A class of double crossed biproducts
Let $H$ be a bialgebra, let $A$ be an algebra and a left $H$-comodule coalgebra, let $B$ be an algebra and a right $H$-comodule coalgebra. Also let $f : H \otimes H \rightarrow A \otimes H, R : H \otimes A \rightarrow A \otimes H$, and $T : B \otimes H \rightarrow H \otimes B$ be linear map...
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| author | Dong, L. H. Li, H. Y. Ma, T. S. Донг, Л. Г. Лі, Г. Я. Ма, Т. С. |
| author_facet | Dong, L. H. Li, H. Y. Ma, T. S. Донг, Л. Г. Лі, Г. Я. Ма, Т. С. |
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| description | Let $H$ be a bialgebra, let $A$ be an algebra and a left $H$-comodule coalgebra, let $B$ be an algebra and a right $H$-comodule coalgebra. Also let $f : H \otimes H \rightarrow A \otimes H, R : H \otimes A \rightarrow A \otimes H$, and $T : B \otimes H \rightarrow H \otimes B$ be linear maps.
We present necessary and sufficient conditions for the one-sided Brzezi´nski’s crossed product algebra $A\#^f_RH_T\#B$ and the two-sided smash coproduct coalgebra $A \times H \times B$ to form a bialgebra, which generalizes the main results from [On
Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. It is clear that both Majid’s double biproduct
[Double-bosonization of braided groups and the construction of $U_q(g)$ // Math. Proc. Cambridge Phil. Soc. – 1999. – 125,
№ 1. – P. 151 – 192] and the Wang – Jiao – Zhao’s crossed product [Hopf algebra structures on crossed products // Communs
Algebra. – 1998. – 26. – P. 1293 – 1303] are obtained as special cases. |
| first_indexed | 2026-03-24T02:10:02Z |
| format | Article |
| fulltext |
UDC 512.5
T. S. Ma, H. Y. Li, L. H. Dong (School Math. and Inform. Sci., Henan Normal Univ., Xinxiang, China)
A CLASS OF DOUBLE CROSSED BIPRODUCTS*
ПРО ОДИН КЛАС ПОДВIЙНИХ ПЕРЕХРЕСНИХ БIДОБУТКIВ
Let H be a bialgebra, let A be an algebra and a left H -comodule coalgebra, let B be an algebra and a right H -comodule
coalgebra. Also let f : H \otimes H - \rightarrow A \otimes H , R : H \otimes A - \rightarrow A \otimes H, and T : B \otimes H - \rightarrow H \otimes B be linear maps.
We present necessary and sufficient conditions for the one-sided Brzeziński’s crossed product algebra A\#f
RHT\#B and
the two-sided smash coproduct coalgebra A \times H \times B to form a bialgebra, which generalizes the main results from [On
Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. It is clear that both Majid’s double biproduct
[Double-bosonization of braided groups and the construction of Uq(g) // Math. Proc. Cambridge Phil. Soc. – 1999. – 125,
№ 1. – P. 151 – 192] and the Wang – Jiao – Zhao’s crossed product [Hopf algebra structures on crossed products // Communs
Algebra. – 1998. – 26. – P. 1293 – 1303] are obtained as special cases.
Нехай H — бiалгебра, A — алгебра та водночас лiва H -комодульна коалгебра, а B — алгебра та водночас права H -
комодульна коалгебра. Крiм того, нехай f : H \otimes H - \rightarrow A\otimes H , R : H \otimes A - \rightarrow A\otimes H та T : B \otimes H - \rightarrow H \otimes B —
лiнiйнi вiдображення. Наведено необхiднi та достатнi умови для того, щоб одностороння алгебра Бжезiнського
A\#f
RHT\#B з перехресним добутком та двостороння коалгебра A\times H \times B зi схрещеним кодобутком утворювали
бiалгебру, що узагальнює основнi результати, отриманi в [On Ranford biproduct // Communs Algebra. – 2015. – 43,
№ 9. – P. 3946 – 3966]. Очевидно, що як подвiйний бiдобуток Маджiда [Double-bosonization of braided groups and the
construction of Uq(g) // Math. Proc. Cambridge Phil. Soc. – 1999. – 125, № 1. – P. 151 – 192], так i перехресний добуток
Ванга – Джао – Жао [Hopf algebra structures on crossed products // Communs Algebra. – 1998. – 26. – P. 1293 – 1303]
можна отримати як частиннi випадки.
1. Introduction and preliminaries. Let H be a Hopf algebra over a field K. S. Majid [10, 11]
made the following conclusion: A is a bialgebra in Yetter – Drinfeld category H
H\scrY \scrD if and only if
A \star H is a Radford biproduct [14]. The Radford biproduct plays an important role in the lifting
method for the classification of finite dimensional pointed Hopf algebras [2]. Let A be a bialgebra
in H
H\scrY \scrD and B a bialgebra in \scrY \scrD H
H . In [9], S. Majid gave the sufficient conditions for a two-sided
smash product algebra A\#H\#B and a two-sided smash coproduct coalgebra A \times H \times B to be
a bialgebra, named the double biproduct and denoted by A\diamondsuit H\diamondsuit B. Some related results about the
double biproduct were recently given in the literature [6, 7, 9, 13].
Let A be an associative and unitary algebra and H a vector space endowed with a distinguished
element 1H . Let f : H\otimes H - \rightarrow A\otimes H and R : H\otimes A - \rightarrow A\otimes H be two linear maps. Let A\#f
RH be
an associative and unitary algebra, with underlying vector space A\otimes H. Following [4], T. Brzeziński
gave the necessary and sufficient conditions for the crossed product A\#f
RH to be an algebra, called
Brzeziński’s crossed product. Brzeziński’s crossed product is an extensive definition that includes
the crossed product A\#\sigma H in [3] and the twisted tensor product A\#RH in [5]. In [6], the authors
replaced the left smash product by the crossed product A\#\sigma H in the double biproduct A\diamondsuit H\diamondsuit B
and obtained a generalized version of A\diamondsuit H\diamondsuit B. And in [7], the authors gave a further extension of
A\diamondsuit H\diamondsuit B via Brzeziński’s crossed product. When the twisted tensor product takes the place of the
right smash product in A\diamondsuit H\diamondsuit B, we want to know under what conditions the resulting structure will
inherit a bialgebra structure. In this paper, we will derive the necessary and sufficient conditions for
* This work was partially supported by China Postdoctoral Science Foundation (No. 2017M611291), Foundati-
on for Young Key Teacher by Henan Province (No. 2015GGJS-088), Natural Science Foundation of Henan Province
(No. 17A110007), and National Natural Science Foundation of China (No. 11801150).
c\bigcirc T. S. MA, H. Y. LI, L. H. DONG, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11 1533
1534 T. S. MA, H. Y. LI, L. H. DONG
the one-sided Brzeziński’s crossed product algebra A\#f
RHT\#B and the two-sided smash coproduct
coalgebra A\times H \times B to be a bialgebra, which we call the Brzeziński’s double biproduct. The main
results in [6, 7] will be included, of course, the celebrated Radford biproduct [14], Majid’s double
biproduct [9], Agore and Militaru’s unified product [1] and Wang – Jiao – Zhao’s crossed product [16]
are all examples of the Brzeziński’s double biproduct.
Throughout the paper, we follow the definitions and terminologies in [12, 15] and all algebraic
systems are over a field K. Let C be a coalgebra. Then we use the simple Sweedler’s notation
for the comultiplication, \Delta (c) = c1 \otimes c2, c \in C. We denote the category of left H -comodules by
H\scrM , for (M,\rho ) \in H\scrM and write \rho (x) = x( - 1) \otimes x(0) \in H \otimes M for all x \in M. We denote the
category of right H -comodules by \scrM H for (M,\psi ) \in \scrM H , write \psi (x) = x[0] \otimes x[1] \in M \otimes H,
for all x \in M. We denote the left-left Yetter – Drinfeld category by H
H\scrY \scrD and the right-right Yetter –
Drinfeld category by \scrY \scrD H
H . Given a K -space M, we write idM for the identity map on M.
Next we recall [4, 9, 12, 14] some basic definitions and results which will be used later.
Brzeziński’s crossed product. Let A be an algebra, H a vector space and 1H \in H. The vector
space A\otimes H is an algebra with unit 1A \otimes 1H and a product such that
(a\otimes 1H)(a\prime \otimes x\prime ) = aa\prime \otimes x\prime
if and only if there exist linear maps f : H \otimes H - \rightarrow A \otimes H
\bigl(
write f(x \otimes x\prime ) = xf \otimes x\prime f for all
x, x\prime \in H
\bigr)
and R : H \otimes A - \rightarrow A\otimes H (write R(x\otimes a) = aR \otimes xR for all x \in H and a \in A) that
satisfy the following conditions:
(A1) aR \otimes 1HR = a\otimes 1H , 1AR \otimes xR = 1A \otimes x;
(A2) (aa\prime )R \otimes xR = aRa
\prime
r \otimes xRr;
(A3) xf \otimes 1Hf = 1H
f \otimes xf = 1A \otimes x;
(A4) x\prime fRxR
g \otimes x\prime \prime fg = xfx\prime f
g \otimes x\prime \prime g ;
(A5) aRrxr
f \otimes x\prime Rf = xfaR \otimes x\prime fR
for all a, a\prime \in A, x, x\prime , x\prime \prime \in H, where g = f and r = R.
The product \mu A\otimes H in A\otimes H explicitly reads
(a\otimes x)(a\prime \otimes x\prime ) = aa\prime RxR
f \otimes x\prime f
for all a, a\prime \in A and x, x\prime \in H. In this case, we call the algebra Brzeziński’s crossed product [4] and
denote it by A\#f
RH.
– Let A be a bialgebra and H a coalgebra with 1H \in H. A Brzeziński’s crossed product,
A\#f
RH, equipped with the usual tensor product coalgebra structure is a bialgebra if and only if the
following conditions hold:
(B1) \Delta H(1H) = 1H \otimes 1H and \varepsilon H(1H) = 1;
(B2) f is a coalgebra map;
(B3) R is a coalgebra map.
Double biproduct. We recall, from [9], the construction of the so-called double biproduct.
Let H be a bialgebra, A a bialgebra in H
H\scrY \scrD , and B a bialgebra in \scrY \scrD H
H . Adopt the following
notation for the structure maps: the counits are \varepsilon A and \varepsilon B, the comultiplications are \Delta A(a) = a1\otimes a2
and \Delta B(b) = b1 \otimes b2, and the actions and coactions are
H \otimes A - \rightarrow A, x\otimes a \mapsto \rightarrow x \triangleleft a,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
A CLASS OF DOUBLE CROSSED BIPRODUCTS 1535
A - \rightarrow H \otimes A, a \mapsto \rightarrow a( - 1) \otimes a(0),
B \otimes H - \rightarrow B, b\otimes x \mapsto \rightarrow b \triangleright x,
B - \rightarrow B \otimes H, b \mapsto \rightarrow b[0] \otimes b[1]
for all x \in H, a \in A, b \in B. Let A\diamondsuit H\diamondsuit B denote the vector space A\otimes H \otimes B, which becomes an
algebra (called the two-sided smash product, A\#H\#B) with unit 1A \otimes 1H \otimes 1B and multiplication
(a\otimes x\otimes b)(a\prime \otimes x\prime \otimes b\prime ) = a(x1 \triangleleft a
\prime )\otimes x2x
\prime
1 \otimes (b \triangleright x\prime 2)b
\prime ,
and a coalgebra (called the two-sided smash coproduct, A \times H \times B) with counit \varepsilon (a \otimes x \otimes b) =
= \varepsilon A(a)\varepsilon H(x)\varepsilon B(b) and comultiplication
\Delta : A\diamondsuit H\diamondsuit B - \rightarrow (A\diamondsuit H\diamondsuit B)\otimes (A\diamondsuit H\diamondsuit B),
\Delta (a\otimes x\otimes b) = a1 \otimes a2( - 1)x1 \otimes b1[0] \otimes a2(0) \otimes x2b1[1] \otimes b2.
Moreover, assume that the following condition holds:
(DB ) b[1] \triangleleft a(0) \otimes b[0] \triangleright a( - 1) = a\otimes b, a \in A, b \in B.
It follows that A\diamondsuit H\diamondsuit B is a bialgebra, called the double biproduct.
Remark 1.1. When A = K (or B = K ), the double biproduct is exactly the right (or left)
variant of Radford biproduct.
2. Main results and its consequence. In this section, we give an extended version of the
structure of the Majid’s double biproduct.
First, we list the right version of twisted tensor product.
Proposition 2.1. Let B and H be two algebras, T : B \otimes H - \rightarrow H \otimes B a linear map. Then
HT\#B (= H \otimes B as a linear space) with the multiplication
(x\otimes b)(x\prime \otimes b\prime ) = xx\prime T \otimes bT b
\prime ,
where x, x\prime \in H, b, b\prime \in B, and unit 1H \otimes 1B becomes an algebra if and only if the following
conditions hold:
(RT1) bT \otimes 1HT = b\otimes 1H , 1BT \otimes xT = 1B \otimes x,
(RT2) xT \otimes (bb\prime )T = xTt \otimes btb
\prime
T ,
(RT3) (xx\prime )T \otimes bT = xTx
\prime
t \otimes bTt,
where x, x\prime \in H, b, b\prime \in B, and t = T. We call this algebra right twisted tensor product algebra
and denote it by HT\#B.
Proof. Straightforward.
Lemma 2.1. Let H be a vector space and 1H \in H and A, B be two algebras. Let f :
H\otimes H - \rightarrow A\otimes H, R : H\otimes A - \rightarrow A\otimes H and T : B\otimes H - \rightarrow H\otimes B be linear maps. If conditions
(RT1), (RT2), (A1) – (A3) and
(BT1) x\prime fRxR
F \otimes x\prime \prime fTF \otimes bT = xfx\prime Tf
F \otimes x\prime \prime tF \otimes bTt;
(BT2) aRrxr
f \otimes x\prime RTf \otimes bT = xfaR \otimes x\prime TfR \otimes bT
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1536 T. S. MA, H. Y. LI, L. H. DONG
are satisfied for all a \in A, b \in B, x, x\prime , x\prime \prime \in H and F = f, r = R, t = T, then A\#f
RHT\#B
(= A\otimes H\otimes B as a vector space) is an associative algebra with unit 1A\otimes 1H\otimes 1B and multiplication
given by
(a\otimes x\otimes b)(a\prime \otimes x\prime \otimes b\prime ) = aa\prime RxR
f \otimes x\prime T f \otimes bT b
\prime ,
where a, a\prime \in A, x, x\prime \in H and b, b\prime \in B. In this case, we call A\#f
RHT\#B the one-sided Brzeziński
crossed product.
Proof. We check associativity as follows. For a, a\prime , a\prime \prime \in A, x, x\prime , x\prime \prime \in H, b, b\prime , b\prime \prime \in B and
F = f, \=R = r = \=r = R,
(a\otimes x\otimes b)
\bigl(
(a\prime \otimes x\prime \otimes b\prime )(a\prime \prime \otimes x\prime \prime \otimes b\prime \prime )
\bigr)
=
= a(a\prime a\prime \prime Rx
\prime
R
f
)rxr
F \otimes x\prime \prime TftF \otimes btb
\prime
T b
\prime \prime (A2)
=
(A2)
= aa\prime ra
\prime \prime
R \=Rx
\prime
R
f
\=rxr \=R\=r
F \otimes x\prime \prime TftF \otimes btb
\prime
T b
\prime \prime (BT1)
=
(BT1)
= aa\prime ra
\prime \prime
R \=Rxr \=R
fx\prime Rtf
F \otimes x\prime \prime T \=tF \otimes bt\=tb
\prime
T b
\prime \prime (BT2)
=
(BT2)
= aa\prime rxr
fa\prime \prime Rx
\prime
tfR
F \otimes x\prime \prime T \=tF \otimes bt\=tb
\prime
T b
\prime \prime (RT2)
=
(RT2)
= aa\prime rxr
fa\prime \prime Rx
\prime
TfR
F \otimes x\prime \prime tF \otimes (bT b
\prime )tb
\prime \prime =
=
\bigl(
(a\otimes x\otimes b)(a\prime \otimes x\prime \otimes b\prime )
\bigr)
(a\prime \prime \otimes x\prime \prime \otimes b\prime \prime ).
It’s obvious that 1A \otimes 1H \otimes 1B is a unit by conditions (A1), (A3) and (RT1).
Remark 2.1. (1) If there is an element \varepsilon B \in \mathrm{H}\mathrm{o}\mathrm{m}(B,K) such that \varepsilon B(1B) = 1 and T is
trivial, i.e., T is the flip map, then the conditions (A4) and (A5) can be obtained by setting b = 1B
and applying \mathrm{i}\mathrm{d}A\otimes \mathrm{i}\mathrm{d}H \otimes \varepsilon B to the condition (BT1) and (BT2), respectively.
(2) If H is an algebra, and there is an element \varepsilon A \in \mathrm{H}\mathrm{o}\mathrm{m}(A,K) such that \varepsilon A(1A) = 1,
and f, F are trivial, then the condition (RT3) can be obtained by setting x = 1H and applying
\varepsilon A \otimes \mathrm{i}\mathrm{d}H \otimes \mathrm{i}\mathrm{d}B to the condition (BT1).
(3) Taking either A = K or B = K, we obtain the right twisted tensor product and Brzeziński
crossed product, respectively.
Theorem 2.1. Let H be a bialgebra, A an algebra and a left H -comodule coalgebra such that
\varepsilon A(1A) = 1, B an algebra and a right H -comodule coalgebra such that \varepsilon B(1B) = 1. Let f :
H \otimes H - \rightarrow A\otimes H, R : H \otimes A - \rightarrow A\otimes H, T : B \otimes H - \rightarrow H \otimes B be linear maps such that
(BP ) a( - 1)x
\prime
T1 \otimes bT1[0]b
\prime
[0] \otimes a(0)RxR
fbT1[1]
F \otimes x\prime T2fb
\prime
[1]tF \otimes bT2t =
= (a( - 1)x
\prime
1)T \otimes b1[0]T b
\prime
[0] \otimes a(0)R(xb1[1])R
f \otimes (x\prime 2b
\prime
[1])tf \otimes b2t
holds for all a \in A, x, x\prime \in H, b, b\prime \in B and F = f, t = T. Then the one-sided Brzeziński crossed
product A\#f
RHT\#B equipped with the two-sided smash coproduct A\times H\times B becomes a bialgebra
if and only if the following conditions hold (a, a\prime \in A, x, x\prime \in H, b, b\prime \in B, F = f and r = R):
(C1) \varepsilon A, \varepsilon B are algebra maps, \varepsilon A(xf )\varepsilon H(x\prime f ) = \varepsilon H(x)\varepsilon H(x\prime );
(C2) \varepsilon A(aR)\varepsilon H(xR) = \varepsilon A(a)\varepsilon H(x), \varepsilon B(bT )\varepsilon H(xT ) = \varepsilon B(b)\varepsilon H(x);
(C3) \Delta A(1A) = 1A \otimes 1A, \Delta B(1B) = 1B \otimes 1B;
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
A CLASS OF DOUBLE CROSSED BIPRODUCTS 1537
(C4) 1A( - 1) \otimes 1A(0) = 1H \otimes 1A, 1B [0] \otimes 1B [1] = 1B \otimes 1H ;
(C5) (bb\prime )1[0] \otimes 1A \otimes (bb\prime )1[1] \otimes (bb\prime )2 = b1[0]b
\prime
1[0] \otimes b1[1]
f \otimes b\prime 1[1]Tf
\otimes b2T b
\prime
2;
(C6) a1 \otimes a2( - 1)x\otimes a2(0) = a1a2( - 1)
f \otimes xf \otimes a2(0);
(C7) (aa\prime )1 \otimes (aa\prime )2( - 1) \otimes (aa\prime )2(0) = a1a
\prime
1Ra2( - 1)R
f \otimes a\prime 2( - 1)f \otimes a2(0)a
\prime
2(0);
(C8) xf 1 \otimes xf 2( - 1)x
\prime
f1 \otimes xf 2(0) \otimes x\prime f2 = x1
f \otimes x\prime 1f \otimes x2
F \otimes x\prime 2F ;
(C9) aR1 \otimes aR2( - 1)xR1 \otimes aR2(0) \otimes xR2 = a1Rx1R
f \otimes a2( - 1)f \otimes a2(0)r \otimes x2r.
In this case, we call the bialgebra Brzeziński double biproduct, and denote it by A\diamondsuit f
RHT\diamondsuit B.
Proof. Sufficiency. It is easy to prove that \varepsilon A\times H\times B is an algebra map. Here we check only that
\Delta A\times H\times B is an algebra map. We have
\Delta A\times H\times B((a\otimes x\otimes b)(a\prime \otimes x\prime \otimes b\prime )) =
= (aa\prime RxR
f )1 \otimes (aa\prime RxR
f )2( - 1)x
\prime
Tf1 \otimes (bT b
\prime )1[0] \otimes (aa\prime RxR
f )2(0) \otimes x\prime Tf2(bT b
\prime )1[1]\otimes
\otimes (bT b
\prime )2
(C6)
= (aa\prime RxR
f )1(aa
\prime
RxR
f )2( - 1)
F \otimes x\prime Tf1F \otimes (bT b
\prime )1[0] \otimes (aa\prime RxR
f )2(0)\otimes
\otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(C7)
=
(C7)
= a1(a
\prime
RxR
f )1 \=Ra2( - 1) \=R
\=f (a\prime RxR
f )2( - 1) \=f
F \otimes x\prime Tf1F \otimes (bT b
\prime )1[0] \otimes a2(0)(a
\prime
RxR
f )2(0)\otimes
\otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(A4)
=
(A4)
= a1(a
\prime
RxR
f )1 \=R(a
\prime
RxR
f )2( - 1)
\=f
r
a2( - 1) \=Rr
F \otimes x\prime Tf1 \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)(a
\prime
RxR
f )2(0)\otimes
\otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(A2)
=
(A2)
= a1((a
\prime
RxR
f )1(a
\prime
RxR
f )2( - 1)
\=f
)ra2( - 1)r
F \otimes x\prime Tf1 \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)(a
\prime
RxR
f )2(0)\otimes
\otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(C7)
=
(C7)
= a1(a
\prime
R1xR
f
1 \=Ra
\prime
R2( - 1) \=R
\=F
xR
f
2( - 1) \=F
\=f
)ra2( - 1)r
F \otimes x\prime Tf1 \=fF \otimes (bT b
\prime )1[0]\otimes
\otimes a2(0)a\prime R2(0)xR
f
2(0) \otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(A4)
=
(A4)
= a1(a
\prime
R1xR
f
1 \=RxR
f
2( - 1)
\=F
\=r
a\prime R2( - 1) \=R\=r
\=f
)ra2( - 1)r
F \otimes x\prime Tf1 \=F \=fF \otimes (bT b
\prime )1[0]\otimes
\otimes a2(0)a\prime R2(0)xR
f
2(0) \otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(A2)
=
(A2)
= a1(a
\prime
R1(xR
f
1xR
f
2( - 1)
\=F
) \=Ra
\prime
R2( - 1) \=R
\=f
)ra2( - 1)r
F \otimes x\prime Tf1 \=F \=fF \otimes (bT b
\prime )1[0]\otimes
\otimes a2(0)a\prime R2(0)xR
f
2(0) \otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(C6)
=
(C6)
= a1(a
\prime
R1xR
f
1 \=Ra
\prime
R2( - 1) \=R
\=f
)ra2( - 1)r
F \otimes (xR
f
2( - 1)x
\prime
Tf1) \=fF \otimes (bT b
\prime )1[0]\otimes
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1538 T. S. MA, H. Y. LI, L. H. DONG
\otimes a2(0)a\prime R2(0)xR
f
2(0) \otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2
(C8)
=
(C8)
= a1(a
\prime
R1xR1
f
\=Ra
\prime
R2( - 1) \=R
\=f
)ra2( - 1)r
F \otimes x\prime T1f \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
R2(0)xR2
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A4)
=
(A4)
= a1(a
\prime
R1a
\prime
R2( - 1)
f
xR1f
\=f )ra2( - 1)r
F \otimes x\prime T1 \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
R2(0)xR2
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A2)
=
(A2)
= a1(a
\prime
R1a
\prime
R2( - 1)
f
)rxR1f
\=f
\=Ra2( - 1)r \=R
F \otimes x\prime T1 \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
R2(0)xR2
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(C6)
=
(C6)
= a1a
\prime
R1r(a
\prime
R2( - 1)xR1)
\=f
\=R
a2( - 1)r \=R
F \otimes x\prime T1 \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
R2(0)xR2
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A4)
=
(A4)
= a1a
\prime
R1ra2( - 1)r
\=f (a\prime R2( - 1)xR1) \=f
F \otimes x\prime T1F \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
R2(0)xR2
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(C9)
=
(C9)
= a1(a
\prime
1Rx1R
f )ra2( - 1)r
\=fa\prime 2( - 1)f \=f
F \otimes x\prime T1F \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A2)
=
(A2)
= a1a
\prime
1Rrx1R
f
\=ra2( - 1)r\=r
\=fa\prime 2( - 1)f \=f
F \otimes x\prime T1F \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A4)
=
(A4)
= a1a
\prime
1Rra2( - 1)r
fx1Rf
\=fa\prime 2( - 1) \=f
F \otimes x\prime T1F \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A5)
=
(A5)
= a1a2( - 1)
fa\prime 1Rx1fR
\=fa\prime 2( - 1) \=f
F \otimes x\prime T1F \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A4)
=
(A4)
= a1a2( - 1)
fa\prime 1Ra
\prime
2( - 1)
\=f
r
x1fRr
F \otimes x\prime T1 \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F\otimes
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(A2)
=
(A2)
= a1a2( - 1)
f (a\prime 1a
\prime
2( - 1)
\=f
)Rx1fR
F \otimes x\prime T1 \=fF \otimes (bT b
\prime )1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F\otimes
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
A CLASS OF DOUBLE CROSSED BIPRODUCTS 1539
\otimes x\prime T2 \=F (bT b
\prime )1[1] \otimes (bT b
\prime )2
(C5)
=
(C5)
= a1a2( - 1)
f (a\prime 1a
\prime
2( - 1)
\=f
)Rx1fR
F \otimes x\prime T1 \=fF \otimes bT1[0]b
\prime
1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F bT1[1]
\=\=F\otimes
\otimes x\prime T2 \=F b
\prime
1[1]t \=\=F
\otimes bT2tb
\prime
2
(C6)
=
(C6)
= a1a2( - 1)
fa\prime 1Rx1fR
F \otimes (a\prime 2( - 1)x
\prime
T1)F \otimes bT1[0]b
\prime
1[0] \otimes a2(0)a
\prime
2(0) \=Rx2 \=R
\=F bT1[1]
\=\=F\otimes
\otimes x\prime T2 \=F b
\prime
1[1]t \=\=F
\otimes bT2tb
\prime
2
(BP )
=
(BP )
= a1a2( - 1)
fa\prime 1Rx1fR
F \otimes (a\prime 2( - 1)x
\prime
1)TF \otimes b1[0]T b
\prime
1[0] \otimes a2(0)a
\prime
2(0) \=R(x2b1[1]) \=R
\=F\otimes
\otimes (x\prime 2b
\prime
1[1])t \=F \otimes b2tb
\prime
2
(C6)
=
(C6)
= a1a
\prime
1R(a2( - 1)x1)R
F \otimes (a\prime 2( - 1)x
\prime
1)TF \otimes b1[0]T b
\prime
1[0] \otimes a2(0)a
\prime
2(0) \=R(x2b1[1]) \=R
\=F\otimes
\otimes (x\prime 2b
\prime
1[1])t \=F \otimes b2tb
\prime
2 = \Delta A\times H\times B(a\otimes x\otimes b)\Delta A\times H\times B(a
\prime \otimes x\prime \otimes b\prime ).
Necessity. Since \varepsilon
A\diamondsuit f
RH\diamondsuit B
is an algebra map, we get
(BA1) \varepsilon A(aa
\prime
RxR
f )\varepsilon H(x\prime T f )\varepsilon B(bT b
\prime ) = \varepsilon A(a)\varepsilon A(a
\prime )\varepsilon H(x)\varepsilon H(x\prime )\varepsilon B(b)\varepsilon B(b
\prime ).
Let x = x = 1H , b = b\prime = 1B, x = x = 1H , a = a\prime = 1A and a = a\prime = 1A, b = b\prime = 1B in
Eq. (BA1), respectively, we obtain (C1). Similarly, (C2) holds.
Apply \mathrm{i}\mathrm{d}A\otimes \varepsilon H \otimes \varepsilon B \otimes \mathrm{i}\mathrm{d}A\otimes \varepsilon H \otimes \varepsilon B (respectively \varepsilon A \otimes \varepsilon H \otimes \mathrm{i}\mathrm{d}B \otimes \varepsilon A \otimes \varepsilon H \otimes \mathrm{i}\mathrm{d}B ) to
(BA2) 1A1 \otimes 1A2( - 1) \otimes 1B1[0] \otimes 1A2(0) \otimes 1B1[1] \otimes 1B2 = 1A \otimes 1H \otimes 1B \otimes 1A \otimes 1H \otimes 1B,
we have (C3). Likewise, we get (C4).
Since \Delta A\times H\times B
\bigl(
(a\otimes x\otimes b)(a\prime \otimes x\prime \otimes b\prime )
\bigr)
= \Delta A\times H\times B(a\otimes x\otimes b)\Delta A\times H\times B(a
\prime \otimes x\prime \otimes b\prime ), we
obtain
(BA3) (aa\prime RxR
f )1 \otimes (aa\prime RxR
f )2( - 1)x
\prime
1f1 \otimes (bT b
\prime )1[0] \otimes (aa\prime RxR
f )2(0)\otimes
\otimes x\prime Tf2(bT b
\prime )1[1] \otimes (bT b
\prime )2 = a1a
\prime
1R(a2( - 1)x1)R
f \otimes (a\prime 2( - 1)x
\prime
1)Tf\otimes
\otimes b1[0]T b\prime 1[0] \otimes a2(0)a
\prime
2(0)r(x2b1[1])r
F \otimes (x\prime 2b
\prime
1[1])tF \otimes b2tb
\prime
2.
Let x = x\prime = 1H and a = a\prime = 1A in Eq. (BA3), we have
1A \otimes 1H \otimes (bb\prime )1[0] \otimes 1A \otimes (bb\prime )1[1] \otimes (bb\prime )2 =
= 1A \otimes 1H \otimes b1[0]b
\prime
1[0] \otimes b1[1]
f \otimes b\prime 1[1]tf \otimes b2tb
\prime
2.
Apply \varepsilon A\otimes \varepsilon H\otimes \mathrm{i}\mathrm{d}B \otimes \mathrm{i}\mathrm{d}A\otimes \mathrm{i}\mathrm{d}H \otimes \mathrm{i}\mathrm{d}B to the above equation, we get (C5). The conditions (C6) –
(C9) can be derived by the similar method.
Remark 2.2. 1. Let x = x\prime = 1H , b
\prime = 1B, R(x\otimes a) = x1 \triangleleft a\otimes x2 and T (b\otimes x) = x1\otimes b \triangleright x2
in Eq. (BP ), we can obtain the condition (DB ). Then Brzeziński double biproduct A\diamondsuit f
RHT\diamondsuit B
is the double biproduct A\diamondsuit H\diamondsuit B when f(x \otimes y) = 1A \otimes xy, R(x \otimes a) = x1 \triangleleft a \otimes x2 and
T (b\otimes x) = x1 \otimes b \triangleright x2 in Theorem 2.1.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1540 T. S. MA, H. Y. LI, L. H. DONG
2. Setting A = K and B = K, we obtain the right version of Radford biproduct in [14] and
Brzeziński crossed biproduct in [6], respectively. Furthermore, if the left comodule coaction is trivial,
and f(x, x\prime ) = \sigma (x1, x
\prime
1) \otimes x2x
\prime
2, R(x \otimes a) = x1 \triangleleft a1 \otimes x2 \triangleright a2 in Brzeziński crossed biproduct,
then we can get Agore and Militaru’s unified product [1].
3. Taking f(x, x\prime ) = \sigma (x1, x
\prime
1)\otimes x2x
\prime
2 in Brzeziński crossed biproduct, we obtain Wang – Jiao –
Zhao’s crossed product in [16].
4. Setting T (b\otimes x) = x1\otimes b \triangleright x2 in Theorem 2.1, we can get the main result in [7] (Theorem 3.2).
And the condition (BP ) here implies the condition (C11) there.
5. Let A = K, the maps f : H \otimes H - \rightarrow A \otimes H and R : H \otimes A - \rightarrow A \otimes H be trivial in
Theorem 2.1, we can obtain that the right twisted tensor product HT\#B equipped with the right
smash coproduct H \times B becomes a bialgebra if and only if the following conditions hold (x \in H,
b, b\prime \in B and T = t):
(D1) \varepsilon B are algebra maps, \varepsilon B(bT )\varepsilon H(xT ) = \varepsilon B(b)\varepsilon H(x);
(D2) 1B [0] \otimes 1B [1] = 1B \otimes 1H , \Delta B(1B) = 1B \otimes 1B;
(D3) (bb\prime )1[0] \otimes (bb\prime )1[1] \otimes (bb\prime )2 = b1[0]b
\prime
1[0] \otimes b1[1]b
\prime
1[1]T
\otimes b2T b
\prime
2;
(D4) xT1 \otimes bT1[0] \otimes xT2bT1[1] \otimes bT2 = x1T \otimes b1[0]T \otimes b1[1]x2t \otimes b2t.
This exactly is the right version of [8] (Corollary 2.5).
References
1. Agore A. L., Militaru G. Extending structures II: The quantum version // J. Algebra. – 2011. – 336. – P. 321 – 341.
2. Andruskiewitsch N., Schneider H.-J. On the classification of finite-dimensional pointed Hopf algebras // Ann. Math. –
2010. – 171, № 1. – P. 375 – 417.
3. Blattner R. J., Cohen M., Montgomery S. Crossed products and inner actions of Hopf algebras // Trans. Amer. Math.
Soc. – 1986. – 289. – P. 671 – 711.
4. Brzeziński T. Crossed products by a coalgebra // Communs Algebra. – 1997. – 25. – P. 3551 – 3575.
5. Caenepeel S., Ion B., Militaru G., Zhu S. L. The factorization problem and the smash biproduct of algebras and
coalgebras // Algebra. Represent Theory. – 2000. – 3. – P. 19 – 42.
6. Ma T. S., Jiao Z. M., Song Y. N. On crossed double biproduct // J. Algebra and Appl. – 2013. – 12, № 5. – 17 p.
7. Ma T. S., Li H. Y. On Radford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966.
8. Ma T. S., Wang S. H. General double quantum groups // Communs Algebra. – 2010. – 38, № 2. – P. 645 – 672.
9. Majid S. Double-bosonization of braided groups and the construction of Uq(g) // Math. Proc. Cambridge Phil. Soc. –
1999. – 125, № 1. – P. 151 – 192.
10. Majid S. Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group // Communs Math.
Phys. – 1993. – 156. – P. 607 – 638.
11. Majid S. Algebras and Hopf algebras in braided categories // Adv. Hopf algebras (Chicago, IL, 1992): Lect. Notes
Pure and Appl. Math. – 1994. –158. – P. 55 – 105.
12. Montgomery S. Hopf algebras and their actions on rings // CBMS Lect. Math. – 1993. – 82.
13. Panaite F., Van Oystaeyen F. L-R-smash biproducts, double biproducts and a braided category of Yetter – Drinfeld –
Long bimodules // Rocky Mountain J. Math. – 2010. – 40, № 6. – P. 2013 – 2024.
14. Radford D. E. The structure of Hopf algebra with a projection // J. Algebra. – 1985. – 92. – P. 322 – 347.
15. Radford D. E. Hopf algebras // KE Ser. Knots and Everything. – New Jersey: World Sci., 2012. – 49.
16. Wang S. H., Jiao Z. M., Zhao W. Z. Hopf algebra structures on crossed products // Communs Algebra. – 1998. – 26. –
P. 1293 – 1303.
Received 21.11.14,
after revision — 18.08.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
|
| id | umjimathkievua-article-1657 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:02Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a9/c102150047447efdedcd9fcd1e621da9.pdf |
| spelling | umjimathkievua-article-16572019-12-05T09:22:19Z A class of double crossed biproducts Про один клас подвiйних перехресних бiдобуткiв Dong, L. H. Li, H. Y. Ma, T. S. Донг, Л. Г. Лі, Г. Я. Ма, Т. С. Let $H$ be a bialgebra, let $A$ be an algebra and a left $H$-comodule coalgebra, let $B$ be an algebra and a right $H$-comodule coalgebra. Also let $f : H \otimes H \rightarrow A \otimes H, R : H \otimes A \rightarrow A \otimes H$, and $T : B \otimes H \rightarrow H \otimes B$ be linear maps. We present necessary and sufficient conditions for the one-sided Brzezi´nski’s crossed product algebra $A\#^f_RH_T\#B$ and the two-sided smash coproduct coalgebra $A \times H \times B$ to form a bialgebra, which generalizes the main results from [On Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. It is clear that both Majid’s double biproduct [Double-bosonization of braided groups and the construction of $U_q(g)$ // Math. Proc. Cambridge Phil. Soc. – 1999. – 125, № 1. – P. 151 – 192] and the Wang – Jiao – Zhao’s crossed product [Hopf algebra structures on crossed products // Communs Algebra. – 1998. – 26. – P. 1293 – 1303] are obtained as special cases. Нехай $H$ — бiалгебра, $A$ — алгебра та водночас лiва $H$-комодульна коалгебра, а $B$ — алгебра та водночас права $H$-комодульна коалгебра. Крiм того, нехай $f : H \otimes H \rightarrow A\otimes H,\; R : H \otimes A \rightarrow A\otimes H$ та $T : B \otimes H \rightarrow H \otimes B$ — лiнiйнi вiдображення. Наведено необхiднi та достатнi умови для того, щоб одностороння алгебра Бжезiнського $A\#^f_RH_T\#B$ з перехресним добутком та двостороння коалгебра $A \times H \times B$ зi схрещеним кодобутком утворювали бiалгебру, що узагальнює основнi результати, отриманi в [On Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. Очевидно, що як подвiйний бiдобуток Маджiда [Double-bosonization of braided groups and the construction of $U_q(g)$ // Math. Proc. Cambridge Phil. Soc. – 1999. – 125, № 1. – P. 151 – 192], так i перехресний добуток Ванга – Джао –Жао [Hopf algebra structures on crossed products // Communs Algebra. – 1998. – 26. – P. 1293 – 1303] можна отримати як частиннi випадки. Institute of Mathematics, NAS of Ukraine 2018-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1657 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 11 (2018); 1533-1540 Український математичний журнал; Том 70 № 11 (2018); 1533-1540 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1657/639 Copyright (c) 2018 Dong L. H.; Li H. Y.; Ma T. S. |
| spellingShingle | Dong, L. H. Li, H. Y. Ma, T. S. Донг, Л. Г. Лі, Г. Я. Ма, Т. С. A class of double crossed biproducts |
| title | A class of double crossed biproducts |
| title_alt | Про один клас подвiйних перехресних бiдобуткiв |
| title_full | A class of double crossed biproducts |
| title_fullStr | A class of double crossed biproducts |
| title_full_unstemmed | A class of double crossed biproducts |
| title_short | A class of double crossed biproducts |
| title_sort | class of double crossed biproducts |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1657 |
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