A lazy 2-cocycle of a Hopf algebra $ H $ is a 2-cocycle $ {\sigma}: H{\otimes} H{\longrightarrow} K $, which commutes with multiplication in the Hopf algebra. The second lazy cohomology group generalizes Sweedler's second cohomology group of a cocommutative Hopf algebra and the Schur multiplier of a group. Let $ B\diamondsuit H $ be a Radford biproduct, where $ H $ is a Hopf algebra and $ B $ is a Hopf algebra in the category of Yetter-Drinfeld modules over $ H $. A group morphism $ H^2_L(B){\longrightarrow} H^2_L(B\diamondsuit H) $ is constructed by Cuadra and Panaite in [1]. In [2], Panaite et al. introduced the concepts of pure and neat lazy 2-cocycle and extended pure and neat lazy cocycles to the Radford biproducts.

The origins of the study of Hom-algebras can be found in [3] by Hartwig, Larsson and Silvestrov, and earlier precursors of Hom-Lie algebras can be found in Hu's paper (see [4]). Subsequently, Hom-type algebra has been studied by many researchers. Especially, in 2014, Li and Ma introduced the notions of Radford biproduct Hom-Hopf algebra $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $ and Hom-Yetter-Drinfeld category $ _H^H{\mathbb{YD}} $ (see [5]), which generalize the corresponding concepts in usual Hopf algebras. In 2017, the authors presented a more general version of $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $ (see [6]).

Radford biproduct Hom-Hopf algebra was given below.

Theorem 1.1 Let $ (H, {\alpha}) $ be a Hom-bialgebra, $ (B, {\beta}) $ a left $ (H, {\alpha}) $-module Hom-algebra with module structure $ \rhd: H{\otimes} B{\longrightarrow} B $ and a left $ (H, {\alpha}) $-comodule Hom-coalgebra with comodule structure $ \rho: B{\longrightarrow} H{\otimes} B $. Then the following conclusions are equivalent.

(i) $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $ is a Hom-bialgebra, where $ (B\# H, {\beta}{\otimes} {\alpha}) $ is a smash product Hom-algebra (see [7]) and $ (B{\times} H, {\beta}{\otimes} {\alpha}) $ is a smash coproduct Hom-coalgebra.

(ii) The following conditions hold ($ \forall\; a,b\in B $ and $ h\in H $)

(R1) $ (B,\rho,{\alpha}) $ is an $ (H,{\beta}) $-comodule Hom-algebra,

(R2) $ (B,\rhd,{\alpha}) $ is an $ (H,{\beta}) $-module Hom-coalgebra,

(R3) $ {\varepsilon}_B $ is a Hom-algebra map and $ {\Delta}_B(1_B) = 1_B{\otimes} 1_B $,

(R4) $ {\Delta}_B(ab) = a_1({\alpha}^2(a_{2-1})\rhd {\beta}^{-1}(b_1)){\otimes} {\beta}^{-1}(a_{20})b_2 $,

(R5) $ h_1{\alpha}(a_{-1}){\otimes} ({\alpha}^3(h_2)\rhd a_0) = ({\alpha}^2(h_1)\rhd a)_{-1}h_2{\otimes} ({\alpha}^2(h_1)\rhd a)_{0} $.

Definition 1.2 Let $ (H, {\alpha}) $ be a Hom-bialgebra, $ (M,\rhd_M, {\alpha}_M) $ a left $ (H,{\alpha}) $-module with action $ \rhd_M: H{\otimes} M{\longrightarrow} M, h{\otimes} m\mapsto h\rhd_M m $ and $ (M,\rho^M, {\alpha}_M) $ a left $ (H,{\alpha}) $-comodule with coaction $ \rho^M: M{\longrightarrow} H{\otimes} M, m\mapsto m_{-1}{\otimes} m_{0} $. Then we call $ (M,\rhd_M, \rho^M,{\alpha}_M) $ a (left-left) Hom-Yetter-Drinfeld module over $ (H,{\alpha}) $ if the following condition holds:

$ \text {(HYD)}\; \; \; h_1{\alpha}(m_{-1}){\otimes} ({\alpha}^3(h_2)\rhd_M m_0) = ({\alpha}^2(h_1)\rhd_M m)_{-1}h_2{\otimes} ({\alpha}^2(h_1)\rhd_M m)_{0}, $

where $ h\in H $ and $ m\in M $.

When $ (H,{\alpha}) $ is a Hom-Hopf algebra, then the condition (HYD) is equivalent to

$ \text {(HYD)}'\; ({\alpha}^4(h)\rhd_M m)_{-1}{\otimes} ({\alpha}^4(h)\rhd_M m)_{0} = {\alpha}^{-2}(h_{11}{\alpha}(m_{-1}))S_H(h_2){\otimes} ({\alpha}^3(h_{12})\rhd_M m_0). $

So it is natural to consider the relations between the 2-cocycles $ {\sigma} $ on $ (B, {\beta}) $ and $ \overline {\sigma} $ on $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $.

In this paper, we mainly investigate the relations between the left 2-cocycles $ {\sigma} $ on $ (B, {\beta}) $ and $ \overline {\sigma} $ on $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $, and also provide two non-trivial examples.

Next we recall some definitions and results in [8] which will be used later.

Definition 1.3 A left 2-cocycle on a Hom-bialgebra $ (H,{\alpha}) $ is a linear map $ {\sigma}:H{\otimes} H{\rightarrow} K $ satisfying

$ \begin{eqnarray} &&{\sigma}\circ({\alpha}{\otimes} {\alpha}) = {\sigma}, \end{eqnarray} $

(1.1)

$ \begin{eqnarray} &&{\sigma}(b_1,c_1){\sigma}({\alpha}^2(a),b_2c_2) = {\sigma}(a_1,b_1){\sigma}(a_2b_2,{\alpha}^2(c)) \end{eqnarray} $

(1.2)

for all $ a,b,c\in H $.

Furthermore, $ {\sigma} $ is normal if $ {\sigma}(1,h) = {\sigma}(h,1) = {\varepsilon}(h) $ for all $ h\in H $.

Remarks (1) Similarly if eq. (1.2) is replaced by

$ {\sigma}({\alpha}^2(a),b_1c_1){\sigma}(b_2,c_2) = {\sigma}(a_1b_1,{\alpha}^2(c)){\sigma}(a_2,b_2), $

then $ {\sigma} $ is a right 2-cocycle.

(2) If $ {\sigma}:H{\otimes} H{\longrightarrow} K $ is a normal and convolution invertible, then $ {\sigma} $ is a left 2-cocycle if and only if $ {\sigma}^{-1} $ is a right 2-cocycle.

Proposition 1.4 Let $ (H,{\alpha}) $ be a Hom-Hopf algebra.

(1) If $ {\sigma} $ is a normal left 2-cocyle on $ (H,{\alpha}) $, for all $ h, g\in H $, define a new multiplication on $ H $ as follows

$ \begin{align} h {\cdot}_{\sigma} g = {\sigma}(h_1,g_1){\alpha}^{-1}(h_2g_2). \end{align} $

(1.3)

Then $ (H,{\cdot}_{\sigma},{\alpha}) $ is a Hom-algebra, we denote the algebra by $ (_{\sigma} H,{\alpha}) $.

(2) If $ {\sigma} $ is a normal right 2-cocyle on $ (H,{\alpha}) $ for all $ h,g\in H $, define multiplication on $ H $ as follows $ h _{\sigma}{\cdot} g = {\alpha}^{-1}(h_1g_1){\sigma}(h_2,g_2). $ Then $ (H,_{\sigma}{\cdot},{\alpha}) $ is also a Hom-algebra, we denote the algebra by $ (H_{\sigma},{\alpha}) $.

Definition 1.5 A left 2-cocyle $ {\sigma} $ on $ (H,{\alpha}) $ is called lazy if for all $ h,g\in H $,

$ \begin{align} {\sigma}(h_1,g_1)h_2g_2 = h_1g_1{\sigma}(h_2,g_2). \end{align} $

(1.4)

Remark A lazy left 2-cocyle on $ (H,{\alpha}) $ is also a right 2-cocyle on $ (H,{\alpha}) $.

Lemma 1.6 Let $ {\gamma}:H{\longrightarrow} K $ be a normal (i.e. $ {\gamma}(1) = 1 $) and convolution invertible linear map such that $ {\gamma}\circ {\alpha} = {\gamma} $, define $ D^1({\gamma}): H{\otimes} H{\longrightarrow} K $ by

$ \begin{align} D^1({\gamma})(h,g) = {\gamma}(h_1){\gamma}(g_1){\gamma}^{-1}(h_2g_2) \end{align} $

(1.5)

for all $ h,g\in H $. Then $ D^1({\gamma}) $ is a normal and convolution invertible left 2-cocycle on $ (H,{\alpha}) $.

Remarks (1) The set $ Reg^1(H,{\alpha}) $ (respectively $ {\rm Re}g^2(H,{\alpha}) $) consisting of normal and convolution invertible linear maps $ {\gamma}:H{\longrightarrow} K $ such that $ {\gamma}\circ {\alpha} = {\gamma} $ (respectively $ {\sigma}:H{\otimes} H{\longrightarrow} K $ such that $ {\sigma}\circ({\alpha}{\otimes} {\alpha}) = {\sigma} $), is a group with respect to the convolution product.

(2) $ {\gamma} $ is lazy if for all $ h\in H $, $ {\gamma}(h_1)h_2 = h_1{\gamma}(h_2) $. The set of all normal and convolution invertible linear maps $ {\gamma}:H{\longrightarrow} K $ satisfying $ {\gamma}\circ {\alpha} = {\gamma} $ is denoted by $ {\rm Re}g^1_L(H) $, which is a group under convolution.

Lemma 1.7 The set of convolution invertible lazy 2-cocycle on $ (H,{\alpha}) $ denoted by $ Z^2_L(H,{\alpha}) $ is a group.

Proposition 1.8 $ D^1:{\rm Re}g^1_L(H,{\alpha}){\longrightarrow} Z^2_L(H,{\alpha}) $ is a group homomorphism, whose image denoted by $ B^2_L(H,{\alpha}) $ (its elements are called lazy 2-coboundary), is contained in the center of $ Z^2_L(H,{\alpha}) $. Thus we call quotient group $ H^2_L(H,{\alpha}): = Z^2_L(H,{\alpha})/B^2_L(H,{\alpha}) $ the second lazy cohomology group of $ H $.

In this section, we investigate the relations between the left 2-cocycles $ {\sigma} $ on $ (B, {\beta}) $ and $ \overline {\sigma} $ on $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $, and also provide two non-trivial examples. In what follows, let $ (H,{\alpha}) $ be a Hom-Hopf algebra with bijective antipode S and $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $ Radford biproduct Hom-Hopf algebra such that $ {\alpha}^2 = id $.

First we give some useful formulas. The Hom-coalgebra structure on $ (B{\otimes} B,{\beta}{\otimes} {\beta}) $ in $ _H^H{\mathbb{YD}} $ is given by

$ \begin{eqnarray} {\Delta}_{B{\otimes} B}(b{\otimes} b')& = &(id{\otimes} C_{B,B}{\otimes} id)\circ ({\Delta}_B{\otimes} {\Delta}_B)(b{\otimes} b')\\ & = &(b_1{\otimes} b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)){\otimes} ({\beta}^{-1}(b_{2(0)}){\otimes} b'_2). \end{eqnarray} $

(2.1)

So by (2.1), if $ {\sigma},\tau:B{\otimes} B{\longrightarrow} K $ are morphisms in $ _H^H{\mathbb{YD}} $, their convolution in $ _H^H{\mathbb{YD}} $ is given by

$ \begin{eqnarray} ({\sigma} \ast\tau)(b, b') = {\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1))\tau({\beta}^{-1}(b_{2(0)}), b'_2). \end{eqnarray} $

(2.2)

Let $ {\sigma}:B{\otimes} B{\longrightarrow} K $ be a morphism in $ _H^H{\mathbb{YD}} $, that is, it satisfies the conditions

$ \begin{eqnarray} &&{\sigma}(h_1{\cdot} b, h_2{\cdot} b') = {\varepsilon}(h){\sigma}(b, b'), \end{eqnarray} $

(2.3)

$ \begin{eqnarray} &&{\sigma}(b_{(0)}, b'_{(0)})b_{(-1)}b'_{(-1)} = {\sigma}(b, b')1_H, \end{eqnarray} $

(2.4)

$ \begin{eqnarray} &&{\sigma}\circ({\beta}{\otimes} {\beta}) = {\sigma} \end{eqnarray} $

(2.5)

for all $ h\in H $ and $ b,b'\in B $.

Lemma 2.1 For a morphism $ {\sigma}:B{\otimes} B{\longrightarrow} K $ in $ _H^H{\mathbb{YD}} $, we can get the following useful formula

$ \begin{eqnarray} {\sigma}(a, {\alpha}(h){\cdot} b) = {\sigma}(S^{-1}(h){\cdot} {\beta}^{-2}(a), b) \end{eqnarray} $

(2.6)

for all $ a,b\in B $ and $ h\in H $.

Proof We can check that as follows

$ \begin{eqnarray*} {\sigma}(a, {\alpha}(h){\cdot} b)& = &{\sigma}((h_{12}S^{-1}(h_{11})){\cdot} {\beta}^{-1}(a), h_2{\cdot} b)\\ & = &{\sigma}({\alpha}(h_{12}){\cdot} (S^{-1}(h_{11}){\cdot} {\beta}^{-2}(a)), h_2{\cdot} b)\\ & = &{\sigma}({\alpha}(h_{21}){\cdot} ({\alpha}(S^{-1}(h_{1})){\cdot} {\beta}^{-2}(a)), {\alpha}^{-1}(h_{22}){\cdot} b)\\ &\stackrel{(2.3)}{ = }&{\sigma}({\alpha}(S^{-1}(h_{1})){\cdot} {\beta}^{-2}(a), b){\varepsilon}(h_2)\\ & = &{\sigma}(S^{-1}(h){\cdot} {\beta}^{-2}(a), b). \end{eqnarray*} $

Definition 2.2 Let $ {\sigma}:B{\otimes} B{\longrightarrow} K $ be a morphism in $ _H^H{\mathbb{YD}} $. Then $ {\sigma} $ is lazy in $ _H^H{\mathbb{YD}} $ if it satisfies the categorical laziness condition (for all $ b,b'\in B $)

$ \begin{eqnarray} {\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)){\beta}^{-1}(b_{2(0)})b'_2 = {\sigma}({\beta}^{-1}(b_{2(0)}), b'_2)b_1(b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)). \end{eqnarray} $

(2.7)

Definition 2.3 Let $ {\sigma}:B{\otimes} B{\longrightarrow} K $ be a morphism in $ _H^H{\mathbb{YD}} $. Then $ {\sigma} $ is a normal left 2-cocycle on $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $ if it is a normal morphism in $ _H^H{\mathbb{YD}} $ and satisfies the categorical left 2-cocycle condition

$ \begin{eqnarray} &&{\sigma}(a_1, a_{2(-1)}{\cdot} {\beta}^{-1}(b_1)){\sigma}({\beta}^{-1}(a_{2(0)})b_2, {\beta}^2(c))\\ & = &{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(c_1)){\sigma}({\beta}^2(a), {\beta}^{-1}(b_{2(0)})c_2) \end{eqnarray} $

(2.8)

for all $ a,b,c\in B $.

Proposition 2.4 If we define a Hom-multiplication $ {\cdot}_{\sigma} $ on $ (B,{\beta}) $ by

$ \begin{eqnarray} b{\cdot}_{\sigma} b' = {\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)){\beta}^{-1}({\beta}^{-1}(b_{2(0)})b'_2) \end{eqnarray} $

(2.9)

for any $ b,b'\in B $, then

(1) $ (_{{\sigma}}B,{\beta}) $ is a Hom-algebra if and only if $ {\sigma} $ is a normal left 2-cocycle in $ _H^H{\mathbb{YD}} $.

(2) $ (_{{\sigma}}B,{\beta}) $ is a left $ (H,{\alpha}) $ Hom-module algebra with the same action as $ (B,{\beta}) $.

Proof (1) For any $ b\in B $, it is easy to check that $ b \cdot_{\sigma} 1_B = {\beta}(b) $ if and only if $ {\sigma}(b,1_B) = {\varepsilon}(b) $ and $ 1_B \cdot_{\sigma} b = {\beta}(b) $ if and only if $ {\sigma}(1_B,b) = {\varepsilon}(b) $. For any $ a,b,c\in B $, we have

$ \begin{eqnarray*} {\beta}(a)\cdot_{\sigma} (b\cdot_{\sigma} c)&\stackrel{(2.9)}{ = }&{\beta}^{-1}(a_{2(0)}){\beta}^{-1}(({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_2))_2){\sigma}({\beta}(a_1), {\alpha}(a_{2(-1)})\\ &&{\cdot} {\beta}^{-1}(({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_2))_1)){\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(c_1)) \end{eqnarray*} $

and

$ \begin{eqnarray*} (a\cdot_{\sigma} b)\cdot_{\sigma} {\beta}(c)&\stackrel{(2.9)}{ = }&{\beta}^{-1}({\beta}^{-1}(({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_2))_{2(0)}){\beta}(c_2)){\sigma}(({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_2))_1, \\ && ({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_{2}))_{2(-1)}{\cdot} c_1){\sigma}(a_1, a_{2(-1)}{\cdot} {\beta}^{-1}(b_1)). \end{eqnarray*} $

Hence, if $ \cdot_{\sigma} $ is Hom-associative, we get

$ \begin{eqnarray*} &&{\beta}^{-1}(a_{2(0)}){\beta}^{-1}(({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_2))_2){\sigma}({\beta}(a_1), {\alpha}(a_{2(-1)}){\cdot} {\beta}^{-1}(({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_2))_1))\\ &&{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(c_1)) = {\beta}^{-1}({\beta}^{-1}(({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_2))_{2(0)}){\beta}(c_2)){\sigma}(({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_2))_1,\\ &&({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_{2}))_{2(-1)}{\cdot} c_1){\sigma}(a_1, a_{2(-1)}{\cdot} {\beta}^{-1}(b_1)). \end{eqnarray*} $

Applying $ {\varepsilon} $ to both sides of the above equation, we get (2.8).

Conversely, if $ {\sigma} $ is a left 2-cocycle in $ _H^H{\mathbb{YD}} $, we have

$ \begin{eqnarray*} {\beta}(a)\cdot_{\sigma} (b\cdot_{\sigma} c)&\stackrel{(2.9)}{ = }&{\beta}^{-1}(a_{2(0)}){\beta}^{-1}(({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_2))_2){\sigma}({\beta}(a_1), {\alpha}(a_{2(-1)})\\ &&{\cdot} {\beta}^{-1}(({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_2))_1)){\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(c_1))\\ &\stackrel{(R4)}{ = }&{\beta}^{-1}(a_{2(0)}){\beta}^{-1}({\beta}^{-1}({\beta}^{-2}(b_{2(0)})_{2(0)}){\beta}^{-1}(c_2)_2){\sigma}({\beta}(a_1), {\alpha}(a_{2(-1)})\\ &&{\cdot}{\beta}^{-1}({\beta}^{-2}(b_{2(0)})_1({\beta}^{-2}(b_{2(0)})_{2(-1)}{\cdot} {\beta}^{-1}({\beta}^{-1}(c_2)_1))))\\ &&{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(c_1))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-4}(b_{2(0)2(0)}){\beta}^{-2}(c_{22})){\sigma}({\beta}(a_1), {\alpha}(a_{2(-1)})\\ &&{\cdot}({\beta}^{-3}(b_{2(0)1})({\alpha}^{-1}(b_{2(0)2(-1)}){\cdot} {\beta}^{-3}(c_{21})))) {\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(c_1))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-4}(b_{22(0)(0)}){\beta}^{-2}(c_{22}))\\ &&{\sigma}({\beta}(a_1), {\alpha}(a_{2(-1)}){\cdot}({\beta}^{-3}(b_{21(0)})({\alpha}^{-1}(b_{22(0)(-1)}){\cdot} {\beta}^{-3}(c_{21}))))\\ && {\sigma}(b_1, (b_{21(-1)}b_{22(-1)}){\cdot} {\beta}^{-1}(c_1)) \\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-3}(b_{22(0)}){\beta}^{-2}(c_{22}))\\ &&{\sigma}({\beta}(a_1), {\alpha}(a_{2(-1)}){\cdot} ({\beta}^{-3}(b_{21(0)})({\alpha}^{-1}(b_{22(-1)2}){\cdot} {\beta}^{-3}(c_{21})))) \\ &&{\sigma}(b_1, (b_{21(-1)}{\alpha}^{-1}(b_{22(-1)1})){\cdot} {\beta}^{-1}(c_1)) \end{eqnarray*} $

$ \begin{eqnarray*} &\stackrel{(2.6)}{ = }&{\beta}^{-1}(a_{2(0)})({\beta}^{-3}(b_{22(0)}){\beta}^{-2}(c_{22}))\\ &&{\sigma}(S^{-1}(a_{2(-1)}){\cdot} {\beta}^{-1}(a_1), {\beta}^{-3}(b_{21(0)})({\alpha}^{-1}(b_{22(-1)2}){\cdot} {\beta}^{-3}(c_{21}))) \\ &&{\sigma}(b_1, {\alpha}(b_{21(-1)}){\cdot} ({\alpha}^{-1}(b_{22(-1)1}){\cdot} {\beta}^{-2}(c_1)))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2}))\\ &&{\sigma}(S^{-1}(a_{2(-1)}){\cdot} {\beta}^{-1}(a_1), {\beta}^{-3}(b_{12(0)})(b_{2(-1)2}{\cdot} {\beta}^{-3}(c_{12}))) \\ &&{\sigma}({\beta}^{-1}(b_{11}), {\alpha}(b_{12(-1)}){\cdot} (b_{2(-1)1}{\cdot} {\beta}^{-3}(c_{11})))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2}))\\ &&{\sigma}({\beta}^2(S^{-1}(a_{2(-1)}){\cdot} {\beta}^{-1}(a_1)), {\beta}^{-1}(b_{12(0)})(b_{2(-1)2}{\cdot} {\beta}^{-1}(c_{12})))\\ &&{\sigma}(b_{11}, b_{12(-1)}{\cdot} ({\alpha}(b_{2(-1)1}){\cdot} {\beta}^{-2}(c_{11})))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2}))\\ &&{\sigma}({\beta}^2(S^{-1}(a_{2(-1)}){\cdot} {\beta}^{-1}(a_1)), {\beta}^{-1}(b_{12(0)})(b_{2(-1)}{\cdot} {\beta}^{-1}(c_{1}))_2)\\ &&{\sigma}(b_{11}, b_{12(-1)}{\cdot} {\beta}^{-1}((b_{2(-1)}{\cdot} {\beta}^{-1}(c_{1}))_1))\\ &\stackrel{(2.8)}{ = }&{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}((S^{-1}(a_{2(-1)}){\cdot} {\beta}^{-1}(a_1))_1, \\ &&(S^{-1}(a_{2(-1)}){\cdot} {\beta}^{-1}(a_1))_{2(-1)}{\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}({\beta}^{-1}((S^{-1}(a_{2(-1)}){\cdot} {\beta}^{-1}(a_1))_{2(0)})b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(S^{-1}(a_{2(-1)})_1{\cdot} {\beta}^{-1}(a_1)_1, \\ &&(S^{-1}(a_{2(-1)})_2 {\cdot} {\beta}^{-1}(a_1)_2)_{(-1)}{\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}({\beta}^{-1}((S^{-1}(a_{2(-1)})_2{\cdot} {\beta}^{-1}(a_1)_2)_{(0)})b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(S^{-1}(a_{2(-1)2}){\cdot} {\beta}^{-1}(a_{11}), \\ &&(S^{-1}(a_{2(-1)1}) {\cdot} {\beta}^{-1}(a_{12}))_{(-1)}{\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}({\beta}^{-1}((S^{-1}(a_{2(-1)1}){\cdot} {\beta}^{-1}(a_{12}))_{(0)})b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ &\stackrel{\text{(HYD)}'}{ = }&{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(S^{-1}(a_{2(-1)2}){\cdot} {\beta}^{-1}(a_{11}), \\ &&((S^{-1}(a_{2(-1)1})_{11}a_{12(-1)})S(S^{-1}(a_{2(-1)1})_2)){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}({\beta}^{-1}({\alpha}(S^{-1}(a_{2(-1)1})_{12}){\cdot} {\beta}^{-1}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(S^{-1}(a_{2(-1)2}){\cdot} {\beta}^{-1}(a_{11}), \\ &&((S^{-1}(a_{2(-1)122})a_{12(-1)})a_{2(-1)11}){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}((S^{-1}(a_{2(-1)121}){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(S^{-1}({\alpha}^{-1}(a_{2(-1)2})){\cdot} {\beta}^{-2}(a_{11}), \\ &&((S^{-1}({\alpha}^{-1}(a_{2(-1)122})){\alpha}^{-1}(a_{12(-1)})){\alpha}^{-1}(a_{2(-1)11})){\cdot} {\beta}^{-2}(b_{11}))\\ &&{\sigma}((S^{-1}(a_{2(-1)121}){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ &\stackrel{(2.6)}{ = }&{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, a_{2(-1)2} \\ &&{\cdot}(((S^{-1}({\alpha}^{-1}(a_{2(-1)122})){\alpha}^{-1}(a_{12(-1)})){\alpha}^{-1}(a_{2(-1)11})){\cdot} {\beta}^{-2}(b_{11})))\\ &&{\sigma}((S^{-1}(a_{2(-1)121}){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1})) \end{eqnarray*} $

$ \begin{eqnarray*} & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, ({\alpha}^{-1}(a_{2(-1)2}) \\ &&((S^{-1}({\alpha}^{-1}(a_{2(-1)122})){\alpha}^{-1}(a_{12(-1)})){\alpha}^{-1}(a_{2(-1)11}))){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}((S^{-1}(a_{2(-1)121}){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1})) \\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, ({\alpha}^{-1}(a_{2(-1)222}) \\ &&((S^{-1}({\alpha}^{-1}(a_{2(-1)221})){\alpha}^{-1}(a_{12(-1)}))a_{2(-1)1})){\cdot} {\beta}^{-1}(b_{11})) \\ &&{\sigma}((S^{-1}({\alpha}(a_{2(-1)21})){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, ((a_{2(-1)222} \\ &&S^{-1}(a_{2(-1)221}))(a_{12(-1)}a_{2(-1)1})){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}((S^{-1}({\alpha}(a_{2(-1)21})){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, {\alpha}(a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\varepsilon}(a_{2(-1)22}){\sigma}((S^{-1}({\alpha}(a_{2(-1)21})){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}, b_{2(-1)}{\cdot} {\beta}(c_{1}))\\ &\stackrel{(2.3)}{ = }&{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, {\alpha}(a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-1}(b_{11})) \\ &&{\sigma}(a_{2(-1)221}{\cdot} ((S^{-1}({\alpha}(a_{2(-1)21})){\cdot} {\beta}^{-2}(a_{12(0)}))b_{12}), a_{2(-1)222}\\ &&{\cdot} (b_{2(-1)}{\cdot} {\beta}(c_{1})))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, {\alpha}(a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}((a_{2(-1)2211}{\cdot} (S^{-1}({\alpha}(a_{2(-1)21})){\cdot} {\beta}^{-2}(a_{12(0)})))(a_{2(-1)2212}{\cdot} b_{12}), \\ &&({\alpha}^{-1}(a_{2(-1)222})b_{2(-1)}){\cdot} {\beta}^2(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, {\alpha}(a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}((({\alpha}^{-1}(a_{2(-1)2211})S^{-1}({\alpha}(a_{2(-1)21}))){\cdot} {\beta}^{-1}(a_{12(0)}))(a_{2(-1)2212}{\cdot} b_{12}), \\ &&({\alpha}^{-1}(a_{2(-1)222})b_{2(-1)}){\cdot} {\beta}^2(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, {\alpha}(a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}((({\alpha}^{-1}(a_{2(-1)2112})S^{-1}({\alpha}^{-1}(a_{2(-1)2111}))){\cdot} {\beta}^{-1}(a_{12(0)}))\\ &&({\alpha}(a_{2(-1)212}){\cdot} b_{12}), (a_{2(-1)22}b_{2(-1)}){\cdot} {\beta}^2(c_{1}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, {\alpha}(a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}(a_{12(0)}(a_{2(-1)21}{\cdot} b_{12}), (a_{2(-1)22}b_{2(-1)}){\cdot} {\beta}^2(c_{1})) \end{eqnarray*} $

and

$ \begin{eqnarray*} (a\cdot_{\sigma} b)\cdot_{\sigma} {\beta}(c)&\stackrel{(2.9)}{ = }&{\beta}^{-1}({\beta}^{-1}(({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_2))_{2(0)}){\beta}(c_2)){\sigma}(({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_2))_1, \\ && ({\beta}^{-2}(a_{2(0)}){\beta}^{-1}(b_{2}))_{2(-1)}{\cdot} c_1){\sigma}(a_1, a_{2(-1)}{\cdot} {\beta}^{-1}(b_1))\\ &\stackrel{(R4)}{ = }&{\beta}^{-2}(({\beta}^{-1}({\beta}^{-2}(a_{2(0)})_{2(0)}){\beta}^{-1}(b_2)_2)_{(0)})c_2\\ &&{\sigma}({\beta}^{-2}(a_{2(0)})_1({\beta}^{-2}(a_{2(0)})_{2(-1)}{\cdot} {\beta}^{-1}({\beta}^{-1}(b_2)_1)), \\ && ({\beta}^{-1}({\beta}^{-2}(a_{2(0)})_{2(0)}){\beta}^{-1}(b_{2})_2)_{(-1)}{\cdot} c_1){\sigma}(a_1, a_{2(-1)}{\cdot} {\beta}^{-1}(b_1))\\ & = &({\beta}^{-5}(a_{2(0)2(0)(0)}){\beta}^{-3}(b_{22(0)}))c_2{\sigma}({\beta}^{-2}(a_{2(0)1})(a_{2(0)2(-1)}{\cdot} {\beta}^{-2}(b_{21})), \end{eqnarray*} $

$ \begin{eqnarray*} && ({\alpha}^{-3}(a_{2(0)2(0)(-1)}){\alpha}^{-1}(b_{22(-1)})){\cdot} c_1){\sigma}(a_1, a_{2(-1)}{\cdot} {\beta}^{-1}(b_1))\\ & = &({\beta}^{-4}(a_{2(0)2(0)}){\beta}^{-3}(b_{22(0)}))c_2{\sigma}({\beta}^{-2}(a_{2(0)1})({\alpha}^{-1}(a_{2(0)2(-1)1}){\cdot} {\beta}^{-2}(b_{21})),\\ && ({\alpha}^{-1}(a_{2(0)2(-1)2}){\alpha}^{-1}(b_{22(-1)})){\cdot} c_1){\sigma}(a_1, a_{2(-1)}{\cdot} {\beta}^{-1}(b_1))\\ & = &({\beta}^{-4}(a_{22(0)(0)}){\beta}^{-3}(b_{22(0)}))c_2{\sigma}({\beta}^{-2}(a_{21(0)})({\alpha}^{-1}(a_{22(0)(-1)1}){\cdot} {\beta}^{-2}(b_{21})),\\ && ({\alpha}^{-1}(a_{22(0)(-1)2}){\alpha}^{-1}(b_{22(-1)})){\cdot} c_1){\sigma}(a_1, (a_{21(-1)} a_{22(-1)}){\cdot} {\beta}^{-1}(b_1))\\ & = &({\beta}^{-3}(a_{22(0)}){\beta}^{-3}(b_{22(0)}))c_2{\sigma}({\beta}^{-2}(a_{21(0)})({\alpha}^{-1}(a_{22(-1)21}){\cdot} {\beta}^{-2}(b_{21})),\\ && ({\alpha}^{-1}(a_{22(-1)22}){\alpha}^{-1}(b_{22(-1)})){\cdot} c_1){\sigma}(a_1, (a_{21(-1)} {\alpha}^{-1}(a_{22(-1)1})){\cdot} {\beta}^{-1}(b_1))\\ & = &({\beta}^{-2}(a_{2(0)}){\beta}^{-2}(b_{2(0)}))c_2{\sigma}({\beta}^{-2}(a_{12(0)})(a_{2(-1)21}{\cdot} {\beta}^{-2}(b_{12})), \\ && (a_{2(-1)22}b_{2(-1)}){\cdot} c_1){\sigma}({\beta}^{-1}(a_{11}), (a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-2}(b_{11}))\\ & = &{\beta}^{-1}(a_{2(0)})({\beta}^{-2}(b_{2(0)}){\beta}^{-1}(c_{2})){\sigma}(a_{11}, {\alpha}(a_{12(-1)}a_{2(-1)1}){\cdot} {\beta}^{-1}(b_{11}))\\ &&{\sigma}(a_{12(0)}(a_{2(-1)21}{\cdot} b_{12}), (a_{2(-1)22}b_{2(-1)}){\cdot} {\beta}^2(c_{1})), \end{eqnarray*} $

then we can get $ {\beta}(a)\cdot_{\sigma} (b\cdot_{\sigma} c) = (a\cdot_{\sigma} b)\cdot_{\sigma} {\beta}(c) $, i.e., $ {\cdot}_{{\sigma}} $ is Hom-associative.

(2) We check that $ (_{{\sigma}}B,{\beta}) $ is a left $ (H,{\alpha}) $-Hom-module algebra. Clearly, $ h{\cdot} 1_B = {\varepsilon}(h)1_B $ for any $ h\in H $. Next we only need to check the identity $ h{\cdot} (b\cdot_{\sigma} b') = (h_1{\cdot} b)\cdot_{\sigma}(h_2{\cdot} b') $ for any $ h\in H $ and $ b,b'\in B $. Indeed, we have

$ \begin{eqnarray*} &&(h_1{\cdot} b)\cdot_{\sigma}(h_2{\cdot} b')\\ &\stackrel{(2.9)}{ = }&{\sigma}(h_{11}{\cdot} b_1, (h_{12}{\cdot} b_2)_{(-1)}{\cdot} {\beta}^{-1}(h_{21}{\cdot} b'_1)){\beta}^{-1}({\beta}^{-1}((h_{12}{\cdot} b_2)_{(0)})(h_{22}{\cdot} b'_2))\\ &\stackrel{\text{(HYD)}'}{ = }&{\sigma}(h_{11}{\cdot} b_1, (h_{1211}{\alpha} (b_{2(-1)}))S(h_{122}){\cdot} {\beta}^{-1}(h_{21}{\cdot} b'_1))\\ &&{\beta}^{-1}({\beta}^{-1}({\alpha}(h_{1212}){\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}(h_{11}{\cdot} b_1, ({\alpha}(h_{121}){\alpha}(b_{2(-1)}))S({\alpha}^{-1}(h_{1222})){\cdot} {\beta}^{-1}(h_{21}{\cdot} b'_1))\\ &&{\beta}^{-1}({\beta}^{-1}({\alpha}(h_{1221}){\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}({\alpha}^{-1}(h_{111}){\cdot} b_1, (({\alpha}(h_{112}){\alpha}(b_{2(-1)}))S(h_{122})){\cdot} {\beta}^{-1}(h_{21}{\cdot} b'_1))\\ &&{\beta}^{-1}({\beta}^{-1}(h_{121}{\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}({\alpha}^{-1}(h_{111}){\cdot} b_1, h_{112}({\alpha}(b_{2(-1)})S({\alpha}^{-1}(h_{122}))){\cdot} {\beta}^{-1}(h_{21}{\cdot} b'_1))\\ &&{\beta}^{-1}({\beta}^{-1}(h_{121}{\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}({\alpha}^{-1}(h_{111}){\cdot} b_1, {\alpha}(h_{112}){\cdot} (({\alpha}(b_{2(-1)})S({\alpha}^{-1}(h_{122}))){\cdot} {\beta}^{-2}(h_{21}{\cdot} b'_1)))\\ &&{\beta}^{-1}({\beta}^{-1}(h_{121}{\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}({\alpha}^{-1}(h_{111}){\cdot} b_1, {\alpha}^{-1}(h_{112}){\cdot} (({\alpha}(b_{2(-1)})S({\alpha}^{-1}(h_{122}))){\cdot} {\beta}^{-2}(h_{21}{\cdot} b'_1)))\\ &&{\beta}^{-1}({\beta}^{-1}(h_{121}{\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ &\stackrel{(2.3)}{ = }&{\sigma}(b_1, {\alpha}(b_{2(-1)})S(h_{12}){\cdot} {\beta}^{-2}(h_{21}{\cdot} b'_1)){\beta}^{-1}({\beta}^{-1}({\alpha}(h_{11}){\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}(b_1, {\alpha}^{-1}({\alpha}(b_{2(-1)})S(h_{12}))h_{21}{\cdot} {\beta}^{-1}(b'_1)){\beta}^{-1}({\beta}^{-1}({\alpha}(h_{11}){\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}(b_1, {\alpha}(b_{2(-1)})(S({\alpha}^{-1}(h_{12})){\alpha}^{-1}(h_{21})){\cdot} {\beta}^{-1}(b'_1)){\beta}^{-1}({\beta}^{-1}({\alpha}(h_{11}){\cdot} b_{2(0)})(h_{22}{\cdot} b'_2))\\ & = &{\sigma}(b_1, {\alpha}(b_{2(-1)})(S(h_{211})h_{212}){\cdot} {\beta}^{-1}(b'_1)){\beta}^{-1}({\beta}^{-1}(h_{1}{\cdot} b_{2(0)})(h_{22}{\cdot} b'_2)) \end{eqnarray*} $

$ \begin{eqnarray*} & = &{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1))(h_{1}{\cdot} {\beta}^{-2}(b_{2(0)}))(h_{2}{\cdot} {\beta}^{-1}(b'_2))\\ & = &h{\cdot} (b\cdot_{\sigma} b'), \end{eqnarray*} $

the proof is completed.

Let $ {\gamma}:B {\longrightarrow} K $ be a morphism in $ _H^H{\mathbb{YD}} $, that is

$ \begin{eqnarray} &&{\gamma}(h{\cdot} b) = {\varepsilon}(h){\gamma}(b), \end{eqnarray} $

(2.10)

$ \begin{eqnarray} &&{\gamma}(b_{(0)})b_{(-1)} = {\gamma}(b)1_H, \end{eqnarray} $

(2.11)

$ \begin{eqnarray} &&{\gamma}\circ{\beta} = {\gamma} \end{eqnarray} $

(2.12)

for all $ h\in H $ and $ b\in B $.

If $ {\gamma} $ is a normal and convolution invertible linear map in $ _H^H{\mathbb{YD}} $, with convolution inverse $ {\gamma}^{-1} $ in $ _H^H{\mathbb{YD}} $, the analogue of the operator $ D^1 $ is given in $ _H^H{\mathbb{YD}} $ by

$ \begin{eqnarray*} D^1({\gamma})(b, b')& = &{\gamma}(b_1){\gamma}(b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)){\gamma}^{-1}({\beta}^{-1}(b_{2(0)})b'_2)\\ &\stackrel{(2.10)}{ = }&{\gamma}(b_1){\gamma}(b'_1){\gamma}^{-1}(b_2b'_2). \end{eqnarray*} $

For a morphism $ {\gamma}:B {\longrightarrow} K $ in $ _H^H{\mathbb{YD}} $, the laziness condition is identical to the usual one: $ {\gamma}(b_1)b_2 = b_1{\gamma}(b_2) $ for all $ b\in B $. Reg$ ^1_L(B,{\beta}) $ is a group in $ _H^H{\mathbb{YD}} $.

Theorem 2.5 (i) For a normal left 2-cocycle $ {\sigma}:B{\otimes} B {\longrightarrow} K $ in $ _H^H{\mathbb{YD}} $, define $ \overline {\sigma}:B^{\#}_{{\times}} H{\otimes} B^{\#}_{{\times}} H{\longrightarrow} K $ by

$ \begin{eqnarray} \overline {\sigma}(b{\otimes} h, b'{\otimes} h') = {\sigma}(b, {\alpha}(h){\cdot} {\beta}^{-1}(b')){\varepsilon}(h'), \end{eqnarray} $

(2.13)

then $ \overline {\sigma} $ is a normal left 2-cocycle on $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $ and we have $ _{{\sigma}}B\# H = _{\overline {\sigma}}(B^{\#}_{{\times}} H) $ as Hom-algebras. Moreover, $ \overline {\sigma} $ is unique with this property.

(ii) If $ \overline {\sigma} $ is a left 2-cocycle on $ (B^{\#}_{{\times}}H,{\beta}{\otimes}{\alpha}) $, then $ {\sigma} $ is a left 2-cocycle on $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $.

(iii) If $ {\sigma} $ is convolution invertible in $ _H^H{\mathbb{YD}} $, then $ \overline {\sigma} $ is convolution invertible, with inverse

$ \begin{eqnarray} \overline {\sigma}^{\; -1}(b{\otimes} h, b'{\otimes} h') = {\sigma}^{-1}(b, {\alpha}(h){\cdot} {\beta}^{-1}(b')){\varepsilon}(h'), \end{eqnarray} $

(2.14)

where $ {\sigma}^{-1} $ is the convolution inverse of $ {\sigma} $ in $ _H^H{\mathbb{YD}} $.

(iv) $ {\sigma} $ is lazy in $ _H^H{\mathbb{YD}} $ if and only if $ \overline {\sigma} $ is lazy.

(v) If $ {\sigma},\tau: B{\otimes} B {\longrightarrow} K $ are lazy 2-cocycles in $ _H^H{\mathbb{YD}} $, then $ \overline{{\sigma}\ast\tau} = \overline{\sigma} \ast \overline\tau $, hence the map $ {\sigma}{\longrightarrow}\overline{\sigma} $ is a group homomorphism from $ Z^2_L(B,{\beta}) $ to $ Z^2_L(B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $.

(vi) If $ {\gamma}:B{\longrightarrow} K $ is a normal and convolution invertible morphism in $ _H^H{\mathbb{YD}} $, define $ \overline {\gamma}:B^{\#}_{{\times}} H{\longrightarrow} K $ by

$ \begin{eqnarray} \overline {\gamma}(b{\otimes} h) = {\gamma}(b){\varepsilon}(h), \end{eqnarray} $

(2.15)

then $ \overline {\gamma} $ is normal and convolution invertible and $ \overline{D^1({\gamma})} = D^1(\overline{\gamma}) $. If $ {\gamma} $ is lazy in $ _H^H{\mathbb{YD}} $, then $ \overline{\gamma} $ is also lazy.

Proof (i) It is easy to see that $ \overline {\sigma} $ is normal. Since $ _{{\sigma}}B\# H $ is a Hom-algebra, we have

$ \begin{eqnarray} (({\beta}{\otimes} {\alpha})(b\# h))((b'\# h')(b''\# h'')) = ((b\# h)(b'\# h'))(({\beta}{\otimes} {\alpha})(b''\# h'')). \end{eqnarray} $

(2.16)

Applying $ {\varepsilon}_B{\otimes} {\varepsilon}_H $ to both sides of (2.16), then $ \overline {\sigma} $ is a normal left 2-cocycle on $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $.

We prove that the multiplication in $ _{{\sigma}}B\# H $ and $ _{\overline {\sigma}}(B^{\#}_{{\times}} H) $ coincide.

$ \begin{eqnarray*} (b\# h)(b'\# h') & = &b{\cdot}(h_1{\cdot} {\beta}^{-1}(b'))\# {\alpha}^{-1}(h_2)h'\\ &\stackrel{(2.9)}{ = }&{\sigma}(b_1, b_{2(-1)}{\cdot} ({\alpha}^{-1}(h_{11}){\cdot} {\beta}^{-2}(b'_1)))\\ &&{\beta}^{-2}(b_{2(0)})({\alpha}^{-1}(h_{12}){\cdot} {\beta}^{-2}(b'_2))\# {\alpha}^{-1}(h_2)h'\\ & = &{\sigma}(b_1, ({\alpha}(b_{2(-1)}){\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1))\\ &&{\beta}^{-2}(b_{2(0)})({\alpha}^{-1}(h_{12}){\cdot} {\beta}^{-2}(b'_2))\# {\alpha}^{-1}(h_2)h'\\ & = &{\sigma}(b_1, ({\alpha}(b_{2(-1)}){\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1)){\varepsilon}(b'_{2(-1)}){\varepsilon}(h'_1)\\ &&({\beta}^{-1}{\otimes}{\alpha}^{-1})({\beta}^{-1}(b_{2(0)})(h_{12}{\cdot} {\beta}^{-2}(b'_{2(0)}))\# h_2h'_2\\ & = &\overline{\sigma}(b_1{\otimes} b_{2(-1)}h_{11},b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_{1}))\\ &&({\beta}^{-1}{\otimes}{\alpha}^{-1})({\beta}^{-1}(b_{2(0)})(h_{12}{\cdot} {\beta}^{-2}(b'_{2(0)}))\# h_2h'_2\\ & = &\overline{\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_{1}),b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_{1}))\\ &&({\beta}^{-1}{\otimes}{\alpha}^{-1})({\beta}^{-1}(b_{2(0)})(h_{21}{\cdot} {\beta}^{-2}(b'_{2(0)}))\# {\alpha}^{-1}(h_{22})h'_2)\\ & = &\overline{\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_{1}),b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_{1}))\\ &&({\beta}^{-1}{\otimes}{\alpha}^{-1})(({\beta}^{-1}(b_{2(0)}){\otimes} h_{2})({\beta}^{-1}(b'_{2(0)}){\otimes} h'_2))\\ & = &\overline{\sigma}((b{\otimes} h)_{1},(b'{\otimes} h')_{1})({\beta}^{-1}{\otimes}{\alpha}^{-1})((b{\otimes} h)_{2}(b'{\otimes} h')_{2})\\ &\stackrel{(1.3)}{ = }&(b{\otimes} h){\cdot}_{\overline{\sigma}}(b'{\otimes} h'). \end{eqnarray*} $

The uniqueness of $ \overline {\sigma} $ follows easily by applying $ {\varepsilon}_B{\otimes} {\varepsilon}_H $ to the multiplications in $ _{{\sigma}}B\# H $ and $ _{\overline {\sigma}}(B^{\#}_{\times} H) $. We check that as follows

$ \begin{eqnarray*} &&({\varepsilon}_B{\otimes} {\varepsilon}_H)((b\# h)(b'\# h'))\\ & = &{\sigma}(b_1, ({\alpha}(b_{2(-1)}){\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1))({\varepsilon}_B{\otimes} {\varepsilon}_H)({\beta}^{-2}(b_{2(0)})({\alpha}^{-1}(h_{12}){\cdot} {\beta}^{-2}(b'_2))\# {\alpha}^{-1}(h_2)h')\\ & = &{\sigma}(b_1, ({\alpha}(b_{2(-1)}){\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1)){\varepsilon}(b_{2(0)}){\varepsilon}(h_{12}){\varepsilon}(b'_2){\varepsilon}(h_2){\varepsilon}(h')\\ & = &{\sigma}({\beta}(b), h{\cdot} b'){\varepsilon}(h')\\ & = &{\sigma}(b, {\alpha}(h){\cdot} {\beta}^{-1}(b')){\varepsilon}(h') \end{eqnarray*} $

and

$ \begin{eqnarray*} &&({\varepsilon}_B{\otimes} {\varepsilon}_H)((b{\otimes} h){\cdot}_{\overline{\sigma}}(b'{\otimes} h'))\\ & = &\overline{\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_{1}),b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_{1}))\\ &&({\varepsilon}_B{\otimes} {\varepsilon}_H)({\beta}^{-1}{\otimes}{\alpha}^{-1})({\beta}^{-1}(b_{2(0)})(h_{21}{\cdot} {\beta}^{-2}(b'_{2(0)}))\# {\alpha}^{-1}(h_{22})h'_2)\\ & = &\overline{\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_{1}),b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_{1})){\varepsilon}(b_{2(0)}){\varepsilon}(h_{21}){\varepsilon}(b'_{2(0)}){\varepsilon}(h_{22}){\varepsilon}(h'_2)\\ & = &\overline{\sigma}({\beta}(b){\otimes} {\alpha}(h),{\beta}(b'){\otimes} {\alpha}(h'))\\ & = &\overline{\sigma}(b{\otimes} h,b'{\otimes} h'). \end{eqnarray*} $

(ii) Let $ a,b,c\in B $ and $ h,g,l\in H $, we have

$ \begin{eqnarray*} &&\overline {\sigma}((a{\otimes} h)_1,(b{\otimes} g)_1)\overline {\sigma}((a{\otimes} h)_2(b{\otimes} g)_2, ({\beta}^2{\otimes}{\alpha}^2)(c{\otimes} l))\\ & = &\overline {\sigma}((b{\otimes} g)_1,(c{\otimes} l)_1)\overline {\sigma}(({\beta}^2{\otimes}{\alpha}^2)(a{\otimes} h),(b{\otimes} g)_2(c{\otimes} l)_2). \end{eqnarray*} $

Then

$ \begin{eqnarray*} LHS & = &\overline {\sigma}(a_1{\otimes} a_{2(-1)}{\alpha}^{-1}(h_1), b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(g_1))\\ &&\overline {\sigma}(({\beta}^{-1}(a_{2(0)}){\otimes} h_2)({\beta}^{-1}(b_{2(0)}){\otimes} g_2),{\beta}^2(c){\otimes} l)\\ & = &\overline {\sigma}(a_1{\otimes} a_{2(-1)}{\alpha}^{-1}(h_1), b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(g_1))\\ &&\overline {\sigma}({\beta}^{-1}(a_{2(0)})(h_{21}{\cdot} {\beta}^{-2}(b_{2(0)})){\otimes} {\alpha}^{-1}(h_{22})g_2, {\beta}^2(c){\otimes} l)\\ &\stackrel{(2.13)}{ = }&{\sigma}(a_1, ({\alpha}(a_{2(-1)})h_1){\cdot} {\beta}^{-1}(b_1))\\ &&{\sigma}({\beta}^{-1}(a_{2(0)})(h_{21}{\cdot} {\beta}^{-1}(b_{2})), (h_{22}g){\cdot} {\beta}(c)){\varepsilon}(l) \end{eqnarray*} $

and

$ \begin{eqnarray*} RHS & = &\overline {\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(g_1), c_1{\otimes} c_{2(-1)}{\alpha}^{-1}(l_1))\\ &&\overline {\sigma}({\beta}^2(a){\otimes} h, ({\beta}^{-1}(b_{2(0)}){\otimes} g_2)({\beta}^{-1}(c_{2(0)}){\otimes} l_2))\\ & = &\overline {\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(g_1), c_1{\otimes} c_{2(-1)}{\alpha}^{-1}(l_1))\\ &&\overline {\sigma}({\beta}^2(a){\otimes} h, {\beta}^{-1}(b_{2(0)})(g_{21}{\cdot} {\beta}^{-2}(c_{2(0)})){\otimes} {\alpha}^{-1}(g_{22})l_2)\\ &\stackrel{(2.13)}{ = }&{\sigma}(b_1, ({\alpha}(b_{2(-1)})g_1){\cdot} {\beta}^{-1}(c_1))\\ &&{\sigma}({\beta}^2(a), {\alpha}(h){\cdot} ({\beta}^{-2}(b_{2(0)})(g_{2}{\cdot} {\beta}^{-2}(c_{2})))){\varepsilon}(l). \end{eqnarray*} $

Let $ h = g = l = 1_H $, we can get (2.8).

(iii)

$ \begin{eqnarray*} &&(\overline {\sigma}\ast\overline {\sigma}^{-1})(b{\otimes} h, b'{\otimes} h')\\ & = &\overline {\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_1),b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_1))\overline {\sigma}^{-1}({\beta}^{-1}(b_{2(0)}){\otimes} h_2,{\beta}^{-1}(b'_{2(0)}){\otimes} h'_2)\\ &\stackrel{(2.13,2.14)}{ = }&{\sigma}(b_1, {\alpha}(b_{2(-1)}{\alpha}^{-1}(h_1)){\cdot} {\beta}^{-1}(b'_1)){\varepsilon}(b'_{2(-1)}){\varepsilon}(h'_1)\\ &&{\sigma}^{-1}({\beta}^{-1}(b_{2(0)}), {\alpha}(h_2){\cdot} {\beta}^{-2}(b'_{2(0)})){\varepsilon}(h'_2)\\ & = &{\sigma}(b_1, b_{2(-1)}{\cdot} (h_1{\cdot} {\beta}^{-2}(b'_1))){\sigma}^{-1}({\beta}^{-1}(b_{2(0)}), {\alpha}(h_2){\cdot} {\beta}^{-1}(b'_{2})){\varepsilon}(h')\\ &\stackrel{(2.2)}{ = }&({\sigma}\ast{\sigma}^{-1})(b, {\alpha}(h){\cdot} {\beta}^{-1}(b')){\varepsilon}(h')\\ & = &{\varepsilon}(b){\varepsilon}(h){\varepsilon}(b'){\varepsilon}(h'). \end{eqnarray*} $

(iv) Now let $ b,b'\in B $ and $ h,h'\in H $ and assume that $ {\sigma} $ is lazy in $ _H^H{\mathbb{YD}} $, then we prove (1.4) for $ \overline {\sigma} $ on $ (B^{\#}_{{\times}} H,{\beta}{\otimes} {\alpha}) $ as follows:

$ \begin{eqnarray*} RHS & = &\overline {\sigma}((b{\otimes} h)_2,(b'{\otimes} h')_2)(b{\otimes} h)_1(b'{\otimes} h')_1\\ & = &\overline {\sigma}({\beta}^{-1}(b_{2(0)}){\otimes} h_2,{\beta}^{-1}(b'_{2(0)}){\otimes} h'_2)(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_1))(b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_1))\\ &\stackrel{(2.13)}{ = }&{\sigma}({\beta}^{-1}(b_{2(0)}), {\alpha}(h_2){\cdot} {\beta}^{-2}(b'_{2(0)}))(b_1((b_{2(-1)1}{\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} ({\alpha}^{-1}(b_{2(-1)2})h_{12})(b'_{2(-1)})h') \end{eqnarray*} $

$ \begin{eqnarray*} & = &{\sigma}({\beta}^{-1}(b_{2(0)}), {\alpha}(h_2){\cdot} {\beta}^{-2}(b'_{2(0)}))(b_1((b_{2(-1)1}{\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} ({\alpha}^{-1}(b_{2(-1)2})h_{12})(b'_{2(-1)}h'))\\ & = &{\sigma}({\beta}^{-2}(b_{2(0)(0)}), {\alpha}(h_2){\cdot} {\beta}^{-2}(b'_{2(0)}))(b_1(({\alpha}(b_{2(-1)}){\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} ({\alpha}^{-1}(b_{2(0)(-1)})h_{12})(b'_{2(-1)}h'))\\ & = &{\sigma}({\beta}^{-2}(b_{2(0)(0)}), h_{22}{\cdot} {\beta}^{-2}(b'_{2(0)}))(b_1(({\alpha}(b_{2(-1)})h_{1}){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} ({\alpha}^{-1}(b_{2(0)(-1)})h_{21})(b'_{2(-1)}h'))\\ & = &{\sigma}({\beta}^{-2}(b_{2(0)(0)}), h_{22}{\cdot} {\beta}^{-2}(b'_{2(0)}))(b_1(({\alpha}(b_{2(-1)})h_{1}){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} ({\alpha}^{-1}(b_{2(0)(-1)})({\alpha}^{-1}(h_{21}){\alpha}^{-1}(b'_{2(-1)}))){\alpha}(h'))\\ & = &{\sigma}(b_{2(0)(0)}, h_{22}{\cdot} b'_{2(0)})(b_1(({\alpha}(b_{2(-1)})h_{1}){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} ({\alpha}^{-1}(b_{2(0)(-1)}){\alpha}^{-1}(h_{21}b'_{2(-1)})){\alpha}(h'))\\ &\stackrel{(R5)}{ = }&{\sigma}({\beta}(b_{2(0)(0)}), (h_{21}{\cdot} {\beta}(b'_{2}))_{(0)})(b_1(({\alpha}(b_{2(-1)})h_{1}){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} ({\alpha}^{-1}(b_{2(0)(-1)}){\alpha}^{-1}(((h_{21}{\cdot} {\beta}(b'_{2}))_{(-1)})h_{22})){\alpha}^{-1}(h'))\\ & = &{\sigma}({\beta}(b_{2(0)(0)}), (h_{21}{\cdot} {\beta}(b'_{2}))_{(0)})(b_1(({\alpha}(b_{2(-1)})h_{1}){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} (({\alpha}^{-2}(b_{2(0)(-1)}){\alpha}^{-1}((h_{21}{\cdot} {\beta}(b'_{2}))_{(-1)}))h_{22}){\alpha}(h'))\\ & = &{\sigma}(b_{2(0)(0)}, {\beta}^{-1}(h_{21}{\cdot} {\beta}(b'_{2}))_{(0)})(b_1(({\alpha}(b_{2(-1)})h_{1}){\cdot} {\beta}^{-1}(b'_1))\\ &&{\otimes} (({\alpha}^{-2}(b_{2(0)(-1)}){\beta}^{-1}(h_{21}{\cdot} {\beta}(b'_{2}))_{(-1)})h_{22}){\alpha}(h'))\\ &\stackrel{(2.4)}{ = }&{\sigma}(b_{2(0)}, {\beta}^{-1}(h_{21}{\cdot} {\beta}(b'_{2})))(b_1(({\alpha}(b_{2(-1)})h_{1}){\cdot} {\beta}^{-1}(b'_1)){\otimes} (1_Hh_{22}){\alpha}(h'))\\ & = &{\sigma}(b_{2(0)}, {\alpha}(h_{12}){\cdot} b'_{2})(b_1(({\alpha}(b_{2(-1)}){\alpha}^{-1}(h_{11})){\cdot} {\beta}^{-1}(b'_1)){\otimes} h_{2}{\alpha}(h'))\\ & = &{\sigma}({\beta}^{-2}(b_{2(0)}), ({\alpha}^{-1}(h_{1}){\cdot} {\beta}^{-2}(b'))_{2})(b_1(b_{2(-1)}{\cdot} ({\alpha}^{-1}(h_{1}){\cdot} {\beta}^{-2}(b'))_1)\\ &&{\otimes} h_{2}{\alpha}(h'))\\ & = &{\sigma}({\beta}^{-1}(b_{2(0)}), (h_{1}{\cdot} {\beta}^{-1}(b'))_{2})(b_1(b_{2(-1)}{\cdot} {\beta}^{-1}((h_{1}{\cdot} {\beta}^{-1}(b'))_1)){\otimes} h_{2}{\alpha}(h')) \end{eqnarray*} $

and

$ \begin{eqnarray*} LHS & = &\overline {\sigma}((b{\otimes} h)_1,(b'{\otimes} h')_1)(b{\otimes} h)_2(b'{\otimes} h')_2\\ & = &\overline {\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_1),b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_1))({\beta}^{-1}(b_{2(0)}){\otimes} h_2)({\beta}^{-1}(b'_{2(0)}){\otimes} h'_2)\\ &\stackrel{(2.13)}{ = }&{\sigma}(b_1, {\alpha}(b_{2(-1)}{\alpha}^{-1}(h_1)){\cdot} {\beta}^{-1}(b'_1))\\ &&{\varepsilon}(b'_{2(-1)}){\varepsilon}(h'_1){\beta}^{-1}(b_{2(0)})(h_{21}{\cdot} {\beta}^{-2}(b'_{2(0)})){\otimes} {\alpha}^{-1}(h_{22})h'_2\\ & = &{\sigma}(b_1, b_{2(-1)}{\cdot} ({\alpha}^{-1}(h_{11}){\cdot} {\beta}^{-2}(b'_1))){\beta}^{-1}(b_{2(0)})(h_{12}{\cdot} {\beta}^{-1}(b'_2)){\otimes} h_2{\alpha}(h')\\ & = &{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}((h_{1}{\cdot} {\beta}^{-1}(b'))_1)){\beta}^{-1}(b_{2(0)})(h_{1}{\cdot} {\beta}^{-1}(b'))_2{\otimes} h_2{\alpha}(h')\\ &\stackrel{(2.7)}{ = }&{\sigma}({\beta}^{-1}(b_{2(0)}), (h_{1}{\cdot} {\beta}^{-1}(b'))_{2})b_1(b_{2(-1)}{\cdot} {\beta}^{-1}((h_{1}{\cdot} {\beta}^{-1}(b'))_1)){\otimes} h_{2}{\alpha}(h'), \end{eqnarray*} $

which proves that $ \overline {\sigma} $ is lazy.

Conversely, if $ \overline {\sigma} $ is lazy, we have

$ \overline {\sigma}((b{\otimes} h)_2,(b'{\otimes} h')_2)(b{\otimes} h)_1(b'{\otimes} h')_1 = \overline {\sigma}((b{\otimes} h)_1,(b'{\otimes} h')_1)(b{\otimes} h)_2(b'{\otimes} h')_2, $

then we can get

$ \begin{eqnarray*} &&{\sigma}({\beta}^{-1}(b_{2(0)}), (h_{1}{\cdot} {\beta}^{-1}(b'))_{2})(b_1(b_{2(-1)}{\cdot} {\beta}^{-1}((h_{1}{\cdot} {\beta}^{-1}(b'))_1)){\otimes} h_{2}{\alpha}(h'))\\ & = &{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}((h_{1}{\cdot} {\beta}^{-1}(b'))_1)){\beta}^{-1}(b_{2(0)})(h_{1}{\cdot} {\beta}^{-1}(b'))_2{\otimes} h_2{\alpha}(h'). \end{eqnarray*} $

Applying $ id{\otimes} {\varepsilon} $ to both sides of the above equation and let $ h = 1_H $, we get $ {\sigma} $ is lazy in $ _H^H{\mathbb{YD}} $.

(v) Using (2.2) for the convolution in $ _H^H{\mathbb{YD}} $, we compute

$ \begin{eqnarray*} &&\overline{({\sigma}\ast\tau)}(b{\otimes} h,b'{\otimes} h')\\ &\stackrel{(2.13)}{ = }&({\sigma}\ast\tau)(b{\otimes} {\alpha}(h){\cdot} {\beta}^{-1}(b')){\varepsilon}(h')\\ &\stackrel{(2.2)}{ = }&{\sigma}(b_1, b_{2(-1)}{\cdot} (h_1{\cdot} {\beta}^{-2}(b'_1))) \tau({\beta}^{-1}(b_{2(0)}), {\alpha}(h_2){\cdot} {\beta}^{-1}(b'_2)){\varepsilon}(h')\\ & = &{\sigma}(b_1,({\alpha}(b_{2(-1)})h_1){\cdot} {\beta}^{-1}(b'_1)){\varepsilon}(b'_{2(-1)}){\varepsilon}(h'_1)\tau({\beta}^{-1}(b_{2(0)}), {\alpha}(h_2){\cdot} {\beta}^{-2}(b'_{2(0)})){\varepsilon}(h'_2)\\ &\stackrel{(2.13)}{ = }&\overline{\sigma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_1),b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_1))\overline\tau({\beta}^{-1}(b_{2(0)}){\otimes} h_2,{\beta}^{-1}(b'_{2(0)}){\otimes} h'_2)\\ & = &\overline{\sigma}((b{\otimes} h)_1,(b'{\otimes} h')_1)\overline\tau((b{\otimes} h)_2,(b'{\otimes} h')_2)\\ & = &(\overline{\sigma} \ast \overline\tau)(b{\otimes} h,b'{\otimes} h'). \end{eqnarray*} $

(vi) Obviously $ \overline{\gamma} $ is normalized, and it is easy to see that its convolution inverse is given by $ \overline{\gamma}^{\; -1}(b{\times} h) = {\gamma}^{-1}(b){\varepsilon}(h) $, where $ {\gamma}^{\; -1} $ is the convolution inverse of $ {\gamma} $ in $ _H^H{\mathbb{YD}} $. Now we compute

$ \begin{eqnarray*} &&\overline{D^1({\gamma})}(b{\otimes} h,b'{\otimes} h')\\ &\stackrel{(2.13)}{ = }&D^1({\gamma})(b, {\alpha}(h){\cdot} {\beta}^{-1}(b')){\varepsilon}(h')\\ & = &{\gamma}(b_1){\gamma}({\alpha}(h_1){\cdot} {\beta}^{-1}(b'_1)){\gamma}^{-1}(b_2({\alpha}(h_2){\cdot} {\beta}^{-1}(b'_2))){\varepsilon}(h')\\ &\stackrel{(2.10)}{ = }&{\gamma}(b_1){\gamma}(b'_1){\gamma}^{-1}(b_2(h{\cdot} {\beta}^{-1}(b'_2))){\varepsilon}(h')\\ & = &{\gamma}(b_1){\varepsilon}(b_{2(-1)}){\varepsilon}(h_1){\gamma}(b'_1){\varepsilon}(b'_{2(-1)}){\varepsilon}(h'_1){\gamma}^{-1}({\beta}^{-1}(b_{2(0)}(h_{21}{\cdot} {\beta}^{-2}(b'_{2(0)}))){\varepsilon}(h_{22}){\varepsilon}(h'_2)\\ & = &{\gamma}(b_1){\varepsilon}(b_{2(-1)}){\varepsilon}(h_1){\gamma}(b'_1){\varepsilon}(b'_{2(-1)}){\varepsilon}(h'_1)\overline{\gamma}^{-1}({\beta}^{-1}(b_{2(0)}(h_{21}{\cdot} {\beta}^{-2}(b'_{2(0)}))\\ &&{\otimes}{\alpha}^{-1}(h_{22})(h'_2)))\\ & = &\overline{\gamma}(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_1))\overline{\gamma}(b'_1{\otimes} b'_{2(-1)}{\alpha}^{-1}(h'_1))\overline{\gamma}^{-1}({\beta}^{-1}(b_{2(0)}{\otimes} h_{2}) ({\beta}^{-1}(b'_{2(0)}){\otimes} h'_{2}))\\ & = &\overline{\gamma}((b{\otimes} h)_1)\overline{\gamma}((b'{\otimes} h')_1)\overline{\gamma}^{-1}((b{\otimes} h)_2(b'{\otimes} h')_2)\\ &\stackrel{(1.5)}{ = }&D^1(\overline{\gamma})(b{\otimes} h,b'{\otimes} h'). \end{eqnarray*} $

Hence we have indeed $ \overline{D^1({\gamma})} = D^1(\overline{\gamma}) $. If $ {\gamma} $ is lazy in $ _H^H{\mathbb{YD}} $, then we have

$ \begin{eqnarray*} \overline{\gamma}((b{\otimes} h)_1)(b{\otimes} h)_2 &\stackrel{(2.15)}{ = }&{\gamma}(b_1)(b_2{\otimes} {\alpha}(h))\\ & = &{\gamma}(b_2)(b_1{\otimes} {\alpha}(h))\\ &\stackrel{(2.11)}{ = }&{\gamma}(b_{2(0)})(b_1{\otimes} b_{2(-1)}h)\\ & = &{\gamma}(b_{2(0)})){\varepsilon}(h_2)(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_1))\\ &\stackrel{(2.15)}{ = }&\overline{\gamma}({\beta}^{-1}(b_{2(0)}){\otimes} h_2)(b_1{\otimes} b_{2(-1)}{\alpha}^{-1}(h_1))\\ & = &\overline{\gamma}((b{\otimes} h)_2)(b{\otimes} h)_1, \end{eqnarray*} $

so $ \overline{\gamma} $ is lazy.

Remarks (1) $ Z^2_L(B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $ is a group by Lemma 1.7.

(2) If $ {\sigma},\tau $ are left lazy 2-cocycles on $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $, we can get $ \overline{\sigma},\overline\tau $ are left lazy 2-cocycles on $ (B^{\#}_{{\times} }H,{\beta}{\otimes}{\alpha}) $ by (i) and (iv). Then $ \overline{\sigma} \ast \overline\tau $ is a left lazy 2-cocycle on $ (B^{\#}_{{\times} }H,{\beta}{\otimes}{\alpha}) $. Combining (ii) with (v), we have $ {\sigma}\ast\tau $ is a left lazy 2-cocycle on $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $.

By (2.13) and (2.14), we have $ \overline {\sigma}^{-1} = \overline {{\sigma}^{-1}} $. If $ \overline {\sigma}\in Z^2_L(B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $, then $ \overline {\sigma}^{-1} = \overline {{\sigma}^{-1}}\in Z^2_L(B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $. Combining (ii) with (iv), then $ {\sigma}^{-1} $ is a lazy 2-cocycle on $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $.

In a word, the set of convolution invertible lazy 2-cocycle on $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $ denoted by $ Z^2_L(B,{\beta}) $ is a group.

Proposition 2.6 $ D^1:{\rm Re}g^1_L(B,{\beta}){\longrightarrow} Z^2_L(B,{\beta}) $ is a group homomorphism in $ _H^H{\mathbb{YD}} $, whose image denoted by $ B^2_L(B,{\beta}) $ (its elements are called lazy 2-coboundary in $ _H^H{\mathbb{YD}} $), is contained in the center of $ Z^2_L(B,{\beta}) $. Thus we call quotient group $ H^2_L(B,{\beta}): = Z^2_L(B,{\beta})/B^2_L(B,{\beta}) $ the second lazy cohomology group of $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $.

Proof It is easy to check that $ D^1:{\rm Re}g^1_L(B,{\beta}){\longrightarrow} Z^2_L(B,{\beta}) $ is a group homomorphism in $ _H^H{\mathbb{YD}} $. Now we prove $ B^2_L(B,{\beta}) $ is contained in the center of $ Z^2_L(B,{\beta}) $.

For all $ {\gamma}\in {\rm Re}g^1_L(B,{\beta}) $ and $ {\sigma}\in Z^2_L(B,{\beta}) $,

$ \begin{eqnarray*} ({\sigma}\ast D^1({\gamma}))(b,b') &\stackrel{(2.2)}{ = }&{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1))D^1({\gamma})({\beta}^{-1}(b_{2(0)}), b'_2)\\ & = &{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)){\gamma}({\beta}^{-1}(b_{2(0)1})){\gamma}(b'_{21}){\gamma}^{-1}({\beta}^{-1}(b_{2(0)2})b'_{22})\\ & = &{\sigma}(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)){\gamma}({\beta}^{-1}(b_{2(0)2})){\gamma}(b'_{22}){\gamma}^{-1}({\beta}^{-1}(b_{2(0)1})b'_{21})\\ & = &{\sigma}(b_1,(b_{21(-1)}b_{22(-1)}){\cdot} {\beta}^{-1}(b'_1))\\ &&{\gamma}({\beta}^{-1}(b_{22(0)})){\gamma}(b'_{22}){\gamma}^{-1}({\beta}^{-1}(b_{21(0)})b'_{21})\\ &\stackrel{(2.11)}{ = }&{\sigma}(b_1,{\alpha}(b_{21(-1)}){\cdot} {\beta}^{-1}(b'_1)){\gamma}(b_{22}){\gamma}(b'_{22}){\gamma}^{-1}({\beta}^{-1}(b_{21(0)})b'_{21})\\ & = &{\sigma}({\beta}^{-1}(b_{11}), {\alpha}(b_{12(-1)}){\cdot} {\beta}^{-2}(b'_{11})){\gamma}(b_{2}){\gamma}(b'_{2}){\gamma}^{-1}({\beta}^{-1}(b_{12(0)})b'_{12})\\ & = &{\sigma}(b_{11}, b_{12(-1)}{\cdot} {\beta}^{-1}(b'_{11})){\gamma}(b_{2}){\gamma}(b'_{2}){\gamma}^{-1}({\beta}^{-1}(b_{12(0)})b'_{12})\\ &\stackrel{(2.7)}{ = }&{\sigma}({\beta}^{-1}(b_{12(0)}), b'_{12}){\gamma}^{-1}(b_{11}(b_{12(-1)}{\cdot} {\beta}^{-1}(b'_{11}))){\gamma}(b_{2}){\gamma}(b'_{2})\\ & = &{\sigma}({\beta}^{-1}(b_{21(0)}), b'_{21}){\gamma}^{-1}({\beta}(b_{1})(b_{21(-1)}{\cdot} b'_{1})){\gamma}(b_{22}){\gamma}(b'_{22})\\ & = &{\sigma}({\beta}^{-1}(b_{22(0)}), b'_{22}){\gamma}^{-1}({\beta}(b_{1})(b_{22(-1)}{\cdot} b'_{1})){\gamma}(b_{21}){\gamma}(b'_{21})\\ & = &{\sigma}(b_{2(0)}, {\beta}(b'_{2})){\gamma}^{-1}(b_{11}({\alpha}(b_{2(-1)}){\cdot} {\beta}^{-1}(b'_{11}))){\gamma}(b_{12}){\gamma}(b'_{12}),\\ (D^1({\gamma})\ast {\sigma})(b,b') &\stackrel{(2.2)}{ = }&D^1({\gamma})(b_1, b_{2(-1)}{\cdot} {\beta}^{-1}(b'_1)){\sigma}({\beta}^{-1}(b_{2(0)}), b'_2)\\ & = &{\gamma}(b_{11}){\gamma}({\alpha}^2(b_{2(-1)1}){\cdot} {\beta}^{-1}(b'_{11})){\gamma}^{-1}(b_{12}(b_{2(-1)2}{\cdot} {\beta}^{-1}(b'_{12})))\\ &&{\sigma}({\beta}^{-1}(b_{2(0)}), b'_2)\\ &\stackrel{(2.10)}{ = }&{\gamma}(b_{11}){\varepsilon}(b_{2(-1)1}){\gamma}(b'_{11}){\gamma}^{-1}(b_{12}(b_{2(-1)2}{\cdot} {\beta}^{-1}(b'_{12}))){\sigma}(b_{2(0)}, {\beta}(b'_2))\\ & = &{\gamma}(b_{11}){\gamma}(b'_{11}){\gamma}^{-1}(b_{12}({\alpha}(b_{2(-1)}){\cdot} {\beta}^{-1}(b'_{12}))){\sigma}({\beta}(b_{2(0)}), {\beta}(b'_2)). \end{eqnarray*} $

Then $ {\sigma}\ast D^1({\gamma}) = D^1({\gamma})\ast {\sigma} $. The proof is completed.

Proposition 2.7 If $ {\sigma} $ is a lazy 2-coboundary for $ (B,{\beta}) $ in $ _H^H{\mathbb{YD}} $, then $ \overline {\sigma} $ is a lazy 2-coboundary for $ (B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}) $, so the group homomorphism $ Z^2_L(B,{\beta}){\longrightarrow} Z^2_L(B^{\#}_{{\times}} H, {\beta}{\otimes} {\alpha}), {\sigma}\mapsto \overline {\sigma} $, factorizes to a group homomorphism $ H^2_L(B,{\beta}){\longrightarrow} H^2_L(B^{\#}_{{\times}} H,{\beta}{\otimes} {\alpha}) $.

Proof It follows immediately from (vi) in Theorem 2.5.

Example 2.8 Let $ A = sp\{1_A,z\} $ and the automorphism $ {\beta}:A{\longrightarrow} A,\; \; {\beta}(1_A) = 1_A,{\beta}(z) = -z $. Then $ (A,{\beta}) $ is a Hom-algebra with multiplication: $ 1_A1_A = 1_A, 1_Az = z1_A = -z,z^2 = 0 $, and $ (A,{\beta}) $ is a Hom-coalgebra with comultiplication and counit

$ \begin{eqnarray*} &&{\Delta}(1_A) = 1_A{\otimes} 1_A, \quad{\varepsilon}(1_A) = 1_k,\\ &&{\Delta}(z) = (-z){\otimes} 1_A+1_A{\otimes} (-z),\quad {\varepsilon}(z) = 0. \end{eqnarray*} $

Let $ H = sp\{1_H,g\} $ be the group Hopf algebra with $ g^2 = 1_H $ and $ {\Delta}(g) = g{\otimes} g,S_H(g) = g = g^{-1} $. Then $ (H,id_H) $ is a Hom-Hopf algebra.

Define $ {\cdot}:H{\otimes} A{\longrightarrow} A $ such that $ 1_H{\cdot} 1_A = 1_A, 1_H{\cdot} z = -z, g{\cdot} 1_A = 1_A $, and $ g{\cdot} z = z $. It is easy to check $ (A,{\beta}) $ is a left $ (H,id_H) $-module Hom-algebra and module Hom-coalgebra.

Define $ {\rho}: A{\longrightarrow} H{\otimes} A $ such that $ {\rho}(1_A) = 1_H{\otimes} 1_A $ and $ {\rho}(x) = g{\otimes} (-z) $. We get $ (A,{\beta}) $ is a left $ (H,id_H) $-comodule Hom-algebra and comodule Hom-coalgebra. Then we can get a Radford biproduct Hom-bialgebra $ (A^{\#}_{{\times}}H,{\beta}{\otimes} id) $ (see [5]).

Define $ {\sigma}:A{\otimes} A{\longrightarrow} K $ by

where $ \forall\; s\in K $.

Then we can check that $ {\sigma} $ is normal left 2-cocycle on $ (A,{\beta}) $ in $ _H^H{\mathbb{YD}} $, and $ {\sigma} $ is lazy in $ _H^H{\mathbb{YD}} $. By Theorem 2.5, $ \overline{\sigma} $ is defined as follows

and $ \overline{\sigma} $ is a normal lazy left 2-cocycle.

Let $ {\gamma}(1_A) = 1,{\gamma}(z) = 0 $. Then $ {\gamma} $ is normal and lazy in $ _H^H{\mathbb{YD}} $. By Theorem 2.5, $ \overline{\gamma} $ is defined as follows

and $ \overline{\gamma} $ is normal and lazy.

Example 2.9 Let $ KZ_2 = K\{1,a\} $ be Hopf group algebra. Then $ (KZ_2, id) $ is a Hom-Hopf algebra. Let $ T_{2,-1} = K\{1,g,x,y|g^2 = 1,x^2 = 0,y = gx,gy = -yg = x\} $ be Taft's Hopf algebra, its coalgebra structure and antipode are given by

$ \begin{eqnarray*} &&{\Delta}(g) = g{\otimes} g, \quad {\Delta}(x) = x{\otimes} g+1{\otimes} x,\quad {\Delta}(y) = y{\otimes} 1+g{\otimes} y;\\ &&{\varepsilon}(g) = 1,\quad {\varepsilon}(x) = 0,\quad {\varepsilon}(y) = 0 \end{eqnarray*} $

and $ S(g) = g, S(x) = y, S(y) = -x. $ Define a linear map $ {\alpha}:T_{2,-1}{\longrightarrow} T_{2,-1} $ by $ {\alpha}(1) = 1, {\alpha}(g) = g, {\alpha}(x) = -x, {\alpha}(y) = -y. $ Then $ {\alpha} $ is an automorphism of Hopf algebra.

So we can get a Hom-Hopf algebra $ H_{{\alpha}} = (T_{2,-1},{\alpha}\circ \mu_{T_{2,-1}},1_{T_{2,-1}},{\Delta}_{T_{2,-1}}\circ{\alpha},{\varepsilon}_{T_{2,-1}},{\alpha}) $. Define module action $ {\triangleright}:KZ_2{\otimes} H_{{\alpha}}{\longrightarrow} H_{{\alpha}} $ by

$ \begin{eqnarray*} &&1_{KZ_2}{\triangleright} 1_{H_{{\alpha}}} = 1_{H_{{\alpha}}},\; \; 1_{KZ_2}{\triangleright} g = g,\; \; 1_{KZ_2}{\triangleright} x = -x,\; \; 1_{KZ_2}{\triangleright} y = -y,\\ &&a{\triangleright} 1_{H_{{\alpha}}} = 1_{H_{{\alpha}}},\; \; a{\triangleright} g = g,\; \; a{\triangleright} x = -x,\; \; a{\triangleright} y = -y. \end{eqnarray*} $

Then by a routine computation we can get $ (H_{{\alpha}},{\triangleright},{\alpha}) $ is a $ (KZ_2,id) $-module Hom-algebra. Therefore, $ (H_{{\alpha}}\# KZ_2,{\alpha}{\otimes} id) $ is a smash product Hom-algebra.

Define comodule action $ {\rho}:H_{{\alpha}}{\longrightarrow} KZ_2{\otimes} H_{{\alpha}} $ by

$ \begin{eqnarray*} &&{\rho}:H_{{\alpha}}{\longrightarrow} KZ_2{\otimes} H_{{\alpha}},\; \; 1_{H_{{\alpha}}}\mapsto 1_{KZ_2}{\otimes} 1_{H_{{\alpha}}},\\ &&g\mapsto 1_{KZ_2}{\otimes} g,\; \; x\mapsto -a{\otimes} x,\; \; y\mapsto -a{\otimes} y. \end{eqnarray*} $

Then we can get $ (H_{{\alpha}},{\rho},{\alpha}) $ is a left $ (KZ_2,id) $-comodule Hom-coalgebra. Therefore $ (H_{{\alpha}}{\times} KZ_2,{\alpha}{\otimes} id) $ is a smash coproduct Hom-coalgebra.

Then we can get a Radford biproduct Hom-bialgebra $ (H^{ \quad \#}_{{\alpha}{\times}}KZ_2,{\alpha}{\otimes} id) $ (see [5]).

Define $ {\sigma}:H_{{\alpha}}{\otimes} H_{{\alpha}}{\longrightarrow} K $ by

where $ \forall\; s\in K $.

Then we can check that $ {\sigma} $ is a normal left 2-cocycle on $ (H_{{\alpha}},{\alpha}) $ in $ _H^H{\mathbb{YD}} $, and $ {\sigma} $ is lazy in $ _H^H{\mathbb{YD}} $. By Theorem 2.5, $ \overline{\sigma} $ is defined as follows

and $ \overline{\sigma} $ is a normal lazy left 2-cocycle on $ (H^{ \quad \#}_{{\alpha}{\times}}KZ_2,{\alpha}{\otimes} id) $.

Let $ {\gamma}(1) = 1,{\gamma}(g) = 1 $, $ {\gamma}(x) = 0,{\gamma}(y) = 0 $. Then $ {\gamma} $ is normal and lazy in $ _H^H{\mathbb{YD}} $. By Theorem 2.5, $ \overline{\gamma} $ is defined as follows

and $ \overline{\gamma} $ is normal and lazy.