))β
−5
(c
222
0
1
))β
−2
(h
11
))
⊗α
−2
(c
211(0)
)⊗((β
−6
(c
212(0)(−1)
)β
−4
(c
221
00
))β
−4
(c
222
0
2
))β
−1
(h
12
)⊗α
−3
(c
212(0)(0)
)
⊗β
−2
(c
222
00
)h
2
=
X
α(c
1
)⊗β
−3
(c
211(−1)
)((β
−5
(c
212(−1)
)(β
−5
(c
221
0
)β
−6
(c
222
0
1
)))β
−2
(h
11
))
⊗α
−2
(c
211(0)
)⊗(β
−5
(c
212(0)(−1)
)(β
−4
(c
221
00
)β
−5
(c
222
0
2
)))β
−1
(h
12
)⊗α
−3
(c
212(0)(0)
)
⊗β
−2
(c
222
00
)h
2
(3)
=
X
α(c
1
)⊗β
−3
(c
211(−1)
)((β
−5
(c
212(−1)
)(β
−6
(c
221(−1)
))β
−5
(c
222
0
1
)))β
−2
(h
11
))
⊗α
−2
(c
211(0)
)⊗(β
−5
(c
212(0)(−1)
)(β
−5
(c
221(0)
0
)β
−5
(c
222
00
1
)))β
−1
(h
12
)⊗α
−3
(c
212(0)(0)
)
⊗(β
−4
(c
221(0)
00
)β
−4
(c
222
00
2
))h
2
=
X
c
11
⊗β
−3
(c
121(−1)
)((β
−6
(c
1221(−1)
)(β
−7
(c
1222(−1)
))β
−3
(c
2
0
1
)))β
−2
(h
11
))
⊗α
−2
(c
121(0)
)⊗(β
−6
(c
1221(0)(−1)
)(β
−6
(c
1222(0)
0
)β
−3
(c
2
00
1
)))β
−1
(h
12
)⊗α
−4
(c
1222(0)(0)
)
⊗(β
−5
(c
1222(0)
00
)β
−2
(c
2
00
2
))h
2
DOI:10.12677/aam.2021.1082942828A^êÆ?Ð
Z
Ñ(α⊗4)4(c⊗h) = (4⊗α)∆(c⊗h).
Ùg,‡ƒ5y².
dα(c)⊗β(h) = (id⊗)∆(c⊗h)
α(c)⊗1
H
=(id⊗)∆(c⊗1
H
)
=(id⊗)(
X
α
−1
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))1
H
⊗α
−2
(c
12(0)
)⊗β
−1
(c
2
00
)1
H
)
=
X
α
−1
(c
11
)⊗β
−3
(c
12(−1)
)β
−1
(c
2
0
)⊗(c
12(0)
)(c
2
00
)
=
X
α
−1
(c
11
)⊗(c
12
)1
H
β
−1
(c
2
0
)⊗(c
2
00
)
=
X
α
−1
(c
11
)⊗(c
12
)c
2
0
(c
2
00
=
X
c
1
⊗c
2
0
(c
2
00
).
†mÓžŠ^(⊗id)
(⊗id)(α(c)⊗1
H
) = (c)⊗1
H
(⊗id)(
X
c
1
⊗c
2
0
(c
2
00
)) =
X
β(c
0
)(c
00
)

(c)⊗1
H
=
X
(c
0
)(c
00
)
dα(c)⊗β(h) = (⊗id)∆(c⊗h)
α(c)⊗1
H
=(⊗id)∆(c⊗1
H
)
=(⊗id)(
X
α
−1
(c
11
)⊗(β
−4
(c
12(−1)
)β
−2
(c
2
0
))1
H
⊗α
−2
(c
12(0)
)⊗β
−1
(c
2
00
)1
H
)
=
X
(c
11
)⊗(c
12(−1)
)(c
2
0
)⊗α
−2
(c
12(0)
)⊗c
2
00
=
X
(c
11
)⊗α
−1
(c
12
)⊗(c
2
0
)c
2
00
=
X
c
1
⊗(c
2
0
)c
2
0
†mÓžŠ^(⊗id)
(⊗id)(α(c)⊗1
H
) = (c)⊗1
H
(⊗id)(
X
c
1
⊗(c
2
0
)c
2
00
) =
X
(c
0
)⊗β(c
00
)
DOI:10.12677/aam.2021.1082942829A^êÆ?Ð
Z

(c)⊗1
H
=
X
(c
0
)c
00
eC⊗
ψ
H´Hom-{“ê,k(α⊗4)4= (4⊗α)∆
(α⊗4)4(c⊗1
H
)
=(α⊗4)(
X
α
−1
(c
11
)⊗β
−3
(c
12(−1)
)β
−1
(c
2
0
)⊗α
−2
(c
12(0)
)⊗c
2
00
)
=
X
c
11
⊗β
−2
(c
12(−1)
)c
2
0
⊗α
−3
(c
12(0)11
)⊗(β
−6
(c
12(0)12(−1)
)β
−4
(c
12(0)2
0
))β
−1
(c
2
00
1
)
⊗α
−4
(c
12(0)12(0)
)⊗β
−3
(c
12(0)2
00
)β
−1
(c
2
00
2
)
(4⊗α)4(c⊗1
H
)
=(4⊗α)(
X
α
−1
(c
11
)⊗β
−3
(c
12(−1)
)β
−1
(c
2
0
)⊗α
−2
(c
12(0)
)⊗c
2
00
)
=
X
α
−2
(c
1111
)⊗(β
−5
(c
1112(−1)
)β
−3
(c
112
0
))(β
−4
(c
12(−1)1
)β
−2
(c
2
0
1
))⊗α
−3
(c
1112(0)
)
⊗β
−2
(c
112
00
)(β
−4
(c
12(−1)2
)β
−2
(c
2
0
2
))⊗α
−1
(c
12(0)
)⊗β(c
2
00
)
éþ¡ü‡ªfÓžŠ^(
c
⊗id)⊗(
c
⊗id)⊗(id⊗
H
)
X
(c
11
)⊗β
−2
(c
12(−1)
)c
2
0
⊗(c
12(0)11
)⊗(β
−6
(c
12(0)12(−1)
)β
−4
(c
12(0)2
0
))
β
−1
(c
2
00
1
)⊗α
−4
(c
12(0)12(0)
)⊗(c
12(0)2
00
)(c
2
00
2
)
=
X
(c
11
)⊗β
−2
(c
12(−1)
)c
2
0
⊗(β
−5
(c
12(0)1(−1)
)(c
12(0)2
)1
H
)c
2
00
⊗α
−3
(c
12(0)1(0)
)
=
X
(c
11
)⊗β
−2
(c
12(−1)
)c
2
0
⊗β
−3
(c
12(0)(−1)
)c
2
00
⊗α
−2
(c
12(0)(0)
)
=
X
β
−1
(c
1(−1)
)c
2
0
⊗β
−2
(c
1(0)(−1)
)c
2
00
⊗α
−1
(c
1(0)(0)
)
X
(c
1111
)⊗(β
−5
(c
1112(−1)
)β
−3
(c
112
0
))(β
−4
(c
12(−1)
)β
−2
(c
2
0
1
)⊗(c
1112(0)
)
⊗β
−2
(c
112
00
)(β
−4
(c
12(−1)2
)β
−2
(c
2
0
2
))⊗α
−1
(c
12(0)
)⊗(c
2
00
)
=
X
((c
111
)β
−2
(c
112
0
))β
−3
(c
12(−1)
)⊗β
−2
(c
112
00
)(β
−4
(c
12(−1)2
)((c
2
)1
H
))⊗α
−1
(c
12(0)
)
=
X
β
−1
(c
11
0
)β
−3
(c
12(−1)1
)⊗β
−1
(c
11
00
)(β
−4
(c
12(−1)2
)((c
2
)1
H
))⊗α
−1
(c
12(0)
)
=
X
c
1
0
β
−2
(c
2(−1)1
)⊗c
1
00
β
−2
(c
2(−1)2
)⊗c
2(0)
DOI:10.12677/aam.2021.1082942830A^êÆ?Ð
Z

X
β
−1
(c
1(−1)
)c
2
0
⊗β
−2
(c
1(0)(−1)
)c
2
00
⊗α
−1
(c
1(0)(0)
)
=
X
c
1
0
β
−2
(c
2(−1)1
)⊗c
1
00
β
−2
(c
2(−1)2
)⊗c
2(0)
=•
X
c
1(−1)
β(c
2
0
)⊗β
−1
(c
1(0)(−1)
)β(c
2
00
)⊗c
1(0)(0)
=
X
β(c
1
0
)β
−1
(c
2(−1)1
)⊗β(c
1
00
)β
−1
(c
2(−1)2
)⊗α(c
2(0)
)
éþ¡ü‡ªfÓžŠ^(
c
⊗id)⊗(
c
⊗id)⊗(
c
⊗id)
X
(c
11
)⊗β
−2
(c
12(−1)
)c
2
0
⊗(c
12(0)11
)⊗(β
−6
(c
12(0)12(−1)
)β
−4
(c
12(0)2
0
))
β
−1
(c
2
00
1
)⊗(c
12(0)12(0)
)⊗β
−3
(c
12(0)2
00
)β
−1
(c
2
00
2
)
=
X
(c
11
)⊗β
−2
(c
12(−1)
)c
2
0
⊗((c
12(0)1
)β
−3
(c
12(0)2
0
))β
−1
(c
2
00
1
)⊗β
−3
(c
12(0)2
00
)β
−1
(c
2
00
2
)
=
X
(c
11
)⊗β
−2
(c
12(−1)
)c
2
0
⊗β
−2
(c
12(0)
0
)β
−1
(c
2
00
1
)⊗β
−2
(c
12(0)
00
)β
−1
(c
2
00
2
)
=
X
β
−1
(c
1(−1)
)c
2
0
⊗β
−1
(c
1(0)
0
)β
−1
(c
2
00
1
)⊗β
−1
(c
1(0)
00
)β
−1
(c
2
00
2
)
X
(c
1111
)⊗(β
−5
(c
1112(−1)
)β
−3
(c
112
0
))(β
−4
(c
12(−1)
)β
−2
(c
2
0
1
))⊗(c
1112(0)
)⊗β
−2
(c
112
00
)
(β
−4
(c
12(−1)2
)β
−2
(c
2
0
2
))⊗(c
12(0)
)⊗β(c
2
00
)
=β
−1
(c
11
0
)((c
121
)β
−1
(c
2
0
1
))⊗β
−1
(c
11
00
)((c
122
)β
−1
(c
2
0
2
))⊗β(c
2
00
)
=β
−1
(c
11
0
)β
−1
(c
2
0
1
)⊗β
−1
(c
11
00
)((c
12
)β
−1
(c
2
0
2
))⊗β(c
2
00
)
=c
1
0
β
−1
(c
2
0
1
)⊗c
1
00
β
−1
(c
2
0
2
)⊗β(c
2
00
)

X
β
−1
(c
1(−1)
)c
2
0
⊗β
−1
(c
1(0)
0
))β
−1
(c
2
00
1
)⊗β
−1
(c
1(0)
00
)β
−1
(c
2
00
2
)
=
X
c
1
0
1
β
−1
(c
2
0
1
)⊗c
1
00
β
−1
(c
2
0
2
))⊗β(c
2
00
)
=•
X
c
1(−1)
β(c
2
0
)⊗c
1(0)
0
c
2
00
2
⊗c
1(0)
00
c
2
00
2
=
X
β(c
1
0
)c
2
0
1
⊗β(c
1
00
)c
2
0
2
⊗β
2
(c
2
00
)
DOI:10.12677/aam.2021.1082942831A^êÆ?Ð
Z
½n1y..
Ún1
1)α,β´ðNž,ùžHom-{ÈC⊗H=•©z[8]¥{È
4(C⊗H) =
X
c
1
⊗c
2(−1)
c
2
0
h
1
⊗c
2(0)
⊗c
2
00
h
2
2)α,β´ðNž,ùž½n1^‡=•©z[8]¥{ÈC⊗H¤{“ê¿‡
^‡
(I)
X
(c
0
)c
00
=
X
c
0
(c
00
) = (c)1
H
(II)
X
c
1(−1)
c
2
0
⊗c
1(0)(−1)
c
2
00
⊗c
1(0)(0)
=
X
c
1
0
c
2(−1)1
)⊗c
1
00
c
2(−1)2
⊗c
2(0)
(III)
X
c
1(−1)
c
2
0
⊗c
1(0)
0
c
2
00
1
⊗c
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