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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(8),2834-2846
PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.108295
BiHom-V“êþRadford
0
s
{È
ZZZ§§§½½½[[[ÂÂÂ
∗
§§§444
úô“‰ŒÆêƆOŽÅ‰ÆÆ,úô7u
ÂvFϵ2021c723F¶¹^Fϵ2021c815F¶uÙFϵ2021c824F
Á‡
•ïÄBiHom-V“êþRadford
0
sVȧÏL$^a'gŽ•{§‰ÑBiHom-{
{“ê±9BiHom-Smash{È½Â§¿BiHom-SmashÈÚBiHom-Smash{È/
¤BiHom-V“ê¿©7‡^‡"
'…c
BiHom-V“ê§BiHom-Smash{ȧBiHom-{{“ê§BiHom-Hopf“ê
Radford
0
sBiproducton
BiHom-Bialgebras
JialinPang,JiafengLv
∗
,LingLiu
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Jul.23
rd
,2021;accepted:Aug.15
th
,2021;published:Aug.24
th
,2021
∗ÏÕŠö"
©ÙÚ^:Z,½[Â,4 .BiHom-V“êþRadford
0
s{È[J].A^êÆ?Ð,2021,10(8):2834-2846.
DOI:10.12677/aam.2021.108295
Z
Abstract
ItwasaimedtostudytheRadford
0
sbiproductoverBiHom-bialgebras.Byapplying
thethoughtofanalogy,thenotionofBiHom-comodulecoalgebraandBiHom-Smash
coproductoverBiHom-bialgebraswasdefined.Further,anecessaryandsufficien-
tconditionfortheBiHom-SmashproductandBiHom-Smashcoproducttoforma
BiHom-bialgebrawasobtained.
Keywords
BiHom-Bialgebra,BiHom-SmashCoproduct,BiHom-ComoduleCoalgebra,
BiHom-HopfAlgebra
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
3C/þfznØÄ:þ§©z[1,2]éSmash{È?10ÚïÄ,Ù½ÂXe:H
´˜‡V“ê,B´†H-{{“ê,KSmash{ÈB×H´3B⊗Hþ½Â“ê,é?¿
b∈B,h∈H,§{¦{´
(b×h) = (b
1
⊗b
2(−1)
h
1
)⊗(b
2(0)
⊗h
2
).
'uSmash{ÈkÃõ/ªí2,©z[3–6]©O‰ÑHom-Smash{ÈÚÜþHom-
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BiHom-V“ê[8,9].
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BiHom-SmashÈÚBiHom-Smash{È/¤BiHom-V“ê¿©7‡^‡.
DOI:10.12677/aam.2021.1082952835A^êÆ?Ð
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éu{“êC,¦^Sweedler.PÒ5L«{¦:é?¿c∈C,∆(B) = c
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:A→A´‚5N.XJ•3‚5N
µ: A⊗A→A,a⊗b7→ab,¦é?¿a,b,c∈A,÷v
α
A
◦β
A
= β
A
◦α
A
,α
A
(ab) = α
A
(a)α
A
(b),
β
A
(ab) = β
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(a)β
A
(b),α
A
(a)(bc) = (ab)β
A
(c),
@o¡(A,µ,α
A
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A
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A
∈A,¦é?¿a∈A,÷v
α
A
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A
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A
,β
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A
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A
,
a1
A
= α
A
(a),1
A
a= β
A
(a),
@o¡(A,µ,α
A
,β
A
)•kü BiHom-(Ü“ê.{P•(A,α
A
,β
A
).
½Â2[7]C´˜‡‚5˜m,ψ
C
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C
:C→C´‚5N.XJ•3‚5N 
∆ : C→C⊗C,÷v
ψ
C
◦ω
C
= ω
C
◦ψ
C
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C
⊗ψ
C
)◦∆ = ∆◦ψ
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,
(ω
C
⊗ω
C
)◦∆ = ∆◦ω
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,(∆⊗ψ
C
)◦∆ = (ω
C
⊗∆)◦∆,
@o¡(C,∆,ψ
C
,ω
C
)•BiHom-{(Ü{“ê.XJ„•3‚5Nε: C→k,÷v
ε◦ψ
C
= ε,ε◦ω
C
= ε,
(id
C
⊗ε)◦∆ = ω
C
,(ε⊗id
C
)◦∆ = ψ
C
,
@o¡(C,∆,ψ
C
,ω
C
)•k{ü BiHom-{(Ü{“ê.{P•(C,ψ
C
,ω
C
).
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H
,β
H
)´BiHom-(Ü“ê,(H,∆,ψ
H
,ω
H
)´BiHom-{(Ü{“ê.
XJéu?¿h,g∈H,÷v
∆(hg) = (h
1
g
1
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2
g
2
),
α
H
◦ψ
H
= ψ
H
◦α
H
,α
H
◦ω
H
= ω
H
◦α
H
,
β
H
◦ψ
H
= ψ
H
◦β
H
,β
H
◦ω
H
= ω
H
◦β
H
,
(α
H
⊗α
H
)◦∆ = ∆◦α
H
,(β
H
⊗β
H
)◦∆ = ∆◦β
H
,
ψ
H
(hg) = ψ
H
(h)ψ
H
(g),ω
H
(hg) = ω
H
(h)ω
H
(g),
@o¡(H,µ,∆,α
H
,β
H
,ψ
H
,ω
H
)•BiHom-V“ê.XJ•3ƒ1
A
∈AÚ‚5Nε
H
:C→
DOI:10.12677/aam.2021.1082952836A^êÆ?Ð
Z
k,÷v
∆(1
H
) = 1
H
⊗1
H
,ε
H
(1
H
) = 1,
ψ
H
(1
H
) = 1
H
,ω
H
(1
H
) = 1
H
,
ε
H
◦α
H
= ε
H
,ε
H
◦β
H
= ε
H
,
ε
H
(hg) = ε
H
(h)ε
H
(g),
@o¡(H,µ,∆,α
H
,β
H
,ψ
H
,ω
H
)•kü Ú{ü BiHom-V“ê.{P•(H,α
H
,β
H
,
ψ
H
,ω
H
).
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H
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H
ψ
H
S
H
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1
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H
ω
H
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2
) = ε
H
(h)1
H
= β
H
ψ
H
(h
1
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H
ω
H
S
H
(h
2
),
@o¡(H,α
H
,β
H
,ψ
H
,ω
H
)•BiHom-Hopf“ê.
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H
,β
H
,ψ
H
,ω
H
)´BiHom-V“ê, (A,α
A
,β
A
)´BiHom-(Ü“ê, α
H
,β
H
,ψ
H
,ω
H
´V.XJ(A,α
A
,β
A
)´†(H,α
H
,β
H
,ψ
H
,ω
H
)-(Š^½Â•H⊗A→A,h⊗a7→(h·a)),
¦é?¿a,b∈A,h,g∈H,÷v
(hg)·β
A
(a) = α
H
(h)·(g·a),
α
A
(h·a) = α
H
(h)·α
A
(a),
β
A
(h·a) = β
H
(h)·β
A
(a),
1
H
·a= β
A
(a),h·1
A
= ε
H
(h)1
A
,
h·(ab) = (α
−1
H
ω
−1
H
(h
1
)·a)(β
−1
H
ψ
−1
H
(h
2
)·b),
@o¡(A,α
A
,β
A
)´†(H,α
H
,β
H
,ψ
H
,ω
H
)-BiHom-“ê.
½Â5[7](H,α
H
,β
H
,ψ
H
,ω
H
)´BiHom-V“ê,(A,α
A
,β
A
)´†(H,α
H
,β
H
,ψ
H
,ω
H
)-
BiHom-“ê, α
H
,β
H
,ψ
H
,ω
H
,α
A
,β
A
´V.XJrBiHom-(Ü“êA⊗H½Â•A]H,…
é?¿a,b∈A,h,g∈H,÷v
(a]h)(b]g) = (a(β
−1
H
ω
−1
H
(h
1
)·β
−1
A
(b)))](ψ
−1
H
(h
2
)g),
@o¡(A]H,α
A
⊗α
H
,β
A
⊗β
H
)•(A,α
A
,β
A
)Ú(H,α
H
,β
H
,ψ
H
,ω
H
)BiHom-SmashÈ.
3.BiHom-R adford
0
sVÈ
̇0BiHom-Smash{È½Â, ¿…‰ÑBiHom-SmashÈÚBiHom-Smash{
È/¤BiHom-V“ê¿©7‡^‡.
DOI:10.12677/aam.2021.1082952837A^êÆ?Ð
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½Â6(H,α
H
,β
H
,ψ
H
,ω
H
)´BiHom-V“ê,(B,∆
B
,ψ
B
,ω
B
)´BiHom-{“ê.e
(B,ρ,ψ
B
,ω
B
)´†H-BiHom-{,Ù¥{Š^½Â•ρ:B→H⊗B,b7→b
(−1)
⊗b
0
=
ω
H
(b
(−1)
)⊗ψ
B
(b
(0)
).¦é?¿b∈B,h∈H,÷v
ψ
H
(b
(−1)
)⊗ψ
B
(b
(0)
) = (ψ
B
(b))
(−1)
⊗(ψ
B
(b))
(0)
,
ω
H
(b
(−1)
)⊗ω
B
(b
(0)
) = (ω
B
(b))
(−1)
⊗(ω
B
(b))
(0)
,
ω
H
(b
1
)⊗b
21
⊗b
22
= b
11
⊗b
12
⊗ψ
H
(b
2
),
β
H
ψ
H
(b
1(−1)
)α
H
ω
H
(b
2(−1)
)⊗b
1(0)
⊗b
2(0)
= α
H
β
H
ω
H
ψ
H
(b
(−1)
)⊗b
(0)1
⊗b
(0)2
,
b
(−1)
ε(b
(0)
) = ε(b)1
H
,b
1
ε(b
2
) = ω(b),ε(b
1
)b
2
= ψ(b).
@o(B,∆
B
,ψ
B
,ω
B
)¡•†(H,α
H
,β
H
,ψ
H
,ω
H
)-BiHom-{{“ê.
·K1(H,α
H
,β
H
,ψ
H
,ω
H
) ´BiHom-V“ê,(B,∆
B
,ψ
B
,ω
B
) ´†(H,α
H
,β
H
,ψ
H
,ω
H
)-
BiHom-{{“ê,α
H
,β
H
,ψ
H
,ω
H
,ψ
B
,ω
B
´V.XJé?¿a,b∈B,h,g∈H,÷v
1)Š•k-˜m,B×H=B⊗H,
2)BiHom-{¦{,=
∆(b×h) = (b
1
×α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
(h
1
))⊗(ψ
−1
B
(b
2(0)
)×h
2
).,
@o(B×H,α
B
×α
H
,β
B
×β
H
)´BiHom-{(Ü{“ê.¡Ù•BiHom-Smash{È.{ü •
ε
B
×ε
H
.
y²dBiHom-{{“êÚBiHom-Smash-{È½Âµ
(∆⊗ψ
B
)∆(b×h)
=(∆⊗ψ
B
)(b
1
⊗α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
(h
1
)⊗ψ
−1
B
(b
2(0)
)⊗h
2
)
=[b
11
⊗α
−1
H
ψ
−1
H
(b
12(−1)
)(β
−1
H
α
−1
H
ψ
−1
H
(b
2(−1)1
)β
−2
H
(h
11
))]
⊗[ψ
−1
B
(b
12(0)
)⊗α
−1
H
ψ
−1
H
(b
2(−1)2
)β
−1
H
(h
12
)⊗(b
2(0)
⊗ψ
H
(h
2
))]
=ω
B
(b
1
)⊗α
−1
H
ψ
−1
H
(b
21(−1)
)(β
−1
H
α
−1
H
ψ
−2
H
(b
22(−1)1
)ω
H
β
−2
H
(h
1
))
⊗ψ
−1
B
(b
21(0)
)⊗α
−1
H
ψ
−2
H
(b
22(−1)2
)β
−1
H
(h
21
)⊗(ψ
−1
B
(b
22(0)
)⊗h
22
)
=ω
B
(b
1
)⊗(α
−2
H
ψ
−1
H
(b
21(−1)
)β
−1
H
α
−1
H
ψ
−2
H
(b
22(−1)1
)ω
H
β
−1
H
(h
1
)
⊗ψ
−1
B
(b
21(0)
)⊗α
−1
H
ψ
−2
H
(b
22(−1)2
)β
−1
H
(h
21
)⊗(ψ
−1
B
(b
22(0)
)⊗h
22
)
=ω
B
(b
1
)⊗(α
−2
H
ψ
−1
H
(b
21(−1)
)β
−1
H
α
−1
H
ψ
−2
H
ω
H
(b
22(−1)
))ω
H
β
−1
H
(h
1
)
⊗ψ
−1
B
(b
21(0)
)⊗α
−1
H
ψ
−2
H
(b
22(0)(−1)
)β
−1
H
(h
21
)⊗(ψ
−2
B
(b
22(0)(0)
)⊗h
22
)
=ω
B
(b
1
)⊗ω
H
α
−1
H
ψ
−1
H
(b
2(−1)
)ω
H
β
−1
H
(h
1
)⊗ψ
−1
B
(b
2(0)1
)
⊗α
−1
H
ψ
−2
H
(b
2(0)2(−1)
)β
−1
H
(h
21
)⊗(ψ
−2
B
(b
2(0)2(0)
)⊗h
22
)
DOI:10.12677/aam.2021.1082952838A^êÆ?Ð
Z
=(ω
H
⊗∆)(b
1
⊗α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
(h
1
)⊗ψ
−1
B
(b
2(0)
)⊗h
2
)
=(ω
H
⊗∆)∆(b×h).
`²(∆⊗ψ
B
)∆(b×h) = (ω
H
⊗∆)∆(b×h).·K1y..
Ún1(H,α
H
,β
H
,ψ
H
,ω
H
)´BiHom-V“ê,(B,α
B
,β
B
)´BiHom-“ê,e
(B,α
B
,β
B
)´†H-BiHom-{,K(B,α
B
,β
B
)¡•†H-BiHom-{“ê,Ù¥{Š^•:
ρ(ab) = a
(−1)
b
(−1)
⊗a
(0)
b
(0)
,ρ(1
B
) = 1
H
⊗1
B
.
Ún2(H,α
H
,β
H
,ψ
H
,ω
H
)´BiHom-V“ê,(B,ψ
B
,ω
B
)´BiHom-{“ê,e
(B,α
B
,β
B
)´†H-BiHom-,K(B,α
B
,β
B
)¡•†H-BiHom-{“ê,Ù¥Š^•:
∆(h·b) = h
1
·b
1
⊗h
2
·b
2
,ε
B
(h·b) = ε
H
(h)ε
B
(b).
½n1(H,α
H
,β
H
,ψ
H
,ω
H
),(B,α
B
,β
B
,ψ
B
,ω
B
)´BiHom-V“ê,α
H
,β
H
,ψ
H
,
ω
H
,α
B
,β
B
,ψ
B
,ω
B
´V.KBiHom-SmashÈ(B]H,α
B
]α
H
,β
B
]β
H
)ÚBiHom-Smash{È
(B×H,α
B
×α
H
,β
B
×β
H
)/¤BiHom-V“ê,…=é?¿a∈A,h∈H,e^‡d:
(I)(B
×
]
H,α
B
×α
H
,β
B
×β
H
)´BiHom-V“ê.
(II)±e^‡¤á,
c
1
)ε
B
´“êÓ…∆
B
(1
B
) = 1
B
⊗1
B
.
c
2
)(B,α
B
,β
B
,ψ
B
,ω
B
)´†(H,α
H
,β
H
,ψ
H
,ω
H
)BiHom-{“ê,
c
3
)(B,α
B
,β
B
,ψ
B
,ω
B
)´†(H,α
H
,β
H
,ψ
H
,ω
H
)BiHom-{“ê,
c
4
)∆
B
(ab) = a
1
(a
2(−1)
·β
−1
(b
1
))⊗β(a
2(0)
)b
2
,(4)
c
5
)h
1
·b
(−1)
⊗h
2
·b
(0)
= α
−1
H
((ω
−1
H
(h
1
)·b)
(−1)
)
α
H
ω
H
β
(−1)
H
ψ
(−1)
H
(h
2
)⊗((ω
−1
H
(h
1
)·b)
(−1)
).
y²Äk,·‚ky²(I) ⇒(II).
Ï•∆
B×H
Úε
B×H
Ñ´“êÓ,é?¿a,b∈B.h,k∈H.·‚k
ε((a×h)(b×k))=ε(a(β
−1
H
ω
−1
H
(h
1
)·β
−1
B
(b))×(ψ
−1
H
(h
2
)k))
=ε
B
(a(β
−1
H
ω
−1
H
(h
1
)·β
−1
B
(b)))ε
H
(h
2
)ε
H
(k)
=ε
B
(a(β
−1
H
(h)·β
−1
B
(b)))ε
H
(k)
ε(a×h)ε(b×k)=ε
B
(a)ε
H
(h)ε
B
(b)ε
H
(k).
-h= k= 1
H
.Œ±ε
B
(ab) = ε
B
(a)ε
B
(b).-a= 1
B
,k= 1
H
,Œ±
ε
B
(1
B
(β
−1
H
(h)·β
−1
B
(b)))ε
H
(1
H
) = ε
B
(h·b) = ε
H
(h)ε
B
(h).
DOI:10.12677/aam.2021.1082952839A^êÆ?Ð
Z
Ï•
∆(1
B
×1
H
)=(1
B1
⊗α
−1
H
ψ
−1
H
(1
B2(−1)
)β
−1
H
(1
H
))⊗(ψ
−1
B
(1
B2(0)
)⊗1
H
)
=(1
B1
×ψ
−1
H
(1
B2(−1)
))⊗(ψ
−1
B
(1
B2(0)
)×1
H
)
=(1
B
×1
H
)⊗(1
B
×1
H
).
†mü>ÓžŠ^Id⊗ε
H
⊗Id⊗ε
H
,Œ±∆(1
B
) = 1
B
⊗1
B
.
†mü>ÓžŠ^ε
B
⊗Id⊗Id⊗ε
H
,Œ±
ε
B
(1
B1
)ψ
−1
H
(1
B2(−1)
)⊗ψ
−1
B
(1
B2(0)
)ε
H
(1
H
) = 1
B(−1)
⊗1
B(0)
= ρ(1
B
) = 1
H
⊗1
B
.
-h= k= 1
H
.Ï•∆((a×1
H
)(b×1
H
)) = ∆(a×1
H
)∆(b×1
H
),¤±·‚Œ±OŽ.
∆((a×1
H
)(b×1
H
))
=∆(a(β
−1
H
ω
−1
H
(1
H
)·β
−1
B
(b))×ψ
−1
H
(1
H
)·1
H
) = ∆(ab×1
H
)
=((ab)
1
×α
−1
H
ψ
−1
H
((ab)
2(−1)
)·β
−1
H
(1
H
))⊗(ψ
−1
B
((ab)
2(0)
)×1
H
)
=((ab)
1
×ψ
−1
H
((ab)
2(−1)
))⊗(ψ
−1
B
((ab)
2(0)
)×1
H
).
∆(a×1
H
)∆(b×1
H
)
=[(a
1
×α
−1
H
ψ
−1
H
(a
2(−1)
)·β
−1
H
(1
H
))⊗(ψ
−1
B
(a
2(0)
)×1
H
)]
[(b
1
×α
−1
H
ψ
−1
H
(b
2(−1)
)·β
−1
H
(1
H
))⊗(ψ
−1
B
(b
2(0)
)×1
H
)]
=[(a
1
×ψ
−1
H
(a
2(−1)
))(b
1
×ψ
−1
H
(b
2(−1)
))]⊗[(ψ
−1
B
(a
2(0)
)
×1
H
)(ψ
−1
B
(b
2(0)
)×1
H
)]
=(a
1
(β
−1
H
ω
−1
H
ψ
−1
H
(a
2(−1)1
)·β
−1
B
(b
1
))×ψ
−2
H
(a
2(−1)2
)·ψ
−1
H
(b
2(−1)
))
⊗(ψ
−1
B
(a
2(0)
)(β
−1
H
ω
−1
H
(1
H
)·β
−1
B
(ψ
−1
B
(b
2(0)
))×ψ
−1
H
(1
H
)·1
H
))
=(a
1
(β
−1
H
ω
−1
H
ψ
−1
H
(a
2(−1)1
)·β
−1
B
(b
1
))×ψ
−2
H
(a
2(−1)2
)·ψ
−1
H
(b
2(−1)
))
⊗ψ
−1
B
(a
2(0)
)(ψ
−1
B
(b
2(0)
)×1
H
)).
Ïd,
((ab)
1
×ψ
−1
H
((ab)
2(−1)
))⊗(ψ
−1
B
((ab)
2(0)
)×1
H
)
=(a
1
(β
−1
H
ω
−1
H
ψ
−1
H
(a
2(−1)1
)·β
−1
B
(b
1
))
×ψ
−2
H
(a
2(−1)2
)·ψ
−1
H
(b
2(−1)
))⊗(ψ
−1
B
(a
2(0)
)(ψ
−1
B
(b
2(0)
)×1
H
)).
DOI:10.12677/aam.2021.1082952840A^êÆ?Ð
Z
ü>ÓžŠ^ε
B
⊗Id⊗Id⊗ε
H
.
ε
B
((ab)
1
)ψ
−1
H
((ab)
2(−1)
)⊗ψ
−1
B
((ab)
2(0)
)ε
H
(1
H
) = (ab)
(−1)
⊗(ab)
(0)
.
ε
B
(a
1
(β
−1
H
ω
−1
H
ψ
−1
H
(a
2(−1)1
)·β
−1
B
(b
1
)))(ψ
−2
H
(a
2(−1)2
)ψ
−1
H
(b
2(−1)
))
⊗(ψ
−1
B
(a
2(0)
)ψ
−1
B
(b
2(0)
))ε
H
(1
H
)
=ε
B
(a
1
)ε
B
(a
2(−1)1
)ε
B
(b
1
)ψ
−2
H
(a
2(−1)2
)ψ
−1
H
(b
2(−1)
)
⊗ψ
−1
B
(a
2(0)
)ψ
−1
H
(b
2(−1)
ψ
−1
B
(b
2(0)
)
=ε
B
(a
1
)ψ
−1
H
(a
2(−1)
)b
(−1)
⊗ψ
−1
B
(a
2(0)
)b
(0)
=a
(−1)
b
(−1)
⊗a
(0)
b
(0)
.
Œ±ρ(ab) = (ab)
(−1)
⊗(ab)
(0)
= a
(−1)
b
(−1)
⊗a
(0)
b
(0)
.
ü>2gÓžŠ^Id⊗ε
H
⊗Id⊗ε
H
.
(ab)
1
ε
H
(ψ
−1
H
((ab)
2(−1)
))⊗ψ
−1
B
((ab)
2(0)
)ε
H
(1
H
)
=(ab)
1
⊗(ab)
2
= ∆(ab).
(a
1
(β
−1
H
ω
−1
H
ψ
−1
H
(a
2(−1)1
)·β
−1
B
(b
1
)))ε
H
(ψ
−1
H
(a
2(−1)2
)
ψ
−1
H
(b
2(−1)
))⊗ψ
−1
B
(a
2(0)
)ψ
−1
B
(b
2(0)
)ε
H
(1
H
)
=(a
1
(β
−1
H
ω
−1
H
ψ
−1
H
(a
2(−1)1
)·β
−1
B
(b
1
)))ε
H
(a
2(−1)2
)ε
H
(b
2(−1)
)
⊗ψ
−1
B
(a
2(0)
)ψ
−1
B
(b
2(0)
)
=(a
1
(β
−1
H
ψ
−1
H
(a
2(−1)
)·β
−1
B
(b
1
)))⊗ψ
−1
B
(a
2(0)
)b
2
.
Œ±∆(ab) = (ab)
1
⊗(ab)
2
= (a
1
(β
−1
H
ψ
−1
H
(a
2(−1)
)·β
−1
B
(b
1
)))⊗ψ
−1
B
(a
2(0)
)b
2
,=c4)¤á.
-a= 1
B
,k= 1
H
.Ï•∆((1
B
×h)(b×1
H
)) = ∆(1
B
×h)∆(b×1
H
),¤±·‚Œ±OŽ.
∆((1
B
×h)(b×1
H
))
=∆(1
B
(β
−1
H
ω
−1
H
(h
1
)·β
−1
B
(b))×ψ
−1
H
(h
2
)·1
H
)
=∆(ω
−1
H
(h
1
)·b×α
H
ψ
−1
H
(h
2
))
=(ω
−1
H
(h
1
)·b)
1
×α
−1
H
ψ
−1
H
((ω
−1
H
(h
1
)·b)
2(−1)
)·β
−1
H
α
H
ψ
−1
H
(h
21
)
⊗ψ
−1
B
((ω
−1
H
(h
1
)·b)
2(0)
)×α
H
ψ
−1
H
(h
22
).
DOI:10.12677/aam.2021.1082952841A^êÆ?Ð
Z
∆(1
B
×h)∆(b×1
H
)
=[1
B1
×α
−1
H
ψ
−1
H
(1
B2(−1)
)·β
−1
H
(h
1
)⊗ψ
−1
B
(1
B2(0)
)×h
2
][b
1
×α
−1
H
ψ
−1
H
(b
2(−1)
)·β
−1
H
(1
H
)⊗ψ
−1
B
(b
2(0)
)×1
H
]
=[(1
B
×h
1
)⊗(1
B
×h
2
)][(b
1
×ψ
−1
H
(b
2(−1)
))⊗(ψ
−1
B
(b
2(0)
)×1
H
)]
=[(1
B
×h
1
)(b
1
×ψ
−1
H
(b
2(−1)
))]⊗[(1
B
×h
2
)(ψ
−1
B
(b
2(0)
)×1
H
)]
=[1
B
(β
−1
H
ω
−1
H
(h
11
)·β
−1
B
(b
1
))×ψ
−1
H
(h
12
)·ψ
−1
H
(b
2(−1)
)]
[1
B
(β
−1
H
ω
−1
H
(h
21
)·β
−1
B
ψ
−1
B
(b
2(0)
))×ψ
−1
H
(h
22
)·1
H
]
=[ω
−1
H
(h
11
)·b
1
×ψ
−1
H
(h
12
)·ψ
−1
H
(b
2(−1)
)][ω
−1
H
(h
21
)·ψ
−1
B
(b
2(0)
)
×α
H
ψ
−1
H
(h
22
)].
Œ±:
(ω
−1
H
(h
1
)·b)
1
×α
−1
H
ψ
−1
H
((ω
−1
H
(h
1
)·b)
2(−1)
)·β
−1
H
α
H
ψ
−1
H
(h
21
)
⊗ψ
−1
B
((ω
−1
H
(h
1
)·b)
2(0)
)×α
H
ψ
−1
H
(h
22
)
=[ω
−1
H
(h
11
)·b
1
×ψ
−1
H
(h
12
)·ψ
−1
H
(b
2(−1)
)][ω
−1
H
(h
21
)·ψ
−1
B
(b
2(0)
)
×α
H
ψ
−1
H
(h
22
)]
ü>ÓžŠ^Id⊗ε
H
⊗Id⊗ε
H
.
(ω
−1
H
(h
1
)·b)
1
ε
H
(α
−1
H
ψ
−1
H
((ω
−1
H
(h
1
)·b)
2(−1)
))ε
H
(β
−1
H
α
H
ψ
−1
H
(h
21
))
⊗ψ
−1
B
((ω
−1
H
(h
1
)·b)
2(0)
)ε
H
(α
H
ψ
−1
H
(h
22
))
=(ω
−1
H
(h
1
)·b)
1
ε
H
((ω
−1
H
(h
1
)·b)
2(−1)
))ε
H
(h
21
)
⊗ψ
−1
B
((ω
−1
H
(h
1
)·b)
2(0)
)ε
H
(h
22
)
=(ω
−1
H
(h
1
)·b)
1
⊗(ω
−1
H
(h
1
)·b)
2
ε
H
(h
2
)
=(h·b)
1
⊗(h·b)
2
.
ω
−1
H
(h
11
)·b
1
ε
H
(ψ
−1
H
(h
12
))ε
H
(ψ
−1
H
(b
2(−1)
))
⊗ω
−1
H
(h
21
)ψ
−1
B
(b
2(0)
)ε
H
(α
H
ψ
−1
H
(h
22
))
=ω
−1
H
(h
11
)·b
1
ε
H
(h
12
)ε
H
(b
2(−1)
)⊗ω
−1
H
(h
21
)ψ
−1
B
(b
2(0)
)ε
H
(h
22
)
=h
1
·b
1
⊗h
2
·b
2
.
Œ±∆(h·b) = (h·b)
1
⊗(h·b)
2
= h
1
·b
1
⊗h
2
·b
2
.
DOI:10.12677/aam.2021.1082952842A^êÆ?Ð
Z
ü>2ÓžŠ^ε
B
⊗Id⊗Id⊗ε
H
.
ε
B
((ω
−1
H
(h
1
)·b)
1
)(α
−1
H
ψ
−1
H
((ω
−1
H
(h
1
)·b)
2(−1)
))ε
H
(β
−1
H
α
H
ψ
−1
H
(h
21
))
⊗ψ
−1
B
((ω
−1
H
(h
1
)·b)
2(0)
)ε
H
(α
H
ψ
−1
H
(h
22
))
=α
−1
H
((ω
−1
H
(h
1
)·b)
(−1)
)α
H
β
−1
H
ψ
−1
H
ω
H
(h
2
)⊗(ω
−1
H
(h
1
)·b)
(0)
.
ε
B
(ω
−1
H
(h
11
)·b
1
)ψ
−1
H
(h
12
)ψ
−1
H
(b
2(−1)
)⊗ω
−1
H
(h
21
)·ψ
−1
B
(b
2(0)
)ε
H
(α
H
ψ
−1
H
(h
22
))
=ε
B
(h
11
)ε
B
(b
1
)ψ
−1
H
(h
12
)ψ
−1
H
(b
2(−1)
)⊗ω
−1
H
(h
21
)ψ
−1
B
(b
2(0)
)ε
H
(h
22
)
=h
1
b
(−1)
⊗h
2
b
(0)
.
Œ±h
1
b
(−1)
⊗h
2
b
(0)
=α
−1
H
((ω
−1
H
(h
1
)·b)
(−1)
)α
H
β
−1
H
ψ
−1
H
ω
H
(h
2
)⊗(ω
−1
H
(h
1
)·b)
(0)
.=c5)
¤á.¤±,c1)−c5)Ѥá.
y3,·‚5y²(II)⇒(I).éN´yε((a×h)(b×k))=ε(a×h)ε(b×k),ε(1
B
×1
H
)=
ε(1
k
),∆(1
B
×1
H
)=(1
B
×1
H
) ⊗(1
B
×1
H
).ŠâBiHom-{(Ü{“ê½ÂŒ•,é?¿
h∈H,·‚k
ω
H
(a
(−1)
)⊗a
(0)(−1)
⊗a
(0)(0)
= a
(−1)1
⊗a
(−1)2
⊗ψ
−1
B
(a
(0)
).(1)
h
11
⊗h
12
⊗h
21
⊗h
22
= h
11
⊗ω
−1
H
(h
121
)⊗ψ
−1
H
(h
122
)⊗ψ
H
(h
2
).(2)
y3·‚OŽ
∆((a×h)(b×k))
=∆(a(β
−1
H
ω
−1
H
(h
1
)·β
−1
B
(b)×ψ
−1
H
(h
2
)k)
=[a(β
−1
H
ω
−1
H
(h
1
)·β
−1
B
(b))]
1
×α
−1
H
ψ
−1
H
([a(β
−1
H
ω
−1
H
(h
1
)·β
−1
B
(b))]
2(−1)
)β
−1
H
(ψ
−1
H
(h
21
)k
1
)
⊗ψ
−1
B
([a(β
−1
H
ω
−1
H
(h
1
)·β
−1
B
(b))]
2(0)
)×ψ
−1
H
(h
22
)k
2
((1),c2)
=a
1
(β
−1
H
ψ
−1
H
(a
2(−1)
)·β
−1
B
(β
−1
H
ω
−1
H
(h
11
)·β
−1
B
(b
1
)))
×α
−1
H
ψ
−1
H
((ψ
−1
B
(a
2(0)
)(β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
)))
(−1)
)β
−1
H
(ψ
−1
H
(h
21
)k
1
)
⊗ψ
−1
B
((ψ
−1
B
(a
2(0)
)(β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
)))
(0)
)×ψ
−1
H
(h
22
)k
2
(c3)
=a
1
(β
−1
H
ψ
−1
H
(a
2(−1)
)·(β
−2
H
ω
−1
H
(h
11
)·β
−2
B
(b
1
)))
×(α
−1
H
ψ
−2
H
(a
2(0)(−1)
)α
−1
H
ψ
−1
H
((β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
))
(−1)
))(β
−1
H
ψ
−1
H
(h
21
)β
−1
H
(k
1
))
⊗(ψ
−2
B
(a
2(0)(0)
)ψ
−1
B
((β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
))
(0)
)×ψ
−1
H
(h
22
)k
2
=a
1
(α
−1
H
β
−1
H
ψ
−1
H
(a
2(−1)
)β
−2
H
ω
−1
H
(h
11
))·β
−1
B
(b
1
))
×[(α
−2
H
ψ
−2
H
(a
2(0)(−1)
)α
−2
H
ψ
−1
H
((β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
))
(−1)
))β
−1
H
ψ
−1
H
(h
21
)]k
1
⊗(ψ
−2
B
(a
2(0)(0)
)ψ
−1
B
((β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
))
(0)
)×ψ
−1
H
(h
22
)k
2
=a
1
((α
−1
H
β
−1
H
ψ
−1
H
(a
2(−1)
)β
−2
H
ω
−1
H
(h
11
))·β
−1
B
(b
1
))
×[α
−1
H
ψ
−2
H
(a
2(0)(−1)
)(α
−2
H
ψ
−1
H
((β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
))
(−1)
)β
−2
H
ψ
−1
H
(h
21
))]k
1
DOI:10.12677/aam.2021.1082952843A^êÆ?Ð
Z
⊗ψ
−2
B
(a
2(0)(0)
)ψ
−1
B
((β
−1
H
ω
−1
H
(h
12
)·β
−1
B
(b
2
))
(0)
)×ψ
−1
H
(h
22
)k
2
(2)
=a
1
((α
−1
H
β
−1
H
ψ
−1
H
(a
2(−1)
)β
−2
H
ω
−1
H
(h
11
))·β
−1
B
(b
1
))
×[α
−1
H
ψ
−2
H
(a
2(0)(−1)
)α
−1
H
ψ
−1
H
(α
−1
H
((β
−1
H
ω
−2
H
(h
121
)·β
−1
B
(b
2
))
(−1)
)α
−1
H
β
−2
H
ψ
−1
H
(h
122
))]k
1
⊗ψ
−2
B
(a
2(0)(0)
)ψ
−1
B
((β
−1
H
ω
−2
H
(h
121
)·β
−1
B
(b
2
))
(0)
)×h
2
k
2
(c5)
=a
1
((α
−1
H
β
−1
H
ψ
−1
H
(a
2(−1)
)β
−2
H
ω
−1
H
(h
11
))·β
−1
B
(b
1
))
×[α
−1
H
ψ
−2
H
(a
2(0)(−1)
)(α
−1
H
ψ
−1
H
β
−1
H
ω
−1
H
(h
121
)α
−1
H
ψ
−1
H
β
−1
B
(b
2(−1)
))]k
1
⊗ψ
−2
B
(a
2(0)(0)
)ψ
−1
B
(β
−1
H
ω
−1
H
(h
122
)·β
−1
B
(b
2(0)
))×h
2
k
2
((1),(2))
=a
1
((α
−1
H
β
−1
H
ψ
−1
H
ω
−1
H
(a
2(−1)
)β
−2
H
ω
−1
H
(h
11
))·β
−1
B
(b
1
))
×[(α
−2
H
ψ
−2
H
(a
2(−1)2
)α
−1
H
ψ
−1
H
β
−1
H
(h
12
))(α
−1
H
ψ
−1
H
(b
2(−1)
))]k
1
⊗ψ
−2
B
(a
2(0)
)(β
−1
H
ω
−1
H
(h
21
)·β
−1
B
ψ
−1
B
(b
2(0)
))×ψ
−1
H
h
22
k
2
=a
1
((α
−1
H
β
−1
H
ψ
−1
H
ω
−1
H
(a
2(−1)
)β
−2
H
ω
−1
H
(h
11
))·β
−1
B
(b
1
))
×(α
−1
H
ψ
−2
H
(a
2(−1)2
)ψ
−1
H
β
−1
H
(h
12
))(α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
(k
1
))
⊗ψ
−1
B
(a
2(0)
)(β
−1
H
ω
−1
H
(h
21
)·β
−1
B
ψ
−1
B
(b
2(0)
))×ψ
−1
H
h
22
k
2
.
∆(a×h)∆(b×h)
=[a
1
×(α
−1
H
ψ
−1
H
(a
2(−1)
)β
−1
H
(h
1
))⊗ψ
−1
B
(a
2(0)
)×h
2
][b
1
×(α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
(k
1
))
⊗ψ
−1
B
(b
2(0)
)×k
2
]
=[(a
1
×α
−1
H
ψ
−1
H
(a
2(−1)
)·β
−1
H
(h
1
))(b
1
×α
−1
H
ψ
−1
H
(b
2(−1)
)·β
−1
H
(k
1
))]
⊗[(ψ
−1
B
(a
2(0)
)×h
2
)(ψ
−1
B
(b
2(0)
)×k
2
)]
=a
1
((β
−1
H
ω
−1
H
(α
−1
H
ψ
−1
H
(a
2(−1)
)·β
−1
H
)
1
)·β
−1
B
(b
1
))
×ψ
−1
H
((α
−1
H
ψ
−1
H
(a
2(−1)
)·β
−1
H
(h
1
))
2
)(α
−1
H
ψ
−1
H
(b
2(−1)
)·β
−1
H
(k
1
))
⊗ψ
−1
B
(a
2(0)
)(β
−1
H
ω
−1
H
(h
21
)·β
−1
B
(ψ
−1
B
(b
2(0)
)))×ψ
−1
H
(h
22
)·k
2
=a
1
((α
−1
H
β
−1
H
ψ
−1
H
ω
−1
H
(a
2(−1)
)β
−2
H
ω
−1
H
(h
11
))·β
−1
B
(b
1
))
×(α
−1
H
ψ
−2
H
(a
2(−1)2
)ψ
−1
H
β
−1
H
(h
12
))(α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
(k
1
))
⊗ψ
−1
B
(a
2(0)
)(β
−1
H
ω
−1
H
(h
21
)·β
−1
B
ψ
−1
B
(b
2(0)
))×ψ
−1
H
h
22
k
2
.
Ïd·‚k∆((a×h)(b×k)) = ∆(a×h)∆(b×h),…(B
×
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H,α
B
×α
H
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B
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H
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´BiHom-V“ê,y²..
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H,α
B
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H
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B
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H
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DOI:10.12677/aam.2021.1082952844A^êÆ?Ð
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·K2(B
×
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B
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B
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H
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0
s-BiHom-V“ê,eH´BiHom-Hopf-“
ê,Ùé4©O´S
H
,S
B
: H→H.(S
B
◦α
B
= α
B
◦S
B
,S
B
◦β
B
= β
B
◦S
B
),K(B
×
]
H,α
B
×α
H
,β
B
×
β
H
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S(b×h) = (1
B
×S
H
(α
−1
H
β
−1
H
ψ
−1
H
(b
(−1)
)β
−2
H
ψ
−1
H
ω
H
(h)))(S
B
(α
B
β
−2
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ψ
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B
ω
B
(b
(0)
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H
).
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(Id∗S)(b×h)
=(b
1
×α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
(h
1
))S
H
(ψ
−1
B
(b
2(0)
)×h
2
)
=(b
1
×α
−1
H
ψ
−1
H
(b
2(−1)
)β
−1
H
)[(1
B
×S
H
(α
−1
H
β
−1
H
ψ
−2
H
(b
2(0)(−1)
)β
−2
H
ψ
−1
H
ω
H
(h
2
)))
(S
B
(α
B
β
−2
B
ψ
−3
B
ω
B
(b
2(0)(0)
))×1
H
)]
=[(α
−1
H
(b
1
)×α
−2
H
ψ
−1
H
(b
2(−1)
)α
−1
H
β
−1
H
(h
1
))(1
B
×S
H
(α
−1
H
β
−1
H
ψ
−2
H
(b
2(0)(−1)
)
β
−2
H
ψ
−1
H
ω
H
(h
2
)))](S
B
α
B
β
−1
B
ψ
−3
B
ω
B
(b
2(0)(0)
)×1
H
)
=[α
−1
H
(b
1
)(β
−1
H
ω
−1
H
((α
−2
H
ψ
−1
H
(b
2(−1)
)α
−1
H
β
−1
H
(h
1
))
1
)·β
−1
B
(1
B
))×ψ
−1
H
((α
−2
H
ψ
−1
H
(b
2(−1)
)
α
−1
H
β
−1
H
(h
1
))
2
)S
H
(α
−1
H
β
−1
H
ψ
−2
H
(b
2(0)(−1)
)β
−2
H
ψ
−1
H
ω
H
(h
2
))]
(S
B
α
B
β
−1
B
ψ
−3
B
ω
B
(b
2(0)(0)
)×1
H
)
=[b
1
×(α
−2
H
ψ
−1
H
(b
2(−1)
)α
−1
H
β
−1
H
(h
1
))(S
H
(α
−1
H
β
−1
H
ψ
−2
H
(b
2(0)(−1)
)β
−2
H
ψ
−1
H
ω
H
(h
2
))]
(S
B
α
B
β
−1
B
ψ
−3
B
ω
B
(b
2(0)(0)
)×1
H
)
=[b
1
×(α
−2
H
ψ
−1
H
(b
2(−1)
))((α
−1
H
β
−1
H
(h
1
)(S
H
α
−1
H
β
−2
H
ψ
−1
H
ω
H
(h
2
)))S
H
β
−2
H
ψ
−2
H
(b
2(0)(−1)
)]
(S
B
α
B
β
−1
B
ψ
−3
B
ω
B
(b
2(0)(0)
)×1
H
)
=(b
1
×(α
−2
H
ψ
−1
H
(b
2(−1)
))S
H
β
−1
H
ψ
−2
H
(b
2(0)(−1)
))(S
B
α
B
β
−1
B
ψ
−3
B
ω
B
(b
2(0)(0)
)×1
H
)ε
H
(h)
=(b
1
×α
−2
H
ψ
−1
H
(b
2(−1)1
)S
H
β
−1
H
ψ
−2
H
(b
2(−1)2
))(S
B
α
B
β
−1
B
ψ
−2
B
ω
B
(b
2(0)
)×1
H
)ε
H
(h)
=(b
1
×1
H
)(S
B
α
B
β
−1
B
ψ
−1
B
ω
B
(b
2
)×1
H
)ε
H
(h)
=(b
1
S
B
α
B
β
−1
B
ψ
−1
B
ω
B
(b
2
)×1
H
)ε
H
(h)
=1
B
×1
H
ε
B
(b)ε
H
(h)
=1
B
×1
H
ε(b×h).
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×
]
H,α
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H
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4.(Š
3c<ïÄÄ:þ,$^a'gŽ•{,²LŒþOŽ,©òSmash{Èí2
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{È/¤Radford
0
s-BiHom-V“ê¿©7‡^‡.3©ïÄÄ:þ,ò5Œ?˜ÚïÄ
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DOI:10.12677/aam.2021.1082952845A^êÆ?Ð
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ë•©z
[1]Molnar,R.K.(1977)Semi-DirectProductsofHopfAlgebras.JournalofAlgebra,47,29-51.
https://doi.org/10.1016/0021-8693(77)90208-3
[2]Andruskiewitsch,N.andSchneider,H.J.(2010)OntheClassificationofFinite-Dimensional
PointedHopfAlgebras.AnnalsofMathematics,171,375-417.
https://doi.org/10.4007/annals.2010.171.375
[3]Chen,D.H.andLi,J.(2010)ConstructionofHom-ComoduleCoalgebrasandHom-Smash
Coproducts.JournalofJianghanUniversity(NaturalSciences),38,5-9.
[4]LingL.andShen,B.(2014)Radford’sBiproductsandYetter-DrinfeldModulesforMonoidal
Hom-HopfAlgebras.JournalofMathematicalPhysics,55,701-725.
[5]Ma,T., Li,H. andTao, Y.(2014) CobraidedSmashProductHom-HopfAlgebras. Colloquium
Mathematicum,134,75-92.https://doi.org/10.4064/cm134-1-3
[6]Abdenacer,M.andSilvestrov,S.D.(2008)Hom-AlgebraStructures.JournalofGeneralized
LieTheoryandApplications,2,51-64.https://doi.org/10.4303/jglta/S070206
[7]Graziani,G.,Makhlouf,A., Menini,C.,etal.(2015) BiHom-Associative Algebras,BiHom Lie
AlgebrasandBiHom-Bialgebras.SymmetryIntegrabilityandGeometry:MethodsandAppli-
cations,11,11-34.
[8]Zhang,J.,Chen,L.Y.andZhang,C.P.(2018)OnSplitRegularBiHom-LieSuperalgebras.
JournalofGeometryandPhysics,128, 38-47.https://doi.org/10.1016/j.geomphys.2018.02.005
[9]Liu,L.,Makhlouf,A.,Menini,C.,etal.(2020)BiHom-NovikovAlgebrasandInfinitesimal
BiHom-Bialgebras.Algebra,560,1146-1172.https://doi.org/10.1016/j.jalgebra.2020.06.012
DOI:10.12677/aam.2021.1082952846A^êÆ?Ð

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