# Covariant Differential Calculus Over Monoidal Hom-Hopf Algebras

SERKAN KARAÇUHA  Department of Mathematics, FCUP, University of Porto, Rua
Campo Alegre 687, 4169-007 Porto, Portugal<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Concepts of first order differential calculus (FODC) on a monoidal Hom-algebra
and left-covariant FODC over a left Hom-quantum space with respect to a
monoidal Hom-Hopf algebra are presented. Then, extension of the universal FODC
over a monoidal Hom-algebra to a universal Hom-differential calculus is
described. Next, concepts of left(right)-covariant and bicovariant FODC over a
monoidal Hom-Hopf algebra are studied in detail. Subsequently, notion of
quantum Hom-tangent space associated to a bicovariant Hom-FODC is introduced
and equipped with an analogue of Lie bracket (commutator) through Woronowicz’
braiding. Finally, it is proven that this commutator satisfies quantum
versions of the antisymmetry relation and Hom-Jacobi identity.

###### Key words and phrases:

covariant Hom-first order differential calculus, universal Hom-differential
calculus, quantum Hom-tangent space, quantum Hom-Lie algebra

##  1\. Introduction

Hom-type algebras were first introduced in the form of Hom-Lie algebras in
[10], where the Jacobi identity is replaced by the so-called Hom-Jacobi
identity via a linear endomorphism. In 2008, Hom-associative algebras were
suggested in [13] to induce a Hom-Lie algebra using the commutator bracket.
Other Hom-type structures such as Hom-coalgebras, Hom-bialgebras, Hom-Hopf
algebras and their properties were further investigated and developed in [1,
7, 8, 14, 15, 16, 21, 22, 23, 24]. In [2], the authors studied the Hom-
bialgebras and Hom-Hopf algebras in the context of tensor categories, and
these objects are featured as monoidal. One can find further research on
monoidal Hom-Hopf algebras and structures on them such as Hom-Yetter-Drinfeld
modules and covariant Hom-bimodules in [3, 4, 9, 11, 12, 17].

The general theory of covariant differential calculi on quantum groups was
introduced by S. L. Woronowicz in [18], [19],[20]. Many results obtained in
this paper in the Hom-setting follow from the classical results appear in the
fundamental reference [20]. In Section 2, after the notions of first order
differential calculus (FODC) on a monoidal Hom-algebra and left-covariant FODC
over a left Hom-quantum space with respect to a monoidal Hom-Hopf algebra are
presented, the left-covariance of a Hom-FODC is characterized. Then, in
Section 3, extension of the universal FODC over a monoidal Hom-algebra to a
universal Hom-differential calculus (Hom-DC) is described as well (for the
classical case, that is, for the extension of a FODC over an algebra $A$ to
the differential envelope of $A$ one should refer to [6], [5]). Thereafter, in
Section 4 and Section 5, the concepts of left-covariant and bicovariant FODC
over a monoidal Hom-Hopf algebra $(H,\alpha)$ are studied in detail. A
subobject $\mathcal{R}$ of $ker\varepsilon$, which is a right Hom-ideal of
$(H,\alpha)$, and a quantum Hom-tangent space are associated to each left-
covariant $(H,\alpha)$-Hom-FODC: It is indicated that left-covariant Hom-FODCs
are in one-to one correspondence with these right Hom-ideals $\mathcal{R}$,
and that the quantum Hom-tangent space and the left coinvariant of the
monoidal Hom-Hopf algebra on Hom-FODC form a nondegenerate dual pair. The
quantum Hom-tangent space associated to a bicovariant Hom-FODC is equipped
with an analogue of Lie bracket (or commutator) through Woronowicz’ braiding
and it is proven that this commutator satisfies quantum versions of the
antisymmetry relation and Hom-Jacobi identity, which is therefore called the
quantum Hom-Lie algebra of that bicovariant Hom-FODC. Throughout, we work with
vector spaces over a field $k$.

##  2\. Left-Covariant FODC over Hom-quantum spaces

######  Definition 2.1.

Let $(B,\beta)$ be a monoidal Hom-bialgebra. A right $(B,\beta)$-Hom-comodule
algebra (or Hom-quantum space) $(A,\alpha)$ is a monoidal Hom-algebra and a
right $(B,\beta)$-Hom-comodule with a Hom-coaction $\rho^{A}:A\to A\otimes
B,\>a\mapsto a_{(0)}\otimes a_{(1)}$ such that $\rho^{A}$ is a Hom-algebra
morphism, i.e., for any $a,a^{\prime}\in A$

$(aa^{\prime})_{(0)}\otimes(aa^{\prime})_{(1)}=a_{(0)}a^{\prime}_{(0)}\otimes
a_{(1)}{a^{\prime}}_{(1)},\>\>\rho^{A}(1_{A})=1_{A}\otimes 1_{B}.$

######  Definition 2.2.

A first order differential calculus over a monoidal Hom-algebra $(A,\alpha)$
is an $(A,\alpha)$-Hom-bimodule $(\Gamma,\gamma)$ with a linear map
$d:A\to\Gamma$ such that

  1. (1)

$d$ satisfies the Leibniz rule, i.e., $d(ab)=a\cdot db+da\cdot b,\forall
a,b\in A$,

  2. (2)

$d\circ\alpha=\gamma\circ d$, which means that $d$ is in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$,

  3. (3)

$\Gamma$ is linearly spanned by the elements of the form $(a\cdot db)\cdot c$
with $a,b,c\in A$.

We call $(\Gamma,\gamma)$ an $(A,\alpha)$-Hom-FODC for short.

######  Remark 2.3.

  1. (1)

In the above definition, the second condition, i.e. $d\circ\alpha=\gamma\circ
d$, is equivalent to the equality $d(1)=0$.

  2. (2)

By the compatibility condition for Hom-bimodule structure of
$(\Gamma,\gamma)$, we have $(a\cdot db)\cdot
c=\alpha(a)\cdot(db\cdot\alpha^{-1}(c))$, which implies that $\Gamma$ is also
linearly spanned by the elements $a\cdot(db\cdot c)$ for all $a,b,c\in A$.
Thus we denote $\Gamma=(A\cdot dA)\cdot A=A\cdot(dA\cdot A)$.

  3. (3)

By using the Leibniz rule and the fact that $d(\alpha(a))=\gamma(da)$ for any
$a\in A$, we get

$\displaystyle(a\cdot db)\cdot c$ $\displaystyle=$
$\displaystyle(d(ab)-da\cdot b)\cdot c=d(ab)\cdot c-(da\cdot b)\cdot c$
$\displaystyle=$ $\displaystyle d(ab)\cdot
c-\gamma(d(a))\cdot(b\alpha^{-1}(c))=d(ab)\cdot
c-d(\alpha(a))\cdot(b\alpha^{-1}(c)),$

and

$\displaystyle\alpha(a)\cdot(db\cdot\alpha^{-1}(c))$ $\displaystyle=$
$\displaystyle\alpha(a)\cdot(d(b\alpha^{-1}(c))-b\cdot d(\alpha^{-1}(c)))$
$\displaystyle=$ $\displaystyle\alpha(a)\cdot
d(b\alpha^{-1}(c))-\alpha(a)\cdot(b\cdot d(\alpha^{-1}(c)))$ $\displaystyle=$
$\displaystyle\alpha(a)\cdot d(b\alpha^{-1}(c))-(ab)\cdot d(c).$

Hence, $\Gamma=A\cdot dA=dA\cdot A$.

######  Definition 2.4.

Let $(H,\beta)$ be a monoidal Hom-bialgebra and $(A,\alpha)$ be a left Hom-
quantum space for $(H,\beta)$ (i.e. a left $(H,\beta)$-Hom-comodule algebra)
with the left Hom-coaction $\varphi:A\to H\otimes A,\>a\mapsto a_{(-1)}\otimes
a_{(0)}$. An $(A,\alpha)$-Hom-FODC $(\Gamma,\gamma)$ is called left-covariant
with respect to $(H,\beta)$ if there is a left Hom-coaction $\phi:\Gamma\to
H\otimes\Gamma,\>\omega\mapsto\omega_{(-1)}\otimes\omega_{(0)}$ of $(H,\beta)$
on $(\Gamma,\gamma)$ such that

  1. (1)

$\phi(\alpha(a)\cdot(\omega\cdot
b))=\varphi(\alpha(a))(\phi(\omega)\varphi(b))$, $\forall a,b\in A$,
$\omega\in\Gamma$,

  2. (2)

$\phi(da)=(id\otimes d)\varphi(a)$, $\forall a\in A$

Condition $(1)$ can equivalently be written as
$\phi((a\cdot\omega)\cdot\alpha(b))=(\varphi(a)\phi(\omega))\varphi(\alpha(b))$
by using the Hom-bimodule compatibility conditions for $(\Gamma,\gamma)$ and
$(H\otimes\Gamma,\beta\otimes\gamma)$, where left and right $(H\otimes
A,\beta\otimes\alpha)$-Hom-module structures of
$(H\otimes\Gamma,\beta\otimes\gamma)$ are respectively given by

$(h\otimes a)(h^{\prime}\otimes\omega)=hh^{\prime}\otimes a\cdot\omega,$
$(h^{\prime}\otimes\omega)(h\otimes a)=hh^{\prime}\otimes\omega\cdot a$

for $h,h\in H$, $a\in A$ and $\omega\in\Gamma$. Condition $(2)$ means that
$d:A\to\Gamma$ is left $(H,\beta)$-colinear, since the equality
$d\circ\alpha=\gamma\circ d$ holds too.

One can see that for a given $(A,\alpha)$-Hom-FODC $(\Gamma,\gamma)$ there
exists at most one morphism $\phi:\Gamma\to H\otimes\Gamma$ in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$ which makes $(\Gamma,\gamma)$ left-
covariant: Indeed, if there is one such $\phi$, then by the conditions $(1)$
and $(2)$ in Definition 2.4 we do the following computation

$\displaystyle\phi\left(\sum_{i}a_{i}\cdot db_{i}\right)$ $\displaystyle=$
$\displaystyle\sum_{i}\phi(\gamma^{-1}(a_{i}\cdot db_{i})\cdot
1_{A})=\sum_{i}\phi((\alpha^{-1}(a_{i})\cdot\gamma^{-1}(db_{i}))\cdot 1_{A})$
$\displaystyle=$
$\displaystyle\sum_{i}(\varphi(\alpha^{-1}(a_{i}))\phi(\gamma^{-1}(db_{i})))\varphi(1_{A})$
$\displaystyle=$
$\displaystyle\sum_{i}([(\beta^{-1}\otimes\alpha^{-1})(\varphi(a_{i}))][(\beta^{-1}\otimes\gamma^{-1})(\phi(db_{i}))])(1_{H}\otimes
1_{A})$ $\displaystyle=$
$\displaystyle\sum_{i}[(\beta^{-1}\otimes\gamma^{-1})(\varphi(a_{i})\phi(db_{i}))](1_{H}\otimes
1_{A})$ $\displaystyle=$
$\displaystyle\sum_{i}\varphi(a_{i})\phi(db_{i})=\sum_{i}\varphi(a_{i})(id\otimes
d)(\varphi(b_{i})),$

showing that $\varphi$ and $d$ describe $\phi$ uniquely.

######  Proposition 2.5.

Let $(\Gamma,\gamma)$ be an $(A,\alpha)$-Hom-FODC. Then the following
statements are equivalent:

  1. (1)

$(\Gamma,\gamma)$ is left-covariant.

  2. (2)

There is a morphism $\phi:\Gamma\to H\otimes\Gamma$ in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$ such that $\phi(a\cdot
db)=\varphi(a)(id\otimes d)(\varphi(b))$ for all $a,b\in A$.

  3. (3)

$\sum_{i}a_{i}\cdot db_{i}=0$ in $\Gamma$ implies that
$\sum_{i}\varphi(a_{i})(id\otimes d)(\varphi(b_{i}))=0$ in $H\otimes\Gamma$.

###### Proof.

$(1)\Rightarrow(2)$ and $(2)\Rightarrow(3)$ are trivial.

$(3)\Rightarrow(1):$ Let $\phi:\Gamma\to H\otimes\Gamma$ be defined by the
equation

$\phi(\sum_{i}a_{i}\cdot db_{i})=\sum_{i}\varphi(a_{i})(id\otimes
d)(\varphi(b_{i}))$

as was obtained in the above computation. By using hypothesis $(3)$ it is
immediate to see that $\phi$ is well-defined. If we write
$\varphi(a)=a_{(-1)}\otimes a_{(0)}$ for any $a\in A$ and
$\phi(\omega)=\omega_{(-1)}\otimes\omega_{(0)}$ for all $\omega\in\Gamma$,
then for $\omega=\sum_{i}a_{i}\cdot db_{i}\in\Gamma$ we have

$\phi(\omega)=\omega_{(-1)}\otimes\omega_{(0)}=\sum_{i}a_{i,(-1)}b_{i,(-1)}\otimes
a_{i,(0)}\cdot db_{i,(0)},$

where we have used the notation $\varphi(a_{i})=a_{i,(-1)}\otimes a_{i,(0)}$.
Now we prove that $\phi$ is a left Hom-coaction of $(H,\beta)$ on
$(\Gamma,\gamma)$:

$\displaystyle\beta^{-1}(\omega_{(-1)})\otimes\phi(\omega_{(0)})$
$\displaystyle=$
$\displaystyle\sum_{i}\beta^{-1}(a_{i,(-1)}b_{i,(-1)})\otimes\phi(a_{i,(0)}\cdot
db_{i,(0)})$ $\displaystyle=$
$\displaystyle\sum_{i}\beta^{-1}(a_{i,(-1)})\beta^{-1}(b_{i,(-1)})\otimes
a_{i,(0)(-1)}b_{i,(0)(-1)}\otimes a_{i,(0)(0)}\cdot db_{i,(0)(0)}$
$\displaystyle=$ $\displaystyle\sum_{i}a_{i,(-1)1}b_{i,(-1)1}\otimes
a_{i,(-1)2}b_{i,(-1)2}\otimes\alpha^{-1}(a_{i,(0)})\cdot
d(\alpha^{-1}(b_{i,(0)}))$ $\displaystyle=$
$\displaystyle\sum_{i}(a_{i,(-1)}b_{i,(-1)})_{1}\otimes(a_{i,(-1)}b_{i,(-1)})_{2}\otimes\gamma^{-1}(a_{i,(0)}\cdot
db_{i,(0)})$ $\displaystyle=$
$\displaystyle\Delta(\omega_{(-1)})\otimes\gamma^{-1}(\omega_{(0)}),$

$\displaystyle\varepsilon(\omega_{(-1)})\omega_{(0)}$ $\displaystyle=$
$\displaystyle\sum_{i}\varepsilon(a_{i,(-1)}b_{i,(-1)})a_{i,(0)}\cdot
db_{i,(0)}$ $\displaystyle=$
$\displaystyle\sum_{i}\varepsilon(a_{i,(-1)})a_{i,(0)}\cdot
d(\varepsilon(b_{i,(-1)})b_{i,(0)})$ $\displaystyle=$
$\displaystyle\sum_{i}\alpha^{-1}(a_{i})\cdot
d(\alpha^{-1}(b_{i}))=\gamma^{-1}(\omega),$
$\displaystyle\phi(\gamma(\sum_{i}a_{i}\cdot db_{i}))$ $\displaystyle=$
$\displaystyle\phi(\sum_{i}\alpha(a_{i})\cdot d(\alpha(b_{i})))$
$\displaystyle=$ $\displaystyle\sum_{i}\varphi(\alpha(a_{i}))(id\otimes
d)(\varphi(\alpha(b_{i})))$ $\displaystyle=$
$\displaystyle(\beta\otimes\alpha)(\varphi(a_{i}))(id\otimes
d)((\beta\otimes\alpha)(\varphi(b_{i})))$ $\displaystyle=$
$\displaystyle\sum_{i}\beta(a_{i,(-1)})\beta(b_{i,(-1)})\otimes\alpha(a_{i,(0)})\cdot
d(\alpha(b_{i,(0)}))$ $\displaystyle=$
$\displaystyle(\beta\otimes\gamma)(\phi(\sum_{i}a_{i}\cdot db_{i})).$

Let $\omega=\sum_{i}a_{i}\cdot db_{i}\in\Gamma$, and $a,b\in A$. Then we have

$\displaystyle\phi(\alpha(a)\cdot(\omega\cdot b))$ $\displaystyle=$
$\displaystyle\phi(\alpha(a)\cdot(\sum_{i}(a_{i}\cdot db_{i})\cdot b))$
$\displaystyle=$ $\displaystyle\phi(\alpha(a)\cdot(\sum_{i}(\alpha(a_{i})\cdot
d(b_{i}\alpha^{-1}(b))-(a_{i}b_{i})\cdot db)))$ $\displaystyle=$
$\displaystyle\phi(\sum_{i}[(a\alpha(a_{i}))\cdot
d(\alpha(b_{i})b)-(a(a_{i}b_{i}))\cdot d(\alpha(b))])$ $\displaystyle=$
$\displaystyle\sum_{i}\varphi(a\alpha(a_{i}))(id\otimes
d)(\varphi(\alpha(b_{i})b))-\sum_{i}\varphi(a(a_{i}b_{i}))(id\otimes
d)(\varphi(\alpha(b)))$ $\displaystyle=$
$\displaystyle\sum_{i}(\varphi(a)\varphi(\alpha(a_{i})))(id\otimes
d)(\varphi(\alpha(b_{i})b))-\sum_{i}(\varphi(a)\varphi(a_{i}b_{i}))(id\otimes
d)(\varphi(\alpha(b)))$ $\displaystyle=$
$\displaystyle\sum_{i}\varphi(\alpha(a))(\varphi(\alpha(a_{i}))(id\otimes
d)(\varphi(b_{i}\alpha^{-1}(b))))-\sum_{i}\varphi(\alpha(a))(\varphi(a_{i}b_{i})(id\otimes
d)(\varphi(b)))$ $\displaystyle=$
$\displaystyle\varphi(\alpha(a))(\sum_{i}\varphi(\alpha(a_{i}))(id\otimes
d)(\varphi(b_{i}\alpha^{-1}(b)))-\sum_{i}\varphi(a_{i}b_{i})(id\otimes
d)(\varphi(b)))$ $\displaystyle=$
$\displaystyle\varphi(\alpha(a))(\sum_{i}\varphi(\alpha(a_{i}))(id\otimes
d)(\varphi(b_{i}\alpha^{-1}(b)))-\sum_{i}(\varphi(a_{i})\varphi(b_{i}))(id\otimes
d)(\varphi(b)))$ $\displaystyle=$
$\displaystyle\varphi(\alpha(a))(\sum_{i}\varphi(\alpha(a_{i}))[(id\otimes
d)(\varphi(b_{i}\alpha^{-1}(b)))-\varphi(b_{i})(id\otimes
d)(\varphi(\alpha^{-1}(b)))])$ $\displaystyle=$
$\displaystyle\varphi(\alpha(a))(\sum_{i}\varphi(\alpha(a_{i}))[((id\otimes
d)(\varphi(b_{i})))\varphi(\alpha^{-1}(b))])$ $\displaystyle=$
$\displaystyle\varphi(\alpha(a))([\sum_{i}\varphi(a_{i})(id\otimes
d)(\varphi(b_{i}))]\varphi(b))$ $\displaystyle=$
$\displaystyle\varphi(\alpha(a))(\phi(\omega)\varphi(b)),$

which is the first condition of Definition 2.4. For any $a\in A$, we get

$\displaystyle\phi(da)$ $\displaystyle=$
$\displaystyle\phi(1_{A}\cdot\gamma^{-1}(da))=\phi(1_{A}\cdot
d(\alpha^{-1}(a)))$ $\displaystyle=$ $\displaystyle\varphi(1_{A})(id\otimes
d)(\varphi(\alpha^{-1}(a)))$ $\displaystyle=$ $\displaystyle(1_{H}\otimes
1_{A})[((id\otimes d)\circ(\beta^{-1}\otimes\alpha^{-1}))(\varphi(a))]$
$\displaystyle=$ $\displaystyle(1_{H}\otimes
1_{A})[((\beta^{-1}\otimes\gamma^{-1})\circ(id\otimes
d))(\varphi(a))]=(id\otimes d)(\varphi(a)),$

which is the second condition of Definition 2.4. ∎

##  3\. Universal Differential Calculus of a Monoidal Hom-Algebra

In the theory of quantum groups, a differential calculus is a substitute of
the de Rham complex of a smooth manifold for arbitrary algebras. In this
section, the definition of differential calculus over a monoidal Hom-algebra
(abbreviated, Hom-DC) is given and the construction of the universal
differential calculus of a monoidal Hom-algebra (universal Hom-DC) is
outlined.

######  Definition 3.1.

A graded monoidal Hom-algebra is a monoidal Hom-algebra $(A,\alpha)$ together
with subobjects $A_{n},n\geq 0$ (that is, for each $k$-submodule
$A_{n}\subseteq A$,
$(A_{n},\alpha|_{A_{n}})\in\widetilde{\mathcal{H}}(\mathcal{M}_{k})$) such
that

$A=\bigoplus_{n\geq 0}A_{n},$

$1\in A_{0}$, and $A_{n}A_{m}\subseteq A_{n+m}$ for all $n,m\geq 0$.

######  Definition 3.2.

A differential calculus over a monoidal Hom-algebra $(A,\alpha)$ is a graded
monoidal Hom-algebra $(\Gamma=\bigoplus_{n\geq 0}\Gamma^{n},\gamma)$ with a
linear map $d:\Gamma\to\Gamma$, in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$,
of degree one (i.e., $d:\Gamma^{n}\to\Gamma^{n+1}$) such that

  1. (1)

$d^{2}=0$,

  2. (2)

$d(\omega\omega^{\prime})=d(\omega)\omega^{\prime}+(-1)^{n}\omega
d(\omega^{\prime})$ for $\omega\in\Gamma^{n},\omega^{\prime}\in\Gamma$ (graded
Leibniz rule),

  3. (3)

$\Gamma^{0}=A$, $\gamma|_{\Gamma^{0}}=\alpha$, and $\Gamma^{n}$ is a linear
span of the elements of the form

$a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))$ with $a_{0},\cdots,a_{n}\in A$,
$n\geq 0$.

A differential Hom-ideal of $(\Gamma,\gamma)$ is a Hom-ideal $\mathcal{I}$ of
the monoidal Hom-algebra $(\Gamma,\gamma)$ (that is, $\mathcal{I}$ is a
subobject of $(\Gamma,\gamma)$ such that
$(\Gamma\mathcal{I})\Gamma=\Gamma(\mathcal{I}\Gamma)\subset\mathcal{I}$) such
that $\mathcal{I}\cap\Gamma^{0}=\\{0\\}$ and $\mathcal{I}$ is invariant under
the differentiation $d$.

Let us write $\gamma^{n}$ for $\gamma|_{\Gamma^{n}}$ for all $n\geq 0$. Then,
the map $d\in\widetilde{\mathcal{H}}(\mathcal{M}_{k})$ means that
$d\circ\gamma^{n}=\gamma^{n+1}\circ d$ for all $n\geq 0$. Let $\mathcal{I}$ be
a differential Hom-ideal of a $(A,\alpha)$-Hom-DC $(\Gamma,\gamma)$. Then,
$\gamma$ induces an automorphism $\bar{\gamma}$ of
$\bar{\Gamma}:=\Gamma/\mathcal{I}$ and $(\bar{\Gamma},\bar{\gamma})$ is a
monoidal Hom-algebra. Since the condition $\mathcal{I}\cap\Gamma^{0}=\\{0\\}$
holds, $\bar{\Gamma}^{0}=\Gamma^{0}=A$. On the other hand, let
$\pi:\Gamma\to\bar{\Gamma}$ be the canonical surjective map and define
$\bar{d}:\bar{\Gamma}\to\bar{\Gamma}$ by
$\bar{d}(\pi(\omega)):=\pi(d(\omega))$ for any $\omega\in\Gamma$. Thus,
$(\bar{\Gamma},\bar{\gamma})$ is again a Hom-DC on $(A,\alpha)$ with
differentiation $\bar{d}$.

In the rest of the section, the construction of the universal differential
calculus on a monoidal Hom-algebra $(A,\alpha)$ is discussed. Let $(A,\alpha)$
be a monoidal Hom-algebra with Hom-multiplication $m_{A}:A\otimes A\to A$. The
linear map $d:A\to A\otimes A$, in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$,
given by

$da:=1\otimes\alpha^{-1}(a)-\alpha^{-1}(a)\otimes 1,\>\>\forall a\in A$

satisfies the Leibniz rule: For $a,b\in A$,

$\displaystyle a\cdot db+da\cdot b$ $\displaystyle=$ $\displaystyle
a\cdot(1\otimes\alpha^{-1}(b)-\alpha^{-1}(b)\otimes
1)+(1\otimes\alpha^{-1}(a)-\alpha^{-1}(a)\otimes 1)\cdot b$ $\displaystyle=$
$\displaystyle\alpha^{-1}(a)1\otimes\alpha(\alpha^{-1}(b))-\alpha^{-1}(a)\alpha^{-1}(b)\otimes
1$ $\displaystyle+$ $\displaystyle
1\otimes\alpha^{-1}(a)\alpha^{-1}(b)-\alpha(\alpha^{-1}(a))\otimes
1\alpha^{-1}(b)$ $\displaystyle=$ $\displaystyle a\otimes
b-\alpha^{-1}(ab)\otimes 1+1\otimes\alpha^{-1}(ab)-a\otimes
b=1\otimes\alpha^{-1}(ab)-\alpha^{-1}(ab)\otimes 1$ $\displaystyle=$
$\displaystyle d(ab).$

For any $a\in A$, we get

$\displaystyle(d\circ\alpha)(a)$ $\displaystyle=$ $\displaystyle
d(\alpha(a))=1\otimes a-a\otimes 1$ $\displaystyle=$
$\displaystyle(\alpha\otimes\alpha)(1\otimes\alpha^{-1}(a)-\alpha^{-1}(a)\otimes
1)=(\alpha\otimes\alpha)(da),$

meaning $d$ is in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$. Let
$\Omega^{1}(A)$ be the $(A,\alpha)$-Hom-subbimodule of $(A\otimes
A,\alpha\otimes\alpha)$ generated by elements of the form $a\cdot db$ for
$a,b\in A$. Then we have

$\Omega^{1}(A)=ker\>m_{A}.$

Indeed, if $a\cdot db\in\Omega^{1}(A)$, then

$m_{A}(a\cdot db)=m_{A}(a\otimes b-\alpha^{-1}(ab)\otimes
1)=ab-\alpha^{-1}(ab)1=0.$

On the other hand, if $\sum_{i}a_{i}\otimes b_{i}\in ker\>m_{A}$ ($\sum_{i}$
denotes a finite sum), then $\sum_{i}a_{i}b_{i}=0$, thus we write

$\sum_{i}a_{i}\otimes b_{i}=\sum_{i}(a_{i}\otimes
b_{i}-\alpha^{-1}(a_{i}b_{i})\otimes
1)=\sum_{i}a_{i}\cdot(1\otimes\alpha^{-1}(b_{i})-\alpha^{-1}(b_{i})\otimes
1)=\sum_{i}a_{i}\cdot db_{i}.$

The left and right $(A,\alpha)$-Hom-module structures of
$(\Omega^{1}(A),\beta)=(\Omega^{1}(A),(\alpha\otimes\alpha)|_{ker\>m_{A}})$
are respectively given by

$a\cdot(b\cdot dc)=(\alpha^{-1}(a)b)\cdot d(\alpha(c))\qquad(a\cdot db)\cdot
c=\alpha(a)\cdot d(b\alpha^{-1}(c))-(ab)\cdot dc,$

for any $a,b,c\in A$. $(\Omega^{1}(A),\beta)$ is called the universal first
order differential calculus of monoidal Hom-algebra $(A,\alpha)$.

Let $\bar{A}:=A/k\cdot 1$ be the quotient space of $A$ by the scalar multiples
of the Hom-unit and let $\bar{a}$ denote the equivalence class $a+k\cdot 1$
for any $a\in A$. $\alpha$ induces an automorphism
$\bar{\alpha}:\bar{A}\to\bar{A},\>\bar{a}\mapsto\bar{\alpha}(\bar{a})=\overline{\alpha(a)}$
and $(\bar{A},\bar{\alpha})\in\widetilde{\mathcal{H}}(\mathcal{M}_{k})$. Let
$A\otimes\bar{A}=\Omega^{1}(A)$ by the identification
$a_{0}\otimes\overline{a_{1}}\mapsto a_{0}da_{1}$. This identification is
well-defined since $d1=0$, and one can easily show that it is an
$(A,\alpha)$-Hom-bimodule isomorphism once the Hom-bimodule structure of
$(A\otimes\bar{A},\alpha\otimes\bar{\alpha})$ is given by, for $b\in A$,

$b(a_{0}\otimes\overline{a_{1}})=\alpha^{-1}(b)a_{0}\otimes\overline{\alpha(a_{1})},\>\>(a_{0}\otimes\overline{a_{1}})b=\alpha(a_{0})\otimes\overline{a_{1}\alpha^{-1}(b)}-a_{0}a_{1}\otimes\bar{b}.$

Now, we set

$\Omega^{n}(A):=\otimes^{(n)}_{A}(\Omega^{1}(A))=\Omega^{1}(A)\otimes_{A}(\otimes^{(n-1)}_{A}(\Omega^{1}(A))).$

Above, $\otimes^{(n)}_{A}(\Omega^{1}(A))$ has been put for

$T^{n}_{A}(\Omega^{1}(A))=\otimes^{t^{n}}_{A}(\Omega^{1}(A),\cdots,\Omega^{1}(A))=\Omega^{1}(A)\otimes_{A}(\Omega^{1}(A)\otimes_{A}(\cdots(\Omega^{1}(A)\otimes_{A}\Omega^{1}(A))\cdots)),$

where $t^{n}$ is a fixed element in the set $T_{n}$ of planar binary trees
with $n$ leaves and one root, which corresponds to the parenthesized monomial
$x_{1}(x_{2}(\cdots(x_{n-1}x_{n})\cdots))$ in $n$ noncommuting variables (see
[Yau0] e.g.). One should also refer to [2, Section 6] for the construction of
tensor Hom-algebra applied to an object
$(M,\mu)\in\widetilde{\mathcal{H}}(\mathcal{M}_{k})$). So, we have, for any
$n\geq 0$,

$\Omega^{n}(A)=A\otimes(\otimes^{(n)}(\bar{A}))=A\otimes(\bar{A}\otimes(\bar{A}\otimes(\cdots(\bar{A}\otimes\bar{A})\cdots)))$

by the correspondence
$(A\otimes\bar{A})\otimes_{A}(A\otimes(\otimes^{(n-1)}(\bar{A})))=A\otimes(\otimes^{(n)}_{A}(\bar{A}))$,
in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$,

$\displaystyle(a_{0}\otimes\overline{a_{1}})\otimes_{A}(a_{2}\otimes(\otimes^{(n-1)}(\overline{a_{3}},\cdots,\overline{a_{n+1}})))$
$\displaystyle\mapsto$
$\displaystyle\alpha(a_{0})\otimes(\otimes^{n}(\overline{\alpha^{-1}(a_{1}a_{2})},\overline{a_{3}},\cdots,\overline{a_{n+1}}))$
$\displaystyle-
a_{0}a_{1}\otimes(\otimes^{n}(\overline{a_{2}},\cdots,\overline{a_{n+1}})),$

where we have used the notation
$\otimes^{(n)}(\overline{a_{1}},\cdots,\overline{a_{n}})$ for
$\overline{a_{1}}\otimes(\overline{a_{2}}\otimes(\cdots(\overline{a_{n-1}}\otimes\overline{a_{n}})\cdots))$.
To the object $A\otimes(\otimes^{(n)}(\bar{A}))$ we associate the automorphism
$\alpha\otimes(\otimes^{(n)}(\bar{\alpha})):A\otimes(\otimes^{(n)}(\bar{A}))\to
A\otimes(\otimes^{(n)}(\bar{A}))$ given by

$a_{0}\otimes(\otimes^{(n)}(\overline{a_{1}},\cdots,\overline{a_{n}}))\mapsto\alpha(a_{0})\otimes(\otimes^{(n)}(\overline{\alpha(a_{1})},\cdots,\overline{\alpha(a_{n})})),$

for $a_{0}\in A$ and $\overline{a_{i}}\in\bar{A}$, $i=1,\cdots,n$.

On $\bigoplus_{n=0}^{\infty}\Omega^{n}(A)$, we define the differential by the
linear mapping $d:A\otimes(\otimes^{(n)}(\bar{A}))\to
A\otimes(\otimes^{(n+1)}(\bar{A}))$ of degree one by

(3.1)
$d(a_{0}\otimes(\overline{a_{1}}\otimes(\cdots(\overline{a_{n-1}}\otimes\overline{a_{n}})\cdots)))=1\otimes(\overline{\alpha^{-1}(a_{0})}\otimes(\cdots(\overline{\alpha^{-1}(a_{n-1})}\otimes\overline{\alpha^{-1}(a_{n})})\cdots)).$

We immediately obtain $d^{2}=0$ from the fact that $\bar{1}=0$. If we start
with $a_{n}\in A$, multiplying on the left and applying $d$ repeatedly gives
us the following

$a_{0}\otimes(\overline{a_{1}}\otimes(\cdots(\overline{a_{n-1}}\otimes\overline{a_{n}})\cdots))=a_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots))),$

where
$a_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots)))=a_{0}\otimes_{A}(da_{1}\otimes_{A}(da_{2}\otimes_{A}(\cdots(da_{n-1}\otimes_{A}da_{n})\cdots)))$.

We make $\bigoplus_{n=0}^{\infty}\Omega^{n}(A)$ an $(A,\alpha)$-Hom-bimodule
as follows. The left $(A,\alpha)$-Hom-module structure is given by, for $b\in
A$ and $a_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots)))\in\Omega^{n}(A)$,
$n\geq 1$,

$\displaystyle b(a_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots))))$
$\displaystyle=$
$\displaystyle(\alpha^{-1}(b)a_{0})(d(\alpha(a_{1}))(d(\alpha(a_{2}))(\cdots(d(\alpha(a_{n-1}))d(\alpha(a_{n}))\cdots))).$

We now get the right $(A,\alpha)$-Hom-module structure: One can show that, for
$b\in A$, $a_{0}da_{1}\in\Omega^{1}(A)$, $a_{0}(da_{1}da_{2})\in\Omega^{2}(A)$
and $a_{0}(da_{1}(da_{2}da_{3}))\in\Omega^{3}(A)$, the following equations
hold:

$(a_{0}da_{1})b=\alpha(a_{0})d(a_{1}\alpha^{-1}(b))-(a_{0}a_{1})db,$

$\displaystyle(a_{0}(da_{1}da_{2}))b$ $\displaystyle=$
$\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))d(a_{2}\alpha^{-2}(b)))-\alpha(a_{0})(d(a_{1}a_{2})d(\alpha^{-1}(b)))$
$\displaystyle+$
$\displaystyle(a_{0}\alpha(a_{1}))(d(\alpha(a_{2}))d(\alpha^{-1}(b))),$
$\displaystyle(a_{0}(da_{1}(da_{2}da_{3})))b$ $\displaystyle=$
$\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))(d(\alpha(a_{2}))d(a_{3}\alpha^{-3}(b))))-\alpha(a_{0})(d(\alpha(a_{1}))(d(a_{2}a_{3})d(\alpha^{-2}(b))))$
$\displaystyle+$
$\displaystyle\alpha(a_{0})(d(a_{1}\alpha(a_{2}))(d(\alpha(a_{3}))d(\alpha^{-2}(b))))-(a_{0}\alpha(a_{1}))(d(\alpha^{2}(a_{2}))(d(\alpha(a_{3}))d(\alpha^{-2}(b)))).$

By induction, one can also prove that the equation

$\displaystyle(a_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots))))b$
$\displaystyle=$
$\displaystyle(-1)^{n}(a_{0}\alpha(a_{1}))(d(\alpha^{2}(a_{2}))(d(\alpha^{2}(a_{3}))(\cdots
d(\alpha^{2}(a_{n-1}))(d(\alpha(a_{n}))d(\alpha^{-(n-1)}(b)))\cdots)))$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n-3}(-1)^{n-i}\alpha(a_{0})(d(\alpha(a_{1}))(\cdots
d(\alpha(a_{i-1}))(d(a_{i}\alpha(a_{i+1}))(d(\alpha^{2}(a_{i+2}))(\cdots$
$\displaystyle
d(\alpha^{2}(a_{n-1}))(d(\alpha(a_{n}))d(\alpha^{-(n-1)}(b)))\cdots)))\cdots))$
$\displaystyle+$ $\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))(\cdots
d(\alpha(a_{n-3}))(d(a_{n-2}\alpha(a_{n-1}))(d(\alpha(a_{n}))d(\alpha^{-(n-1)}(b))))\cdots))$
$\displaystyle-$ $\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))(\cdots
d(\alpha(a_{n-2}))(d(a_{n-1}a_{n})d(\alpha^{-(n-1)}(b)))\cdots))$
$\displaystyle+$
$\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))(\cdots(d(\alpha(a_{n-1}))d(a_{n}\alpha^{-n}(b)))\cdots))$

holds for
$a_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots)))\in\Omega^{n}(A)$, $n\geq
4$.

Next, we define the Hom-multiplication between any two parenthesized
monomials, by using the right Hom-module structure given above, as

$\displaystyle[a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))][a_{n+1}(da_{n+2}(\cdots(da_{n+k-1}da_{n+k})\cdots))]$
$\displaystyle=$
$\displaystyle[(\alpha^{-1}(a_{0})(d(\alpha^{-1}(a_{1}))(\cdots(d(\alpha^{-1}(a_{n-1}))d(\alpha^{-1}(a_{n}))\cdots))))a_{n+1}]$
$\displaystyle[d(\alpha(a_{n+2}))(\cdots(d(\alpha(a_{n+k-1}))d(\alpha(a_{n+k}))\cdots))],$

for $\omega_{n}=a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))\in\Omega^{n}(A)$
and $\omega_{k-1}=a_{n+1}(da_{n+2}(\cdots(da_{n+k-1}da_{n+k})\cdots))$
$\in\Omega^{k-1}(A)$. For any $n\geq 4$, we explicitly write the above
multiplication:

$\displaystyle\omega_{n}\omega_{k-1}$ $\displaystyle=$
$\displaystyle[a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))][a_{n+1}(da_{n+2}(\cdots(da_{n+k-1}da_{n+k})\cdots))]$
$\displaystyle=$
$\displaystyle[(\alpha^{-1}(a_{0})(d(\alpha^{-1}(a_{1}))(\cdots(d(\alpha^{-1}(a_{n-1}))d(\alpha^{-1}(a_{n}))\cdots))))a_{n+1}]$
$\displaystyle[d(\alpha(a_{n+2}))(\cdots(d(\alpha(a_{n+k-1}))d(\alpha(a_{n+k}))\cdots))]$
$\displaystyle=$
$\displaystyle[(-1)^{n}(\alpha^{-1}(a_{0})a_{1})(d(\alpha(a_{2}))(d(\alpha(a_{3}))(\cdots
d(\alpha(a_{n-1}))(d(a_{n})d(\alpha^{-(n-1)}(a_{n+1})))\cdots)))$
$\displaystyle+$ $\displaystyle\sum_{i=1}^{n-3}(-1)^{n-i}a_{0}(d(a_{1})(\cdots
d(a_{i-1})(d(\alpha^{-1}(a_{i})a_{i+1})(d(\alpha(a_{i+2}))(\cdots$
$\displaystyle
d(\alpha(a_{n-1}))(d(a_{n})d(\alpha^{-(n-1)}(a_{n+1})))\cdots)))\cdots))$
$\displaystyle+$ $\displaystyle a_{0}(d(a_{1})(\cdots
d(a_{n-3})(d(\alpha^{-1}(a_{n-2})a_{n-1})(d(a_{n})d(\alpha^{-(n-1)}(a_{n+1}))))\cdots))$
$\displaystyle-$ $\displaystyle a_{0}(d(a_{1})(\cdots
d(a_{n-2})(d(\alpha^{-1}(a_{n-1}a_{n}))d(\alpha^{-(n-1)}(a_{n+1})))\cdots))$
$\displaystyle+$ $\displaystyle
a_{0}(d(a_{1})(\cdots(d(a_{n-1})d(\alpha^{-1}(a_{n})\alpha^{-n}(a_{n+1})))\cdots))]$
$\displaystyle[d(\alpha(a_{n+2}))(\cdots(d(\alpha(a_{n+k-1}))d(\alpha(a_{n+k}))\cdots))]$
$\displaystyle=$
$\displaystyle(-1)^{n}(a_{0}\alpha(a_{1}))(d(\alpha^{2}(a_{2}))(\cdots
d(\alpha^{2}(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots)))\cdots))$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n-3}(-1)^{n-i}\alpha(a_{0})(d(\alpha(a_{1}))(\cdots
d(\alpha(a_{i-1}))(d(a_{i}\alpha(a_{i+1}))(d(\alpha^{2}(a_{i+2}))(\cdots
d(\alpha^{2}(a_{n-1}))$
$\displaystyle(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots)))\cdots)))\cdots))$
$\displaystyle+$ $\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))(\cdots
d(\alpha(a_{n-3}))(d(a_{n-2}\alpha(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))$
$\displaystyle-$ $\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))\cdots
d(\alpha(a_{n-2}))(d(a_{n-1}a_{n-1})(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))\cdots))$
$\displaystyle+$ $\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))(\cdots
d(\alpha(a_{n-2}))(d(\alpha(a_{n-1}))(d(\alpha^{-1}(a_{n})\alpha^{-n}(a_{n+1}))(d(\alpha^{-(n-1)}(a_{n+2}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))$

On the other hand, we have the following computations for $\omega_{n}$ and
$\omega_{k-1}$ given above:

$\displaystyle d\omega_{n}\omega_{k-1}$ $\displaystyle=$
$\displaystyle[da_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))][a_{n+1}(da_{n+2}(\cdots(da_{n+k-1}da_{n+k})\cdots))]$
$\displaystyle=$
$\displaystyle[(d(\alpha^{-1}(a_{0}))(d(\alpha^{-1}(a_{1}))(\cdots(d(\alpha^{-1}(a_{n-1}))d(\alpha^{-1}(a_{n}))\cdots))))a_{n+1}]$
$\displaystyle[d(\alpha(a_{n+2}))(\cdots(d(\alpha(a_{n+k-1}))d(\alpha(a_{n+k})))\cdots)]$
$\displaystyle=$
$\displaystyle(-1)^{n+1}\alpha(a_{0})(d(\alpha(a_{1}))(\cdots(d(\alpha(a_{n-1}))(da_{n}(d(\alpha^{-n}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-n}(a_{n+k-1}))d(\alpha^{-n}(a_{n+k})))\cdots))))\cdots))$
$\displaystyle+$
$\displaystyle(-1)^{n}d(a_{0}\alpha(a_{1}))(d(\alpha^{2}(a_{2}))(\cdots(d(\alpha^{2}(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))$
$\displaystyle+$
$\displaystyle\sum_{i=2}^{n-2}(-1)^{n+1-i}d(\alpha(a_{0}))(\cdots(d(\alpha(a_{i-2}))(d(a_{i-1}\alpha(a_{i}))(d(\alpha^{2}(a_{i+1}))(\cdots(d(\alpha^{2}(a_{n-1}))$
$\displaystyle(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))))\cdots)$
$\displaystyle+$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-3}))(d(a_{n-2}\alpha(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots)))))\cdots)$
$\displaystyle-$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-2}))(d(a_{n-1}a_{n})(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots)$
$\displaystyle+$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-1}))(d(\alpha^{-1}(a_{n})\alpha^{-n}(a_{n+1}))(d(\alpha^{-(n-1)}(a_{n+2}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots)$

and

$\displaystyle\omega_{n}d\omega_{k-1}$ $\displaystyle=$
$\displaystyle[a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))][da_{n+1}(\cdots(da_{n+k-1}da_{n+k})\cdots)]$
$\displaystyle=$
$\displaystyle\alpha(a_{0})([da_{1}(\cdots(da_{n-1}da_{n})\cdots)][d(\alpha^{-1}(a_{n+1}))(\cdots(d(\alpha^{-1}(a_{n+k-1}))d(\alpha^{-1}(a_{n+k})))\cdots)])$
$\displaystyle=$
$\displaystyle\alpha(a_{0})(d(\alpha(a_{1}))(\cdots(d(\alpha(a_{n-1}))(d(a_{n})(d(\alpha^{-n}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-n}(a_{n+k-1}))d(\alpha^{-n}(a_{n+k})))\cdots))))\cdots)).$

Thus, the equation below holds:

$\displaystyle d\omega_{n}\omega_{k-1}+(-1)^{n}\omega_{n}d\omega_{k-1}$
$\displaystyle=$
$\displaystyle(-1)^{n}d(a_{0}\alpha(a_{1}))(d(\alpha^{2}(a_{2}))(\cdots(d(\alpha^{2}(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))$
$\displaystyle+$
$\displaystyle\sum_{i=2}^{n-2}(-1)^{n+1-i}d(\alpha(a_{0}))(\cdots(d(\alpha(a_{i-2}))(d(a_{i-1}\alpha(a_{i}))(d(\alpha^{2}(a_{i+1}))(\cdots(d(\alpha^{2}(a_{n-1}))$
$\displaystyle(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))))\cdots)$
$\displaystyle+$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-3}))(d(a_{n-2}\alpha(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots)))))\cdots)$
$\displaystyle-$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-2}))(d(a_{n-1}a_{n})(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots)$
$\displaystyle+$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-1}))(d(\alpha^{-1}(a_{n})\alpha^{-n}(a_{n+1}))(d(\alpha^{-(n-1)}(a_{n+2}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots)$
$\displaystyle=$
$\displaystyle(-1)^{n}d(a_{0}\alpha(a_{1}))(d(\alpha^{2}(a_{2}))(\cdots(d(\alpha^{2}(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n-3}(-1)^{n-i}d(\alpha(a_{0}))(\cdots(d(\alpha(a_{i-1}))(d(a_{i}\alpha(a_{i+1}))(d(\alpha^{2}(a_{i+2}))(\cdots(d(\alpha^{2}(a_{n-1}))$
$\displaystyle(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots))))\cdots)$
$\displaystyle+$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-3}))(d(a_{n-2}\alpha(a_{n-1}))(d(\alpha(a_{n}))(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots)))))\cdots)$
$\displaystyle-$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-2}))(d(a_{n-1}a_{n})(d(\alpha^{-(n-1)}(a_{n+1}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots)$
$\displaystyle+$ $\displaystyle
d(\alpha(a_{0}))(\cdots(d(\alpha(a_{n-1}))(d(\alpha^{-1}(a_{n})\alpha^{-n}(a_{n+1}))(d(\alpha^{-(n-1)}(a_{n+2}))(\cdots$
$\displaystyle(d(\alpha^{-(n-1)}(a_{n+k-1}))d(\alpha^{-(n-1)}(a_{n+k})))\cdots))))\cdots)$
$\displaystyle=$ $\displaystyle d(\omega_{n}\omega_{k-1}),$

which is the graded Leibniz rule. Next, we verify by induction that the
following identity holds:

(3.3)
$d(a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots)))=da_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))$

using the graded Leibniz rule and the equation $d^{2}=d\circ d=0$. For
$a_{0}da_{1}\in\Omega^{1}(A)$,

$d(a_{0}da_{1})=da_{0}da_{1}+(-1)^{0}a_{0}d(d(a_{1}))=da_{0}da_{1}.$

Suppose now that the identity

$d(a_{0}(da_{1}(\cdots(da_{n-2}da_{n-1})\cdots)))=da_{0}(da_{1}(\cdots(da_{n-2}da_{n-1})\cdots))$

holds for $a_{0}(da_{1}(\cdots(da_{n-2}da_{n-1})\cdots))\in\Omega^{n-1}(A)$,
that is, if we replace $a_{i}$ with $a_{i+1}$ for $i=0,\cdots,n-1$, we have
$d(a_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots)))=da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots))$.
Thus, for $a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))\in\Omega^{n}(A)$,

$\displaystyle d(a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots)))$
$\displaystyle=$ $\displaystyle
da_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots)))+(-1)^{0}a_{0}d(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots)))$
$\displaystyle=$ $\displaystyle
da_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots)))+(-1)^{0}a_{0}d(d(a_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots))))$
$\displaystyle=$ $\displaystyle
da_{0}(da_{1}(da_{2}(\cdots(da_{n-1}da_{n})\cdots))).$

Let $(\Gamma,\gamma)$ be another Hom-DC on $(A,\alpha)$ with differential
$\tilde{d}$ and let the morphism $\psi:\Omega(A)\to\Gamma$, in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$, be given by

$\psi(a)=a\>\mbox{and}\>\psi(a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots)))=a_{0}(\tilde{d}a_{1}(\cdots(\tilde{d}a_{n-1}\tilde{d}a_{n})\cdots)),\>n\geq
1$

for $a\in A$, $a_{0}(da_{1}(\cdots(da_{n-1}da_{n})\cdots))\in\Omega^{n}(A)$.
Clearly, $\psi$ is surjective by its definition. Now, let
$\mathcal{N}:=ker\psi$ be the kernel of $\psi$. From the equations (3) and
(3.3) it is concluded that $\mathcal{N}$ is a differential Hom-ideal of
$\Omega(A)$. Thus, $\Gamma$ is identified with $\Omega(A)/\mathcal{N}$ showing
the universality of $\Omega(A)$.

##  4\. Left-Covariant FODC over Monoidal Hom-Hopf Algebras

###  4.1. Left-Covariant Hom-FODC and Their Right Hom-ideals

Let $(H,\alpha)$ be a monoidal Hom-Hopf algebra with a bijective antipode
throughout the section. $(H,\alpha)$ is a left Hom-quantum space for itself
with respect to the Hom-comultiplication $\Delta:H\to H\otimes H,h\mapsto
h_{1}\otimes h_{2}$. Thus, by applying Definition 2.4 to the monoidal Hom-Hopf
algebra $(H,\alpha)$ we obtain the following

######  Definition 4.1.

A FODC $(\Gamma,\gamma)$ over the monoidal Hom-Hopf algebra $(H,\alpha)$ is
said to be left-covariant if $(\Gamma,\gamma)$ is a left-covariant FODC over
the left Hom-quantum space $(H,\alpha)$ with left Hom-coaction
$\varphi=\Delta$ in Definition 2.4.

######  Remark 4.2.

According to Proposition 2.5, an $(H,\alpha)$-Hom-FODC $(\Gamma,\gamma)$ is
left-covariant if and only if there exists a morphism $\phi:\Gamma\to
H\otimes\Gamma$ in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$ such that, for
$h,g\in H$,

(4.4) $\phi(h\cdot dg)=\Delta(h)(id\otimes d)(\Delta(g)).$

In the proof of Proposition 2.5, it has been shown that if there is such a
morphism $\phi$, it defines a left Hom-comodule structure of $(\Gamma,\gamma)$
on $(H,\alpha)$ and satisfies

$\phi(\alpha(h)\cdot(\omega\cdot g))=\Delta(\alpha(h))(\phi(\omega)\Delta(g))$

for $h,g\in H$ and $\omega\in\Gamma$. From this it follows that
$(\Gamma,\gamma)$ is a left-covariant $(H,\alpha)$-Hom-bimodule.

Let $(\Gamma,\gamma)$ be a left-covariant $(H,\alpha)$-Hom-FODC with
derivation $d:H\to\Gamma$. By the above remark $(\Gamma,\gamma)$ is a left-
covariant $(H,\alpha)$-Hom-bimodule, and then by adapting the structure theory
of left-covariant Hom-bimodules, which is discussed in Lemma 4.7 and
Proposition 4.9 in [11], to $(\Gamma,\gamma)$ we summarize the following
results. We have the unique projection
$P_{L}:(\Gamma,\gamma)\to(^{coH}\Gamma,\gamma|_{{}^{coH}\Gamma})$ given by
$P_{L}(\varrho)=S(\varrho_{(-1)})\varrho_{(0)}$, for all $\varrho\in\Gamma$,
such that

$P_{L}(h\cdot\varrho)=\varepsilon(h)\gamma(P_{L}(\varrho)),\qquad\varrho=\varrho_{(-1)}P_{L}(\varrho_{(0)})$

and

$P_{L}(\varrho\cdot
h)=\widetilde{ad}_{R}(h)(P_{L}(\varrho))=:P_{L}(\varrho)\lhd h$

for any $h\in H$ and $\varrho\in\Gamma$. Let us now define a linear mapping
$\omega_{\Gamma}:H\overset{d}{\rightarrow}\Gamma\overset{P_{L}}{\rightarrow}\>^{coH}\Gamma$
by

$\omega_{\Gamma}(h)=P_{L}(dh),\>\forall h\in H.$

Obviously, it is in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$, that is,
$\omega_{\Gamma}\circ\alpha=\gamma\circ\omega_{\Gamma}$. Since
$\phi(dh)=(dh)_{(-1)}\otimes(dh)_{(0)}=(id\otimes d)(\Delta(h))=h_{1}\otimes
dh_{2}$ by the above remark, we obtain

(4.5) $\omega_{\Gamma}(h)=P_{L}(dh)=S(h_{1})\cdot dh_{2},\>\forall h\in H.$

On the other hand, we can write $dh=(dh)_{(-1)}\cdot
P_{L}((dh)_{(0)})=h_{1}\cdot P_{L}(dh_{2})$, that is,

(4.6) $dh=h_{1}\cdot\omega_{\Gamma}(h_{2}),\>\forall h\in H.$

We will drop the subscript $\Gamma$ from $\omega_{\Gamma}(\cdot)$. By
definition, for any $h\in H$, $\omega(h)\in\>^{coH}\Gamma$. Conversely, let
$\varrho=\sum_{i}h_{i}\cdot dg_{i}\in\>^{coH}\Gamma$ for $h_{i},g_{i}\in H$.
Then

$\varrho=P_{L}(\varrho)=\sum_{i}\varepsilon(h_{i})\gamma(P_{L}(dg_{i}))=\sum_{i}\varepsilon(h_{i})\gamma(\omega(g_{i}))=\sum_{i}\varepsilon(h_{i})\omega(\alpha(g_{i})),$

showing that $\rho\in\omega(H)$. Thus, we get $\omega(H)=\>^{coH}\Gamma$ which
implies that $\Gamma=H\cdot\omega(H)=\omega(H)\cdot H$ and hence any
$k$-linear basis of $\omega(H)$ is a left $(H,\alpha)$-Hom-module basis and a
right $(H,\alpha)$-Hom-module basis for $(\Gamma,\gamma)$.

For $h,g\in H$, we get

$\displaystyle\omega(h)\lhd g$ $\displaystyle=$ $\displaystyle
P_{L}(\omega(h)\cdot g)=P_{L}((S(h_{1})\cdot dh_{2})\cdot g)$ $\displaystyle=$
$\displaystyle
P_{L}(S(\alpha(h_{1}))\cdot(dh_{2}\cdot\alpha^{-1}(g)))=\varepsilon(S(\alpha(h_{1})))\gamma(P_{L}(dh_{2}\cdot\alpha^{-1}(g)))$
$\displaystyle=$
$\displaystyle\varepsilon(h_{1})\gamma(P_{L}(dh_{2}\cdot\alpha^{-1}(g)))=\gamma(P_{L}(d(\alpha^{-1}(h))\cdot\alpha^{-1}(g)))$
$\displaystyle=$
$\displaystyle\gamma(P_{L}(d(\alpha^{-1}(hg))-\alpha^{-1}(h)\cdot
d(\alpha^{-1}(g))))$ $\displaystyle=$
$\displaystyle\gamma(\omega(\alpha^{-1}(hg)))-\gamma(\varepsilon(\alpha^{-1}(h))\gamma(P_{L}(d(\alpha^{-1}(g)))))$
$\displaystyle=$
$\displaystyle\omega(hg)-\varepsilon(h)\gamma^{2}(\omega(\alpha^{-1}(g)))=\omega(hg)-\varepsilon(h)\omega(\alpha(g))$
$\displaystyle=$
$\displaystyle\omega(hg-\varepsilon(h)\alpha(g))=\omega((h-\varepsilon(h)1)g).$

Thus, by setting the notation $\bar{h}:=h-\varepsilon(h)1$, we have

(4.7) $\omega(h)\lhd g=\widetilde{ad}_{R}(g)(\omega(h))=\omega(\bar{h}g),$

and we rewrite the $(H,\alpha)$-Hom-bimodule structure as

(4.8)
$g^{\prime}\cdot(g\cdot\omega(h))=(\alpha^{-1}(g^{\prime})g)\cdot\omega(\alpha(h)),$
(4.9) $(g^{\prime}\cdot\omega(h))\cdot g=(g^{\prime}g_{1})\cdot(\omega(h)\lhd
g_{2})=(g^{\prime}g_{1})\cdot\omega(\bar{h}g_{2}),$

for $g,g^{\prime},h\in H$.

In the following example we introduce the universal FODC over monoidal Hom-
Hopf-algebra $(H,\alpha)$.

######  Example 4.3.

We define $(\Omega^{1}(H),\beta):=(H\otimes
ker\varepsilon,\alpha\otimes\alpha^{\prime})$, where
$\alpha^{\prime}=\alpha|_{ker\varepsilon}$. Let us denote the element
$1\otimes\alpha^{-1}(\bar{g})=1\otimes\overline{\alpha^{-1}(g)}$, for $g\in
H$, by $\omega(g)$. Thus we identify $g\otimes\bar{h}\in\Omega^{1}(H)$, where
$g,h\in H$, with $g\cdot\omega(h)$. We then introduce the Hom-bimodule
structure of $\Omega^{1}(H)$ as in (4.8) and (4.9), for all $g,g^{\prime},h\in
H$,

$g^{\prime}\cdot(g\cdot\omega(h)):=(\alpha^{-1}(g^{\prime})g)\cdot\omega(\alpha(h)),$
$(g^{\prime}\cdot\omega(h))\cdot
g:=(g^{\prime}g_{1})\cdot\omega(\bar{h}g_{2}),$

and a linear mapping

$d:H\to\Omega^{1}(H),\>h\mapsto
h_{1}\otimes\overline{h_{2}}=h_{1}\cdot\omega(h_{2}).$

For any $g,h\in H$,

$\displaystyle g\cdot dh+dg\cdot h$ $\displaystyle=$ $\displaystyle
g\cdot(h_{1}\cdot\omega(h_{2}))+(g_{1}\cdot\omega(g_{2}))\cdot h$
$\displaystyle=$
$\displaystyle(\alpha^{-1}(g)h_{1})\cdot\omega(\alpha(h_{2}))+(g_{1}h_{1})\cdot\omega(\overline{g_{2}}h_{2})$
$\displaystyle=$
$\displaystyle(\alpha^{-1}(g)h_{1})\cdot\omega(\alpha(h_{2}))+(g_{1}h_{1})\cdot\omega(g_{2}h_{2})-(g_{1}h_{1})\omega((\varepsilon(g_{2})1)h_{2})$
$\displaystyle=$
$\displaystyle(\alpha^{-1}(g)h_{1})\cdot\omega(\alpha(h_{2}))+(g_{1}h_{1})\cdot\omega(g_{2}h_{2})-(\alpha^{-1}(g)h_{1})\cdot\omega(\alpha(h_{2}))$
$\displaystyle=$
$\displaystyle(g_{1}h_{1})\cdot\omega(g_{2}h_{2})=(gh)_{1}\omega((gh)_{2})$
$\displaystyle=$ $\displaystyle d(gh),$

showing that $d$ satisfies the Leibniz rule.

$d(\alpha(h))=\alpha(h_{1})\cdot\omega(\alpha(h_{2}))=\alpha(h_{1})\cdot\beta(\omega(h_{2}))=\beta(h_{1}\cdot\omega(h_{2}))=\beta(dh),$

which means that $d\in\widetilde{\mathcal{H}}(\mathcal{M}_{k})$.

$\displaystyle\omega(h)$ $\displaystyle=$
$\displaystyle\omega(\alpha(\varepsilon(h_{1})h_{2}))=\varepsilon(h_{1})\omega(\alpha(h_{2}))$
$\displaystyle=$
$\displaystyle\varepsilon(h_{1})\beta(\omega(h_{2}))=(\varepsilon(h_{1})1)\cdot\omega(h_{2})$
$\displaystyle=$
$\displaystyle(S(h_{11})h_{12})\cdot\omega(h_{2})=(S(\alpha^{-1}(h_{1}))h_{21})\cdot\omega(\alpha(h_{22}))$
$\displaystyle=$
$\displaystyle\alpha(S(\alpha^{-1}(h_{1})))\cdot(h_{21}\cdot\beta^{-1}(\omega(\alpha(h_{22}))))$
$\displaystyle=$ $\displaystyle
S(h_{1})\cdot(h_{21}\cdot\omega(h_{22}))=S(h_{1})\cdot d(h_{2}),$

which proves that $\Omega^{1}(H)=H\cdot dH$. Therefore,
$(\Omega^{1}(H),\beta)$ is an $(H,\alpha)$-Hom-FODC.

For another $(H,\alpha)$-Hom-FODC $(\Gamma,\gamma)$ with differentiation
$\bar{d}:H\to\Gamma$, let us define the linear map
$\psi:\Omega^{1}(H)\to\Gamma$ by $\psi(h\cdot dg)=h\cdot\bar{d}g$, where
$g,h\in H$. It is well-defined: Suppose that $\sum_{i}h_{i}\cdot dg_{i}=0$ in
$\Omega^{1}(H)$, where $h_{i},g_{i}\in H$. Then we have

$\displaystyle\sum_{i}h_{i}\cdot dg_{i}$ $\displaystyle=$
$\displaystyle\sum_{i}h_{i}\cdot(g_{i,1}\otimes\overline{g_{i,2}})=\sum_{i}\alpha^{-1}(h_{i})g_{i,1}\otimes\alpha(\overline{g_{i,2}})$
$\displaystyle=$
$\displaystyle\sum_{i}\alpha^{-1}(h_{i})g_{i,1}\otimes\overline{\alpha(g_{i,2})}=\sum_{i}\alpha^{-1}(h_{i})g_{i,1}\otimes(\alpha(g_{i,2})-\varepsilon(\alpha(g_{i,2}))1)$
$\displaystyle=$
$\displaystyle\sum_{i}[\alpha^{-1}(h_{i})g_{i,1}\otimes\alpha(g_{i,2})-\alpha^{-1}(h_{i})g_{i,1}\varepsilon(g_{i,2})\otimes
1)]$ $\displaystyle=$
$\displaystyle\sum_{i}[\alpha^{-1}(h_{i})g_{i,1}\otimes\alpha(g_{i,2})-\alpha^{-1}(h_{i}g_{i})\otimes
1)]=0.$

So, by applying $(m\otimes id)\circ\tilde{a}^{-1}\circ(id\otimes S\otimes
id)\circ(id\otimes\Delta)$ to

$\sum_{i}h_{i}\cdot
dg_{i}=\sum_{i}[\alpha^{-1}(h_{i})g_{i,1}\otimes\alpha(g_{i,2})-\alpha^{-1}(h_{i}g_{i})\otimes
1)]=0,$

we acquire the equality $\sum_{i}(h_{i}\otimes
g_{i}-\alpha^{-1}(h_{i}g_{i})\otimes 1)=0$. Thus
$\sum_{i}h_{i}\cdot\bar{d}g_{i}=0$ in $\Gamma$ concluding that $\psi$ is well-
defined. On the other hand we prove that
$\psi\in\widetilde{\mathcal{H}}(\mathcal{M}_{k})$:

$\displaystyle\psi(\beta(h\cdot dg))$ $\displaystyle=$
$\displaystyle\psi(\alpha(h)\cdot\beta(dg))=\psi(\alpha(h)\cdot d(\alpha(g)))$
$\displaystyle=$
$\displaystyle\alpha(h)\cdot\bar{d}(\alpha(g))=\alpha(h)\cdot\gamma(\bar{d}(g))=\gamma(h\cdot\bar{d}g)=\gamma(\psi(h\cdot
dg)).$

The subobject $(ker\psi,\beta|_{ker\psi})=(\mathcal{N},\beta^{\prime})$ is an
$(H,\alpha)$-Hom-subbimodule of $(\Omega^{1}(H),\beta)$: Indeed, for
$h^{\prime}\in H$ and $h\cdot dg\in\mathcal{N}$,

$\displaystyle\psi(h^{\prime}\cdot(h\cdot dg))$ $\displaystyle=$
$\displaystyle\psi((\alpha^{-1}(h^{\prime})h)\cdot
d(\alpha(g)))=(\alpha^{-1}(h^{\prime})h)\cdot\bar{d}(\alpha(g))$
$\displaystyle=$ $\displaystyle
h^{\prime}\cdot(h\cdot\bar{d}g)=h^{\prime}\cdot\psi(h\cdot dg)=0,$

$\displaystyle\psi((h\cdot dg)\cdot h^{\prime})$ $\displaystyle=$
$\displaystyle\psi(\alpha(h)\cdot d(g\alpha^{-1}(h^{\prime}))-(hg)\cdot
dh^{\prime})$ $\displaystyle=$
$\displaystyle\alpha(h)\cdot\bar{d}(g\alpha^{-1}(h^{\prime}))-(hg)\cdot\bar{d}h^{\prime}=\alpha(h)\cdot\bar{d}(g\alpha^{-1}(h^{\prime}))-\alpha(h)\cdot(g\cdot\bar{d}(\alpha^{-1}(h^{\prime})))$
$\displaystyle=$
$\displaystyle\alpha(h)\cdot(\bar{d}g\cdot\alpha^{-1}(h^{\prime}))=(h\cdot\bar{d}g)\cdot
h^{\prime}=0.$

Hence we have the quotient object $(\Omega^{1}(H)/\mathcal{N},\bar{\beta})$ as
$(H,\alpha)$-Hom-bimodule, where the automorphism $\bar{\beta}$ is induced by
$\beta$ and define the $(H,\alpha)$-bilinear map
$\bar{\psi}:\Omega^{1}(H)/\mathcal{N}\to\Gamma,\>h\cdot dg\mapsto
h\cdot\bar{d}g$, which is surjective by definition. Since
$ker\bar{\psi}=\mathcal{N}$, $\bar{\psi}$ is 1-1, showing that $\Gamma$ is
isomorphic to the quotient $\Omega^{1}(H)/\mathcal{N}$. Therefore
$(\Omega^{1}(H),\beta)$ is the universal Hom-FODC over $(H,\alpha)$.

We define the subobject

(4.10) $\mathcal{R}_{\Gamma}=\\{h\in ker\varepsilon|\>\omega_{\Gamma}(h)=0\\}$

of $(ker\varepsilon,\alpha|_{ker\varepsilon})$ for a given left-covariant
$(H,\alpha)$-Hom-FODC $(\Gamma,\gamma)$, which is clearly a Hom-ideal of
$(H,\alpha)$. We now prove that there is a one-to-one correspondence between
left-covariant $(H,\alpha)$-Hom-FODC’s and right Hom-ideals $\mathcal{R}$.

######  Proposition 4.4.

  1. (1)

Let $(\mathcal{R},\alpha^{\prime\prime})$ be a right Hom-ideal of $(H,\alpha)$
which is a subobject of $(ker\varepsilon,\alpha^{\prime})$, where
$\alpha^{\prime\prime}=\alpha|_{\mathcal{R}}$. Then
$\mathcal{N}:=H\cdot\omega_{\Omega^{1}(H)}(\mathcal{R})$ is an
$(H,\alpha)$-Hom-subbimodule of $(\Omega^{1}(H),\beta)$. Furthermore,
$(\Gamma,\gamma):=(\Omega^{1}(H)/\mathcal{N},\bar{\beta})$ is a left-covariant
Hom-FODC over $(H,\alpha)$ such that $\mathcal{R}_{\Gamma}=\mathcal{R}$.

  2. (2)

For a given left-covariant $(H,\alpha)$-Hom-FODC $(\Gamma,\gamma)$,
$\mathcal{R}_{\Gamma}$ is a right Hom-ideal of $(H,\alpha)$ and $\Gamma$ is
isomorphic to
$\Omega^{1}(H)/H\cdot\omega_{\Omega^{1}(H)}(\mathcal{R}_{\Gamma})$.

###### Proof.

  1. (1)

For any $h\in\mathcal{R}$ and $g\in H$, we have

$\displaystyle\omega(h)\cdot g$ $\displaystyle=$
$\displaystyle(1g_{1})\cdot(\beta^{-1}(\omega(h))\lhd
g_{2})=\alpha(g_{1})\cdot(\omega(\alpha^{-1}(h))\lhd g_{2})$ $\displaystyle=$
$\displaystyle\alpha(g_{1})\cdot\omega(\alpha^{-1}(\bar{h})g_{2})=\alpha(g_{1})\cdot\omega(\alpha^{-1}(h)g_{2}),$

which is in $H\cdot\omega_{\Omega^{1}(H)}(\mathcal{R})$, and hence
$\mathcal{N}=H\cdot\omega_{\Omega^{1}(H)}(\mathcal{R})$ is an
$(H,\alpha)$-Hom-subbimodule of
$\Omega^{1}(H)=H\cdot\omega_{\Omega^{1}(H)}(H)$. So,
$(\Gamma=\Omega^{1}(H)/\mathcal{N},\bar{\beta})$ is a $(H,\alpha)$-Hom-FODC
with differentiation
$\bar{d}:H\to\Gamma,\>h\mapsto\bar{d}h=\pi(dh)=h_{1}\cdot\omega(h_{2})+\mathcal{N}$,
where $\pi:\Omega^{1}(H)\to\Omega^{1}(H)/\mathcal{N}$ is the natural
projection.

Let $\phi:\Omega^{1}(H)\to
H\otimes\Omega^{1}(H),\>h\cdot\omega(g)\mapsto\alpha(h_{1})\otimes
h_{2}\cdot\omega(\alpha^{-1}(g))$ be the Hom-coaction for the left-covariant
Hom-FODC $(\Omega^{1}(H),\beta)$. Since, for $h\cdot\omega(r)\in\mathcal{N}$
we have

$\phi(h\cdot\omega(r))=\alpha(h_{1})\otimes
h_{2}\cdot\omega(\alpha^{-1}(r))\>\in H\otimes\mathcal{N},$

that is, $\phi(\mathcal{N})\subseteq H\otimes\mathcal{N}$, $\phi$ passes to a
left Hom-action of $(H,\alpha)$ on $(\Gamma,\bar{\beta})$ as
$\bar{\phi}(h\cdot\omega(g)+\mathcal{N})=\alpha(h_{1})\otimes(h_{2}\cdot\omega(\alpha^{-1}(g))+\mathcal{N})$.
For $g,h\in H$, we get

$\displaystyle\Delta(g)(id\otimes\bar{d})(\Delta(h))$ $\displaystyle=$
$\displaystyle(g_{1}\otimes g_{2})(h_{1}\otimes\bar{d}h_{2})$ $\displaystyle=$
$\displaystyle g_{1}h_{1}\otimes
g_{2}\cdot(h_{21}\cdot\omega(h_{22})+\mathcal{N})=g_{1}h_{1}\otimes(g_{2}\cdot(h_{21}\cdot\omega(h_{22}))+\mathcal{N})$
$\displaystyle=$ $\displaystyle
g_{1}h_{1}\otimes((\alpha^{-1}(g_{2})h_{21})\cdot\omega(\alpha(h_{22}))+\mathcal{N})$
$\displaystyle=$ $\displaystyle
g_{1}\alpha(h_{11})\otimes((\alpha^{-1}(g_{2})h_{12})\cdot\omega(h_{2})+\mathcal{N})=\bar{\phi}(\alpha^{-1}(g)h_{1}\cdot\omega(\alpha(h_{2}))+\mathcal{N})$
$\displaystyle=$
$\displaystyle\bar{\phi}(g\cdot(h_{1}\cdot\omega(h_{2}))+\mathcal{N})=\bar{\phi}(g\cdot\bar{d}h),$

proving the left-covariance of $(\Gamma,\bar{\beta})$ with respect to
$(H,\alpha)$. Thus, we have the projection
$\overline{P}_{L}:\Gamma\to\>^{coH}\Gamma$ given by

$\overline{P}_{L}(h\cdot\omega(g)+\mathcal{N})=\varepsilon(h)\omega(\alpha(g))+\mathcal{N}$

for $h\cdot\omega(g)\in\Omega^{1}(H)$.

For $h\in\mathcal{R}$,

$\omega_{\Gamma}(h)=\overline{P}_{L}(\bar{d}h)=\overline{P}_{L}(h_{1}\cdot\omega(h_{2})+\mathcal{N})=\varepsilon(h_{1})\omega(\alpha(h_{2}))+\mathcal{N}=\omega(h)+\mathcal{N}=\mathcal{N}=0_{\Gamma},$

implying that $\mathcal{R}\subseteq\mathcal{R}_{\Gamma}$. On the contrary, if
$\omega_{\Gamma}(h)=0_{\Gamma}$ for some $h\in ker\varepsilon$, then
$\omega(h)\in\mathcal{N}=H\cdot\omega(\mathcal{R})$, that is,
$h\in\mathcal{R}$, i.e., $\mathcal{R}_{\Gamma}\subseteq\mathcal{R}$.
Therefore, $\mathcal{R}=\mathcal{R}_{\Gamma}$.

  2. (2)

Since $(\Gamma,\gamma)$ is a left-covariant Hom-FODC,
$\widetilde{ad}_{R}(g)(\omega(h))=\omega(\bar{h}g)$ holds for $g,h\in H$.
Hence, for $h\in\mathcal{R}_{\Gamma}$ and $g\in ker\varepsilon$, we have
$\omega_{\Gamma}(hg)=\omega_{\Gamma}(\bar{h}g)=\widetilde{ad}_{R}(g)(\omega_{\Gamma}(h))=0$
since $\omega_{\Gamma}(h)=0$. Therefore, $\mathcal{R}_{\Gamma}$ is a subobject
of $ker\varepsilon$ which is a right Hom-ideal of $(H,\alpha)$. Thus,
$\Gamma\simeq\Omega^{1}(H)/H\cdot\omega_{\Omega^{1}(H)}(\mathcal{R}_{\Gamma})$
by (1).

∎

###  4.2. Quantum Hom-Tangent Space

In the theory of Lie groups, if $A=C^{\infty}(G)$ is the algebra of smooth
functions on a Lie group $G$ and $\mathcal{R}$ is the ideal of $A$ consisting
of all functions vanishing with first derivatives at the neutral element of
$G$, then the vector space of all linear functionals on $A$ annihilating $1$
of $A$ and $\mathcal{R}$ is identified with the tangent space at the neutral
element, i.e., with the Lie algebra of $G$. In the theory of quantum groups,
this consideration gives rise to the notion of quantum tangent space
associated to a left-covariant FODC $\Gamma$ on a Hopf algebra $A$, which is
defined as the vector space

$\mathcal{T}_{\Gamma}=\\{X\in A^{\prime}|\>X(1)=0,\>X(a)=0,\forall
a\in\mathcal{R}_{\Gamma}\\},$

where $\mathcal{R}_{\Gamma}=\\{a\in ker\varepsilon_{A}|\>P_{L}(da)=0\\}$. In
what follows, we study the Hom-version of the quantum tangent space.

We recall that the dual monoidal Hom-algebra $(H^{\prime},\bar{\alpha})$ of
$(H,\alpha)$ consists of functionals $X:H\to k$ and is equipped with the
convolution product $(XY)(h)=X(h_{1})Y(h_{2})$, for $X,Y\in H^{\prime}$ and
$h\in H$, as Hom-multiplication and with the Hom-unit $\varepsilon:H\to k,$
where automorphism $\bar{\alpha}:H^{\prime}\to H^{\prime}$ is given by
$\bar{\alpha}(X)=X\circ\alpha^{-1}$. The morphism

$H^{\prime}\otimes H\to H,X\otimes h\mapsto X\bullet
h:=\alpha^{2}(h_{1})X(\alpha(h_{2})),$

in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$, makes $(H,\alpha)$ a left
$(H^{\prime},\bar{\alpha})$-Hom-module.

######  Definition 4.5.

Let $(\Gamma,\gamma)$ be a left-covariant $(H,\alpha)$-Hom-FODC. Then the
subobject

(4.11) $\mathcal{T}_{\Gamma}=\\{X\in H^{\prime}|\>X(1)=0,\>X(h)=0,\forall
h\in\mathcal{R}_{\Gamma}\\}$

of $(H^{\prime},\bar{\alpha})$, in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$,
is said to be the quantum Hom-tangent space to $(\Gamma,\gamma)$.

######  Proposition 4.6.

Let $(\Gamma,\gamma)$ be a left-covariant $(H,\alpha)$-Hom-FODC and
$(\mathcal{T}_{\Gamma},\bar{\alpha}^{\prime})$ be the quantum Hom-tangent
space to it, where
$\bar{\alpha}^{\prime}=\bar{\alpha}|_{\mathcal{T}_{\Gamma}}$. Then, there is a
unique bilinear form $<\cdot,\cdot>:\mathcal{T}_{\Gamma}\times\Gamma\to k$ in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$ such that

(4.12) $<X,h\cdot dg>=\varepsilon(h)X(g),\>\forall g,h\in
H,\>X\in\mathcal{T}_{\Gamma}.$

With respect to this bilinear form,
$(\mathcal{T}_{\Gamma},\bar{\alpha}^{\prime})$ and
$(\>^{coH}\Gamma,\gamma^{\prime})=(\omega(H),\gamma^{\prime})$ form a
nondegenerate dual pairing, where $\gamma^{\prime}=\gamma|_{{}^{coH}\Gamma}$.
Moreover, we have

(4.13) $<X,\omega(h)>=X(\alpha^{-1}(h)),\forall h\in
H,X\in\mathcal{T}_{\Gamma}.$

###### Proof.

We define
$<X,\varrho>:=X(\sum_{i}\varepsilon(h_{i})g_{i})=\sum_{i}\varepsilon(h_{i})X(g_{i})$
for $X\in H^{\prime}$ and $\varrho=\sum_{i}h_{i}\cdot dg_{i}\in\Gamma$.
Suppose that $\varrho=\sum_{i}h_{i}\cdot dg_{i}=0$. Then

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
P_{L}(\gamma^{-1}(\rho))=\sum_{i}P_{L}(\alpha^{-1}(h_{i})d(\alpha^{-1}(g_{i})))$
$\displaystyle=$
$\displaystyle\sum_{i}\varepsilon(\alpha^{-1}(h_{i}))\gamma(P_{L}(d(\alpha^{-1}(g_{i}))))$
$\displaystyle=$
$\displaystyle\sum_{i}\varepsilon(h_{i})\gamma(\omega(\alpha^{-1}(g_{i})))$
$\displaystyle=$
$\displaystyle\omega\left(\sum_{i}\varepsilon(h_{i})g_{i}\right),$

hence $\omega(\sum_{i}\varepsilon(h_{i})\overline{g_{i}})=0$, which implies
that $\sum_{i}\varepsilon(h_{i})\overline{g_{i}}\in\mathcal{R}_{\Gamma}$.
Thus, by the definition of $\mathcal{T}_{\Gamma}$ we get

$\displaystyle<X,\rho>$ $\displaystyle=$ $\displaystyle
X\left(\sum_{i}\varepsilon(h_{i})g_{i}\right)=X\left(\sum_{i}(\varepsilon(h_{i})\overline{g_{i}}+\varepsilon(h_{i})\varepsilon(g_{i})1)\right)$
$\displaystyle=$ $\displaystyle
X\left(\sum_{i}\varepsilon(h_{i})\overline{g_{i}}\right)+\sum_{i}\varepsilon(h_{i})\varepsilon(g_{i})X(1)=0,$

which proves that the bilinear form $<\cdot,\cdot>$ is well-defined.
Uniqueness comes immediately from the fact that $\Gamma=H\cdot dH$. Since

$\displaystyle<\bar{\alpha}(X),\gamma(\varrho)>$ $\displaystyle=$
$\displaystyle(X\circ\alpha^{-1})\left(\sum_{i}\varepsilon(\alpha(h_{i}))\alpha(g_{i})\right)=\sum_{i}\varepsilon(\alpha(h_{i}))(X\circ\alpha^{-1})(\alpha(g_{i}))$
$\displaystyle=$ $\displaystyle
X\left(\sum_{i}\varepsilon(h_{i})g_{i}\right)=<X,\varrho>,$

the bilinear form $<\cdot,\cdot>$ is in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$. For any $h\in H$,
$<X,\omega(h)>=<X,S(h_{1})\cdot
dh_{2}>=\varepsilon(h_{1})X(h_{2})=X(\varepsilon(h_{1})h_{2})=X(\alpha^{-1}(h))$,
which is the formula (4.13). For any $h\in ker\varepsilon$, if
$<X,\omega(h)>=X(\alpha^{-1}(h))=0,\>\forall X\in\mathcal{T}_{\Gamma}$, then
$\alpha^{-1}(h)\in\mathcal{R}_{\Gamma}$: Suppose that the element
$0\neq\alpha^{-1}(h)\in ker\varepsilon$ is not contained in
$\mathcal{R}_{\Gamma}$. Then we can extend $\alpha^{-1}(h)$ to a basis of
$ker\varepsilon$ and find a functional $X\in\mathcal{T}_{\Gamma}$ such that
$X(\alpha^{-1}(h))\neq 0$, which contradicts with the hypothesis of the
statement. So we have $h\in\mathcal{R}_{\Gamma}$ since
$\omega\circ\alpha^{-1}=\gamma^{-1}\circ\omega$. On the other hand
$<X,\omega(h)>=X(\alpha^{-1}(h))=\bar{\alpha}(X)(h)=0$ for all
$\omega(h)\in\omega(H)$ implies $\bar{\alpha}(X)=0$, that is, $X=0$. Hence,
$(\mathcal{T}_{\Gamma},\bar{\alpha}^{\prime})$ and
$(\>^{coH}\Gamma,\gamma^{\prime})=(\omega(H),\gamma^{\prime})$ form a
nondegenerate dual pairing with respect to $<\cdot,\cdot>$. ∎

Let $\\{X_{i}\\}_{i\in I}$ be a linear basis of $\mathcal{T}_{\Gamma}$ and
$\\{\omega_{i}\\}_{i\in I}$ be the dual basis of ${}^{coH}\Gamma$, that is,
$<X_{i},\omega_{j}>=\delta_{ij}$ for $i,j\in I$. Also, from Theorem (4.17) in
[11], recall the family of functionals $\\{f^{i}_{j}\\}_{i,j\in I}$ in the
definition of the Hom-action ${}^{coH}\Gamma\otimes
H\to\>^{coH}\Gamma,\omega_{i}\otimes h\mapsto\omega_{i}\lhd
h=f^{i}_{j}(h)\omega_{j}$, where all but finitely many $f^{i}_{j}(h)$ vanish
and Einstein summation convention is used. These functionals satisfy, for all
$h,g\in H$ and $i,j\in I$,

$f^{i}_{j}(hg)=(\bar{\gamma}^{i}_{k}f^{k}_{l})(h)f^{l}_{j}(\alpha(g)),\>\>f^{i}_{j}(1)=\gamma^{i}_{j},$

where $\gamma^{\prime}(\omega_{i})=\gamma^{i}_{j}\omega_{j}$ and
$\gamma^{\prime-1}(\omega_{i})=\bar{\gamma}^{i}_{j}\omega_{j}$ such that
$\gamma^{i}_{j}\bar{\gamma}^{j}_{k}=\delta_{ik}=\bar{\gamma}^{i}_{j}\gamma^{j}_{k}$.

######  Proposition 4.7.

For $h,g\in H$, we have

(4.14) $dh=(X_{i}\bullet\alpha^{-2}(h))\cdot\omega_{i},$ (4.15)
$X_{i}(hg)=\varepsilon(h)(\gamma_{i}^{j}X_{j})(g)+X_{k}(h)(\gamma_{i}^{l}\bar{f}_{l}^{k})(g),$

where $\bar{f}^{k}_{l}=\bar{\gamma}^{k}_{p}f^{p}_{l}$.

###### Proof.

By the formula 4.13, we have $<X_{i},\omega(h)>=X_{i}(\alpha^{-1}(h))$
implying $\omega(h)=X_{i}(\alpha^{-1}(h))\omega_{i}$. Thus,
$dh=h_{1}\cdot\omega(h_{2})=h_{1}\cdot(X_{i}(\alpha^{-1}(h_{2})\omega_{i}))=(X_{i}\bullet\alpha^{-2}(h))\cdot\omega_{i}$
which is the formula 4.14. By using this formula and the Leibniz rule, we
obtain

$\displaystyle(X_{l}\bullet\alpha^{-2}(hg))\cdot\omega_{l}$ $\displaystyle=$
$\displaystyle d(hg)=dh\cdot g+h\cdot dg$ $\displaystyle=$
$\displaystyle((X_{j}\bullet\alpha^{-2}(h))\cdot\omega_{j})\cdot
g+h\cdot((X_{i}\bullet\alpha^{-2}(g))\cdot\omega_{i})$ $\displaystyle=$
$\displaystyle\alpha(X_{j}\bullet\alpha^{-2}(h))\cdot(\omega_{j}\cdot\alpha^{-1}(g))+(\alpha^{-1}(h)(X_{i}\bullet\alpha^{-2}(g)))\cdot\gamma^{\prime}(\omega_{i})$
$\displaystyle=$
$\displaystyle\alpha(X_{j}\bullet\alpha^{-2}(h))\cdot((\bar{f}^{j}_{k}\bullet\alpha^{-2}(g))\cdot\omega_{k})+(\alpha^{-1}(h)(X_{i}\bullet\alpha^{-2}(g)))\cdot(\gamma^{i}_{k}\omega_{k})$
$\displaystyle=$
$\displaystyle((X_{j}\bullet\alpha^{-2}(h))(\bar{f}^{j}_{k}\bullet\alpha^{-2}(g)))\cdot(\gamma^{k}_{l}\omega_{l})+(\alpha^{-1}(h)(X_{i}\bullet\alpha^{-2}(g)))\cdot(\gamma^{i}_{l}\omega_{l})$
$\displaystyle=$
$\displaystyle[(X_{j}\bullet\alpha^{-2}(h))((\gamma^{k}_{l}\bar{f}^{j}_{k})\bullet\alpha^{-2}(g))+\alpha^{-1}(h)((\gamma^{i}_{l}X_{i})\bullet\alpha^{-2}(g))]\cdot\omega_{l},$

hence, by replacing $\alpha^{-2}(h)$ and $\alpha^{-2}(g)$ by $h$ and $g$,
respectively, we get

$X_{l}\bullet(hg)=\alpha(h)((\gamma^{i}_{l}X_{i})\bullet g)+(X_{j}\bullet
h)((\gamma^{k}_{l}\bar{f}^{j}_{k})\bullet g),l\in I.$

By applying $\varepsilon$ to the both sides of this equation we acquire

$X_{l}(hg)=\varepsilon(h)(\gamma_{i}^{l}X_{i})(g)+X_{j}(h)(\gamma_{k}^{l}\bar{f}_{j}^{k})(g),$

since, for any $h\in H$ and $f\in H^{\prime}$, the equality
$\varepsilon(f\bullet
h)=\varepsilon(\alpha^{2}(h_{1}))f(\alpha(h_{2}))=\varepsilon(h_{1})f(\alpha(h_{2}))=f(\alpha(\varepsilon(h_{1})h_{2}))=f(h)$
holds. ∎

Let $(A,\alpha)$ be a monoidal Hom-algebra. Then we consider
$A^{\prime}\otimes A^{\prime}$, where $A^{\prime}=Hom(A,k)$, as a linear
subspace of $(A\otimes A)^{\prime}$ by identifying $f\otimes g\in
A^{\prime}\otimes A^{\prime}$ with the linear functional on $A\otimes A$
specified by $(f\otimes g)(a\otimes a^{\prime}):=f(a)g(a^{\prime})$ for
$a,a^{\prime}\in A$. For $f\in H^{\prime}$, let us define
$\Delta(f)\in(A\otimes A)^{\prime}$ by $\Delta(f)(a\otimes b):=f(ab)$ for
$a,b\in A$. We now denote, by $A^{\circ}$, the set of all functionals $f\in
A^{\prime}$ such that $\Delta(f)\in A^{\prime}\otimes A^{\prime}$, i.e., it is
written as a finite sum

$\Delta(f)=\sum_{p=1}^{P}f_{p}\otimes g_{p}$

for some functionals $f_{p},g_{p}\in A^{\prime},\>p=1,...,P$, where $P$ is a
natural number so that we have $f(ab)=\sum_{p}f_{p}(a)g_{p}(b)$. Then
$(A^{\circ},\alpha^{\circ})$ is a monoidal Hom-coalgebra with Hom-
comultiplication given above and the Hom-counit is defined by
$\varepsilon(f)=f(1_{A})$, where $\alpha^{\circ}(f)=f\circ\alpha^{-1}$ for any
$f\in A^{\circ}$: Let $f\in A^{\circ}$ and $\Delta(f)=\sum_{p}f_{p}\otimes
g_{p}$ such that the functionals $\\{f_{p}\\}_{p=1}^{P}$ are chosen to be
linearly independent. So, one can find $a_{q}\in A$ such that
$f_{p}(a_{q})=\delta_{pq}$. Thus we get

$\displaystyle g_{q}(ab)$ $\displaystyle=$
$\displaystyle\sum_{p}\delta_{qp}g_{p}(ab)=\sum_{p}f_{p}(a_{q})g_{p}(ab)=f(a_{q}(ab))$
$\displaystyle=$ $\displaystyle
f((\alpha^{-1}(a_{q})a)\alpha(b))=\sum_{p}f_{p}(\alpha^{-1}(a_{q})a)g_{p}(\alpha(b)),$

showing that $g_{q}\in A^{\circ}$, and analogously $f_{q}\in A^{\circ}$, and
hence $\Delta(f)\in A^{\circ}\otimes A^{\circ}$. Let $f\in A^{\circ}$ and
$a,b,c\in A$. Then we have the Hom-coassociativity of $\Delta$:

$(\bar{\alpha}^{-1}\otimes\Delta)(\Delta(f))(a\otimes b\otimes
c)=f(\alpha(a)(bc))=f((ab)\alpha(c))=(\Delta\otimes\bar{\alpha}^{-1})(\Delta(f))(a\otimes
b\otimes c).$

On the other hand,

$(id\otimes\varepsilon)(\Delta(f))(h)=\left(\sum_{p}\bar{\alpha}(f_{p})g_{p}(1_{A})\right)(h)=\sum_{p}f_{p}(\alpha^{-1}(h))g_{p}(1_{A})=f(h)$

shows that Hom-counity is satisfied.

Suppose that $(A,\alpha)$ is a monoidal Hom-bialgebra, then the monoidal Hom-
coalgebra $(A^{\circ},\alpha^{\circ})$ endowed with the convolution product,
as in the argument before Lemma (4.16) in [11], is as well a monoidal Hom-
bialgebra with the Hom-unit given by the Hom-counit $\varepsilon$ of the
monoidal Hom-coalgebra $(A,\alpha)$: One can easily check the compatibility
condition between Hom-comultiplication and Hom-multiplication of
$(A^{\circ},\alpha^{\circ})$ which follows from that of $(A,\alpha)$. So, it
suffices to verify that for any $f,g\in A^{\circ}$, $fg$ is also in
$A^{\circ}$: If we put $\Delta(f)=\sum_{p}f_{p}\otimes g_{p}$ and
$\Delta(g)=\sum_{q}h_{q}\otimes k_{q}$, then we get

$(fg)(ab)=\Delta(fg)(a\otimes
b)=\sum_{p,q}f_{p}h_{q}(a)g_{p}k_{q}(b)=\left(\sum_{p,q}f_{p}h_{q}\otimes
g_{p}k_{q}\right)(a\otimes b),$

so that $fg\in A^{\circ}$.

If $(A,\alpha)$ is a monoidal Hom-Hopf algebra, then so is
$(A^{\circ},\alpha^{\circ})$ with antipode defined by $S(f)(a)=f(S(a))$ for
$f\in A^{\circ}$ and $a\in A$: Set $\Delta(f)=\sum_{p}f_{p}\otimes g_{p}$, and
then we obtain

$\Delta(S(f))(a\otimes
b)=S(f)(ab)=f(S(ab))=\sum_{p}S(f_{p})(b)S(g_{p})(a)=\left(\sum_{p}S(g_{p})\otimes
S(f_{p})\right)(a\otimes b),$

implying $S(f)\in A^{\circ}$. Lastly, for $a\in A$, we have

$((m(S\otimes
id)\Delta)(f))(a)=\sum_{p}(S(f_{p})g_{p})(a)=\varepsilon(a)f(1)=1_{A^{\circ}}(a)\varepsilon_{A^{\circ}}(f)=((\eta\circ\varepsilon)(f))(a),$

similarly we get $((m(id\otimes
S)\Delta)(f))(a)=((\eta\circ\varepsilon)(f))(a)$.

We then call the monoidal Hom-coalgebra (respectively, Hom-bialgebra, Hom-Hopf
algebra) $A^{\circ}$ above the dual monoidal Hom-coalgebra (respectively, Hom-
bialgebra, Hom-Hopf algebra). Suppose now that the vector space
$\mathcal{T}_{\Gamma}$ is finite dimensional. Then we assert from Theorem
(4.17) in [11] and (4.15) that the functionals $f^{i}_{j}$ and $X_{l}$ are in
the dual monoidal Hom-Hopf algebra $H^{\circ}$ and we have the following
equations, where there is summation over repeating indices,

(4.16) $\Delta(f^{i}_{j})=\bar{f}^{i}_{l}\otimes f^{l}_{j}\circ\alpha,$ (4.17)
$\Delta(X_{l})=X_{j}\otimes\gamma^{k}_{l}\bar{f}^{j}_{k}+\varepsilon\otimes\gamma^{l}_{i}X_{i}$

in $H^{\circ}$.

##  5\. Bicovariant FODC over Monoidal Hom-Hopf Algebras

###  5.1. Right-Covariant Hom-FODC

######  Definition 5.1.

Let $(H,\beta)$ be a monoidal Hom-bialgebra. A FODC $(\Gamma,\gamma)$ over a
right Hom-quantum space $(A,\alpha)$ with right Hom-coaction $\varphi:A\to
A\otimes H,\>a\mapsto a_{[0]}\otimes a_{[1]}$ is called right-covariant with
respect to $(H,\beta)$ if there exists a right Hom-coaction
$\phi:\Gamma\to\Gamma\otimes H,\>\omega\mapsto\omega_{[0]}\otimes\omega_{[1]}$
of $(H,\beta)$ on $(\Gamma,\gamma)$ such that

  1. (1)

$\phi(\alpha(a)\cdot(\omega\cdot
b))=\varphi(\alpha(a))(\phi(\omega)\varphi(b))[=(\varphi(a)\phi(\omega))\varphi(\alpha(b))=\phi(a\cdot\omega)\cdot
b]$, $\forall a,b\in A$, $\omega\in\Gamma$,

  2. (2)

$\phi(da)=(d\otimes id)(\varphi(a))$, $\forall a\in A$

Let $(H,\alpha)$ be a monoidal Hom-Hopf algebra with an invertible antipode
$S$. Since $(H,\alpha)$ is a right Hom-quantum space for itself with respect
to the Hom-comultiplication $\Delta:H\to H\otimes H,h\mapsto h_{1}\otimes
h_{2}$, the above definition induces the following definition.

######  Definition 5.2.

A $(H,\alpha)$-Hom-FODC $(\Gamma,\gamma)$ is said to be right-covariant if
$(\Gamma,\gamma)$ is a right-covariant FODC over the right Hom-quantum space
$(H,\alpha)$ with right Hom-coaction $\varphi=\Delta$ in the above definition,
or in an equivalent way if there is a morphism $\phi:\Gamma\to\Gamma\otimes H$
in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$ such that, for $h,g\in H$,

(5.18) $\phi(h\cdot dg)=\Delta(h)(d\otimes id)(\Delta(g)).$

If we modify the Proposition 2.5 to the right-covariant case, we conclude that
the right-covariant $(H,\alpha)$-Hom-FODC $(\Gamma,\gamma)$ is a right-
covariant $(H,\alpha)$-Hom-bimodule. Thus, by using the unique projection
$P_{R}:(\Gamma,\gamma)\to(\Gamma^{coH},\gamma|_{\Gamma^{coH}}),\>P_{R}(\rho)=\omega_{[0]}\cdot
S(\omega_{[1]})$ we define the linear mapping

$\eta_{\Gamma}:H\to\Gamma^{coH},\>\eta(h):=P_{R}(dh),$

for any $h\in H$, in $\widetilde{\mathcal{H}}(\mathcal{M}_{k})$, for which
$\eta(H)\Gamma^{coH}$. Since $\phi(dh)=dh_{1}\otimes h_{2}$, we have, for
$h\in H$

$\eta(h)=dh_{1}\cdot S(h_{2})\>and\>dh=\eta(h_{1})\cdot h_{2}.$

###  5.2. Bicovariant Hom-FODC

######  Definition 5.3.

A $(H,\alpha)$-Hom-FODC $(\Gamma,\gamma)$ is said to be bicovariant if it is
both left-covariant and right-covariant FODC.

######  Remark 5.4.

By the Remark 4.2 and the Definition 5.2, a $(H,\alpha)$-Hom-FODC
$(\Gamma,\gamma)$ is bicovariant if and only if there exist morphisms
$\phi_{L}:\Gamma\to H\otimes\Gamma$ and $\phi_{R}:\Gamma\to\Gamma\otimes H$ in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$, satisfying the equations 4.4 and
5.18, respectively. So, if $(\Gamma,\gamma)$ is a bicovariant
$(H,\alpha)$-Hom-FODC with Hom-coactions $\phi_{L}$ and $\phi_{R}$ satisfying
4.4 and 5.18 we get, for $h,g\in H$,

$(id\otimes\phi_{R})(\phi_{L}(h\cdot
dg))=(id\otimes\phi_{R})(h_{1}g_{1}\otimes h_{2}\cdot
dg_{2})=h_{1}g_{1}\otimes(h_{21}\cdot dg_{21}\otimes h_{22}g_{22}),$
$\displaystyle(\tilde{a}\circ(\phi_{L}\otimes id))(\phi_{R}(h\cdot dg))$
$\displaystyle=$ $\displaystyle(\tilde{a}\circ(\phi_{L}\otimes id))(h_{1}\cdot
dg_{1}\otimes h_{2}g_{2})$ $\displaystyle=$
$\displaystyle\alpha(h_{11}g_{11})\otimes(h_{12}\cdot
dg_{12}\otimes\alpha^{-1}(h_{2}g_{2}))$ $\displaystyle=$ $\displaystyle
h_{1}g_{1}\otimes(h_{21}\cdot dg_{21}\otimes h_{22}g_{22}).$

Thus, $(\Gamma,\gamma)$ is a bicovariant $(H,\alpha)$-Hom-bimodule and the
whole structure theory of bicovariant Hom-bimodules can be applied to it.

######  Lemma 5.5.

Let $(H,\alpha)$ be a monoidal Hom-Hopf algebra. Then

  1. (1)

the linear mapping $\widetilde{Ad}_{R}:H\to H\otimes H$ given by

$\widetilde{Ad}_{R}(h)=\alpha(h_{12})\otimes
S(h_{11})\alpha^{-1}(h_{2})=\alpha(h_{21})\otimes S(\alpha^{-1}(h_{1}))h_{22}$

is a right Hom-coaction of $(H,\alpha)$ on itself.

  2. (2)

The linear mapping $\widetilde{Ad}_{L}:H\to H\otimes H$ given by

$\widetilde{Ad}_{L}(h)=\alpha(h_{11})S(\alpha^{-1}(h_{2}))\otimes\alpha(h_{12})=\alpha^{-1}(h_{1})S(h_{22})\otimes\alpha(h_{21})$

is a left Hom-coaction of $(H,\alpha)$ on itself. $\widetilde{Ad}_{R}$ and
$\widetilde{Ad}_{L}$ are called adjoint right Hom-coaction and adjoint left
Hom-coaction of $(H,\alpha)$ on itself, respectively

###### Proof.

  1. (1)

If we write $\widetilde{Ad}_{R}(h)=h_{[0]}\otimes h_{[1]}$ for $h\in H$, then
the Hom-coassociativity follows from

$\displaystyle\alpha^{-1}(h_{[0]})\otimes\Delta(h_{[1]})$ $\displaystyle=$
$\displaystyle\alpha^{-1}(\alpha(h_{12}))\otimes\Delta(S(h_{11})\alpha^{-1}(h_{2}))$
$\displaystyle=$ $\displaystyle h_{12}\otimes
S(h_{112})\alpha^{-1}(h_{21})\otimes S(h_{111})\alpha^{-1}(h_{22})$
$\displaystyle=$ $\displaystyle\alpha^{2}(h_{1212})\otimes
S(\alpha(h_{1211}))h_{122}\otimes S(\alpha^{-1}(h_{11}))\alpha^{-2}(h_{22})$
$\displaystyle=$ $\displaystyle\alpha^{2}(h_{1212})\otimes
S(\alpha(h_{1211}))h_{122}\otimes\alpha^{-1}(S(h_{11})\alpha^{-1}(h_{22}))$
$\displaystyle=$ $\displaystyle h_{[0][0]}\otimes
h_{[0][1]}\otimes\alpha^{-1}(h_{[1]}),$

where in the third step we have used

$h_{11}\otimes\alpha(h_{1211})\otimes
h_{1212}\otimes\alpha^{-1}(h_{122})\otimes h_{2}=\alpha(h_{111})\otimes
h_{112})\otimes\alpha^{-2}(h_{12})\otimes\alpha^{-2}(h_{21})\otimes\alpha(h_{22}),$

which results from

$\displaystyle((id_{H}\otimes\tilde{a}_{H,H,H})\otimes
id_{H})\circ((id_{H}\otimes(\Delta\otimes id_{H}))\otimes id_{H})$
$\displaystyle\circ((id_{H}\otimes\Delta)\otimes id_{H})\circ(\Delta\otimes
id_{H})\circ\Delta$ $\displaystyle=$ $\displaystyle(\tilde{a}_{H,H,H\otimes
H}\otimes id_{H})\circ((\Delta\otimes id_{H\otimes H})\otimes
id_{H})\circ(\tilde{a}_{H,H,H}\otimes id_{H})$
$\displaystyle\circ\tilde{a}^{-1}_{H\otimes H,H,H}\circ(id_{H\otimes
H}\otimes\Delta)\circ(\Delta\otimes id_{H})\circ\Delta.$

Hom-unity condition: For any $h\in H$,

$\displaystyle h_{[0]}\varepsilon(h_{[1]})$ $\displaystyle=$
$\displaystyle\alpha(h_{12})\varepsilon(S(h_{11})\alpha^{-1}(h_{2}))=\alpha(h_{12})\varepsilon(h_{11})\varepsilon(h_{2})$
$\displaystyle=$
$\displaystyle\alpha(\varepsilon(h_{11})h_{12})\varepsilon(h_{2})=h_{1}\varepsilon(h_{2})=\alpha^{-1}(h),$

and one can also easily show that
$\widetilde{Ad}_{R}\circ\alpha=(\alpha\otimes\alpha)\circ\widetilde{Ad}_{R}$.
Thus $\widetilde{Ad}_{R}$ is a right Hom-action of $(H,\alpha)$ onto itself.

  2. (2)

In a similar manner, it can be proven that $\widetilde{Ad}_{L}$ is a left Hom-
action of $(H,\alpha)$ onto itself.

∎

With the next lemma we describe the right Hom-coaction $\phi_{R}$ on a left-
invariant form $\omega_{\Gamma}(h)$ and the left Hom-coaction $\phi_{L}$ on a
right-invariant form $\eta_{\Gamma}(h)$ by means of $\widetilde{Ad}_{R}$ and
$\widetilde{Ad}_{L}$, respectively.

######  Lemma 5.6.

For $h\in H$, we have the formulas

  1. (1)

$\phi_{R}(\omega(h))=(\omega\otimes id)(\widetilde{Ad}_{R}(h)),$

  2. (2)

$\phi_{L}(\eta(h))=(id\otimes\eta)(\widetilde{Ad}_{L}(h)).$

###### Proof.

  1. (1)

For $h\in H$,

$\displaystyle\phi_{R}(\omega(h))$ $\displaystyle=$
$\displaystyle\Delta(S(h_{1}))(d\otimes id)(\Delta(h_{2}))$ $\displaystyle=$
$\displaystyle(S(h_{12})\otimes S(h_{11}))(dh_{21}\otimes
h_{22})=S(h_{12})\cdot dh_{21}\otimes S(h_{11})h_{22}$ $\displaystyle=$
$\displaystyle S(\alpha(h_{121}))\cdot d(\alpha(h_{122}))\otimes
S(h_{11})\alpha^{-1}(h_{2})$ $\displaystyle=$
$\displaystyle\omega(\alpha(h_{12}))\otimes
S(h_{11})\alpha^{-1}(h_{2})=(\omega\otimes id)(\alpha(h_{12})\otimes
S(h_{11})\alpha^{-1}(h_{2}))$ $\displaystyle=$ $\displaystyle(\omega\otimes
id)(\widetilde{Ad}_{R}(h)).$

  2. (2)

Similarly, one can show that the equality
$\phi_{L}(\eta(h))=(id\otimes\eta)(\widetilde{Ad}_{L}(h))$ holds.

∎

######  Proposition 5.7.

Suppose that $(\Gamma,\gamma)$ is a left-covariant $(H,\alpha)$-Hom-FODC with
associated right Hom-ideal $\mathcal{R}_{\Gamma}$. Then $(\Gamma,\gamma)$ is a
bicovariant $(H,\alpha)$-Hom-FODC if and only if
$\widetilde{Ad}_{R}(\mathcal{R})\subseteq\mathcal{R}\otimes H$, that is,
$\mathcal{R}$ is $\widetilde{Ad}_{R}$-invariant.

###### Proof.

If $(\Gamma,\gamma)$ is a bicovariant Hom-FODC, then the equation obtained in
Lemma (5.6) holds. It implies that
$\widetilde{Ad}_{R}(\mathcal{R})\subseteq\mathcal{R}\otimes H$ since
$\mathcal{R}=\left(h\in ker\varepsilon|\>\omega(h)=0\right)$. On the contrary,
suppose that $\widetilde{Ad}_{R}(\mathcal{R})\subseteq\mathcal{R}\otimes H$.
We know that the universal Hom-FODC $\Omega^{1}(H)$ is bicovariant. So, by
applying Lemma (5.6) to the bicovariant Hom-FODC $\Omega^{1}(H)$ and using the
$Ad_{R}$-invariance of $\mathcal{R}$, we conclude that theright Hom-action of
$\Omega^{1}(H)$ passes to the quotient $\Omega^{1}(H)/\mathcal{N}$, where
$\mathcal{N}:=H\omega_{\Omega^{1}(H)}(\mathcal{(}R))$, which is right-
covariant. Hence, from Proposition (4.4), $(\Gamma,\gamma)$ is right-covariant
as well. ∎

###  5.3. Quantum Monoidal Hom-Lie Algebra

Let $(\Gamma,\gamma)$ be a bicovariant $(H,\alpha)$-Hom-FODC with associated
right Hom-ideal $\mathcal{R}$ and finite dimensional quantum Hom-tangent space
$(\mathcal{T},\tau)$, where $\tau=\bar{\alpha}|_{\mathcal{T}}$.

We define a linear mapping $[-,-]:\mathcal{T}\otimes\mathcal{T}\to\mathcal{T}$
by setting, for $X,Y\in\mathcal{T}$,

(5.19) $[X,Y](h)=(X\otimes Y)(\widetilde{Ad}_{R}(h)),\>\forall h\in H.$

$[X,Y]\in\mathcal{T}$: Indeed, since
$\widetilde{Ad}_{R}(\mathcal{R})\subseteq\mathcal{R}\otimes H$ by the previous
proposition and any element of $\mathcal{T}$ annihilates $\mathcal{R}$ by the
definition of quantum Hom-tangent space, $(X\otimes
Y)(\widetilde{Ad}_{R}(h))=0$ for all $h\in\mathcal{R}$, i.e.,
$[X,Y](h)=0,\>\forall h\in\mathcal{R}$. We also obtain $[X,Y](1)=0$ since
$X(1)=0=Y(1)$. Thus $[X,Y]\in\mathcal{T}$. Besides, we have

$\displaystyle[\tau(X),\tau(Y)](h)$ $\displaystyle=$
$\displaystyle(X\circ\alpha^{-1}\otimes
Y\circ\alpha^{-1})(\widetilde{Ad}_{R}(h))$ $\displaystyle=$ $\displaystyle
X(h_{12})Y(S(\alpha^{-1}(h_{11}))\alpha^{-2}(h_{2}))$ $\displaystyle=$
$\displaystyle(X\otimes
Y)(\widetilde{Ad}_{R}(\alpha^{-1}(h)))=[X,Y](\alpha^{-1}(h))$ $\displaystyle=$
$\displaystyle\tau([X,Y])(h),$

for any $h\in H$, which means
$[-,-]:\mathcal{T}\otimes\mathcal{T}\to\mathcal{T}$ is a morphism in
$\widetilde{\mathcal{H}}(\mathcal{M}_{k})$.

We now fix some notation. Suppose that
$<\cdot,\cdot>:\mathcal{T}\times\>^{coH}\Gamma\to k$ is the bilinear form in
the Proposition 4.6. There exists a unique bilinear form
$<\cdot,\cdot>_{2}\>:(\mathcal{T}\otimes\mathcal{T})\times\>^{coH}(\Gamma\otimes_{H}\Gamma)\to
k$ defined by

(5.20) $<X\otimes Y,u\otimes v>_{2}\>=<X,u><Y,v>$

for $X,Y\in\mathcal{T}$ and $u,v\in\>^{coH}\Gamma$, which is nondegenerate as
the bilinear form $<\cdot,\cdot>$ is. If we put
$B:\Gamma\otimes_{H}\Gamma\to\Gamma\otimes_{H}\Gamma$ for the Woronowicz’
braiding, then, for $h,g\in H$, we compute

(5.21) $\displaystyle B(\omega(h)\otimes_{H}\omega(g))$ $\displaystyle=$
$\displaystyle\gamma(\omega(\alpha(g_{12})))\otimes_{H}\gamma^{-1}(\omega(h))\lhd(S(g_{11})\alpha^{-1}(g_{2}))$
$\displaystyle=$
$\displaystyle\omega(\alpha^{2}(g_{12}))\otimes_{H}\gamma^{-1}(\omega(h)\lhd(S(\alpha(g_{11}))g_{2}))$
$\displaystyle=$
$\displaystyle\omega(\alpha^{2}(g_{12}))\otimes_{H}\gamma^{-1}(\omega(\bar{h}(S(\alpha(g_{11}))g_{2})))$
$\displaystyle=$
$\displaystyle\omega(\alpha^{2}(g_{12}))\otimes_{H}\omega(\overline{\alpha^{-1}(h)}(S(g_{11})\alpha^{-1}(g_{2}))).$

With respect to the nondegenerate bilinear form $<\cdot,\cdot>_{2}$, we define
the transpose $B^{t}$ of $B$ as a linear map acting on
$\mathcal{T}\otimes\mathcal{T}$ such that

$<B^{t}(X\otimes Y),u\otimes v>_{2}\>=<X\otimes Y,B(u\otimes v)>_{2}.$

We now recall that the dual monoidal Hom-Hopf algebra
$(H^{\circ},\alpha^{\circ})$ of $(H,\alpha)$ consists of functionals $f\in
H^{\prime}$ for which $\Delta_{H^{\circ}}(f)=f_{1}\otimes f_{2}\in
H^{\prime}\otimes H^{\prime}$ and the Hom-counit is given by
$\varepsilon_{H^{\circ}}(f)=f(1_{H})$. Since, also $\Delta(f)(h\otimes
g):=f(hg)$ for $\Delta(f)\in(H\otimes H)^{\prime}$ and $h,g\in H$, we have
$f(hg)=f_{1}(h)f_{2}(g)$. $\alpha^{\circ}$ is given by
$\alpha^{\circ}(f)=f\circ\alpha^{-1}$ for $f\in H^{\circ}$. Hom-multiplication
$m_{H^{\circ}}$ is the convolution, i.e., $m_{H^{\circ}}(f\otimes
f^{\prime})(h)=(ff^{\prime})(h)=f(h_{1})f^{\prime}(h_{2})$ for
$f,f^{\prime}\in H^{\prime}$, $h\in H$ and the Hom-unit is $\varepsilon_{H}$.
The antipode is given by $S(f)(h)=f(S(h))$ for $f\in H^{\circ}$ and $h\in H$.
Since we assumed that $\mathcal{T}_{\Gamma}$ is finite dimensional,
$\mathcal{T}_{\Gamma}$ is contained in $H^{\circ}$. Thus we have the following
theorem in $(H^{\circ},\alpha^{\circ})$.

######  Theorem 5.8.

For any $X,Y,Z\in\mathcal{T}_{\Gamma}$ we have

  1. (1)

$[X,Y]=\widetilde{ad}_{R}(Y)(X)=XY-m_{H^{\circ}}(B^{t}(X\otimes Y))$.

  2. (2)

Let $\chi=\sum_{i}X_{i}\otimes Y_{i}$ for $X_{i},Y_{i}\in\mathcal{T}$ such
that $B^{t}(\chi)=\chi$, then $\sum_{i}[X_{i},Y_{i}]=0$.

  3. (3)

$[\tau(X),[Y,\tau^{-1}(Z)]]=[[X,Y],Z]-\sum_{i}[[X,\tau^{-1}(Z_{i})],\tau(Y_{i})]$,
where $Y_{i},Z_{i}\in\mathcal{T}$ such that $B^{t}(Y\otimes
Z)=\sum_{i}Z_{i}\otimes Y_{i}$.

###### Proof.

  1. (1)

For $h\in H$,

$\displaystyle\widetilde{ad}_{R}(Y)(X)(h)$ $\displaystyle=$
$\displaystyle((S(Y_{1})\tau^{-1}(X))\tau(Y_{2}))(h)$ $\displaystyle=$
$\displaystyle(S(Y_{1})\tau^{-1}(X))(h_{1})\tau(Y_{2}))(h_{2})=S(Y_{1})(h_{11})\tau^{-1}(X)(h_{12})\tau(Y_{2}))(h_{2})$
$\displaystyle=$ $\displaystyle
Y_{1}(S(h_{11}))X(\alpha(h_{12}))Y_{2}(\alpha^{-1}(h_{2}))=X(\alpha(h_{12}))Y_{1}(S(h_{11}))Y_{2}(\alpha^{-1}(h_{2}))$
$\displaystyle=$ $\displaystyle
X(\alpha(h_{12}))Y(S(h_{11})\alpha^{-1}(h_{2}))=(X\otimes
Y)(\alpha(h_{12})\otimes S(h_{11})\alpha^{-1}(h_{2}))$ $\displaystyle=$
$\displaystyle(X\otimes Y)(\widetilde{Ad}_{R}(h))=[X,Y](h),$

which gives us the first equality. If we set the finite sum for
$B^{t}(X\otimes Y)=\sum_{i}Y_{i}\otimes X_{i}$ with
$X_{i},Y_{i}\in\mathcal{T}$, then, for any $h,g\in H$,

$\displaystyle B^{t}(X\otimes Y)(h\otimes g)$ $\displaystyle=$
$\displaystyle\sum_{i}Y_{i}(h)X_{i}(g)=\sum_{i}<\tau^{-1}(Y_{i}),\omega(h)><\tau^{-1}(X_{i}),\omega(g)>$
$\displaystyle=$
$\displaystyle<\sum_{i}\tau^{-1}(Y_{i})\otimes\tau^{-1}(X_{i}),\omega(h)\otimes_{H}\omega(g)>_{2}$
$\displaystyle=$
$\displaystyle<B^{t}(\tau^{-1}(X)\otimes\tau^{-1}(Y)),\omega(h)\otimes_{H}\omega(g)>_{2}$
$\displaystyle=$
$\displaystyle<\tau^{-1}(X)\otimes\tau^{-1}(Y),B(\omega(h)\otimes_{H}\omega(g))>_{2}$
$\displaystyle=$
$\displaystyle<\tau^{-1}(X)\otimes\tau^{-1}(Y),\omega(\alpha^{2}(g_{12}))\otimes_{H}\omega(\overline{\alpha^{-1}(h)}(S(g_{11})\alpha^{-1}(g_{2})))>_{2}$
$\displaystyle=$
$\displaystyle<\tau^{-1}(X),\omega(\alpha^{2}(g_{12}))><\tau^{-1}(Y),\omega(\overline{\alpha^{-1}(h)}(S(g_{11})\alpha^{-1}(g_{2})))>$
$\displaystyle=$ $\displaystyle
X(\alpha^{2}(g_{12}))Y(\overline{\alpha^{-1}(h)}(S(g_{11})\alpha^{-1}(g_{2})))$
$\displaystyle=$ $\displaystyle
X(\alpha^{2}(g_{12}))Y_{1}(\overline{\alpha^{-1}(h)})Y_{2}(S(g_{11})\alpha^{-1}(g_{2}))$
$\displaystyle=$ $\displaystyle
Y_{1}(\overline{\alpha^{-1}(h)})X(\alpha^{2}(g_{12}))Y_{21}(S(g_{11}))Y_{22}(\alpha^{-1}(g_{2}))$
$\displaystyle=$
$\displaystyle\overline{\tau(Y_{1})}(h)S(Y_{21})(g_{11})X(\alpha^{2}(g_{12}))Y_{22}(\alpha^{-1}(g_{2}))$
$\displaystyle=$
$\displaystyle\overline{\tau(Y_{1})}(h)(S(Y_{21})\tau^{-2}(X))(g_{1})\tau(Y_{22})(g_{2})$
$\displaystyle=$
$\displaystyle\overline{\tau(Y_{1})}(h)[(S(Y_{21})\tau^{-2}(X))\tau(Y_{22})](g)$
$\displaystyle=$
$\displaystyle(\overline{\tau(Y_{1})}\otimes\widetilde{ad}_{R}(Y_{2})(\tau^{-1}(X)))(h\otimes
g),$

where in the sixth equality we have used the equation 5.21. So, we have
$B^{t}(X\otimes
Y)=\overline{\tau(Y_{1})}\otimes\widetilde{ad}_{R}(Y_{2})(\tau^{-1}(X))$.
Hence, we make the following computation

$\displaystyle m_{H^{\prime}}(B^{t}(X\otimes Y))$ $\displaystyle=$
$\displaystyle\overline{\tau(Y_{1})}\widetilde{ad}_{R}(Y_{2})(\tau^{-1}(X))$
$\displaystyle=$
$\displaystyle(\tau(Y_{1})-\varepsilon_{H^{\circ}}(\tau(Y_{1}))1_{H^{\circ}})\widetilde{ad}_{R}(Y_{2})(\tau^{-1}(X))$
$\displaystyle=$
$\displaystyle\tau(Y_{1})[(S(Y_{21})\tau^{-2}(X))\tau(Y_{22})]-(\varepsilon_{H^{\circ}}(Y_{1})1_{H^{\circ}})[(S(Y_{21})\tau^{-2}(X))\tau(Y_{22})]$
$\displaystyle=$
$\displaystyle\tau(Y_{1})[S(\tau(Y_{21}))(\tau^{-2}(X)Y_{22})]-(\varepsilon_{H^{\circ}}(Y_{1})1_{H^{\circ}})((S(Y_{21})\tau^{-2}(X))\tau(Y_{22}))$
$\displaystyle=$
$\displaystyle(Y_{1}S(\tau(Y_{21})))\tau(\tau^{-2}(X)Y_{22})-(\varepsilon_{H^{\circ}}(Y_{1})1_{H^{\circ}})((S(Y_{21})\tau^{-2}(X))\tau(Y_{22}))$
$\displaystyle=$
$\displaystyle(\tau(Y_{11})S(\tau(Y_{12})))(\tau^{-1}(X)Y_{2})-1_{H^{\circ}}([S(\varepsilon_{H^{\circ}}(Y_{11})Y_{12})\tau^{-2}(X)]Y_{2})$
$\displaystyle=$
$\displaystyle(\varepsilon_{H^{\circ}}(Y_{1})1_{H^{\circ}})(\tau^{-1}(X)Y_{2})-(S(Y_{1})\tau^{-1}(X))\tau(Y_{2})$
$\displaystyle=$ $\displaystyle
1_{H^{\circ}}(\tau^{-1}(XY))-\widetilde{ad}_{R}(Y)(X)$ $\displaystyle=$
$\displaystyle XY-\widetilde{ad}_{R}(Y)(X),$

that is, we get $\widetilde{ad}_{R}(Y)(X)=XY-m_{H^{\prime}}(B^{t}(X\otimes
Y))$.

  2. (2)

It immediately follows from (1) that
$\sum_{i}[X_{i},Y_{i}]=\sum_{i}X_{i}Y_{i}-m_{H^{\prime}}(B^{t}(\chi))=\sum_{i}X_{i}Y_{i}-\sum_{i}X_{i}Y_{i}=0$.

  3. (3)

Let us first set $[X,Y]=\widetilde{ad}_{R}(Y)(X)=X\lhd Y$. Then,

$[[X,Y],Z]=[X,Y]\lhd Z=(X\lhd Y)\lhd
Z=\tau(X)\lhd(Y\tau^{-1}(Z))=[\tau(X),Y\tau^{-1}(Z)].$

Since, by (1), $YZ=[Y,Z]+\sum_{i}Z_{i}Y_{i}$ for $B^{t}(Y\otimes
Z)=\sum_{i}Z_{i}\otimes Y_{i}$, we have

$\displaystyle[[X,Y],Z]$ $\displaystyle=$
$\displaystyle[\tau(X),Y\tau^{-1}(Z)]=[\tau(X),[Y,\tau^{-1}(Z)]+\sum_{i}\tau^{-1}(Z_{i})Y_{i}]$
$\displaystyle=$
$\displaystyle[\tau(X),[Y,\tau^{-1}(Z)]]+\sum_{i}[\tau(X),\tau^{-1}(Z_{i})Y_{i}]$
$\displaystyle=$
$\displaystyle[\tau(X),[Y,\tau^{-1}(Z)]]+\sum_{i}[[X,\tau^{-1}(Z_{i})],\tau(Y_{i})],$

that is,
$[\tau(X),[Y,\tau^{-1}(Z)]]=[[X,Y],Z]-\sum_{i}[[X,\tau^{-1}(Z_{i})],\tau(Y_{i})]$
holds.

∎

######  Remark 5.9.

If we take the braiding $B$ as the flip operator, then $B^{t}$ is the flip on
$\mathcal{T}\otimes\mathcal{T}$ by its definition. In this case, we obtain

$[X,Y]=XY-YX,\qquad[X,Y]+[Y,X]=0,\forall X,Y\in\mathcal{T}$

and

$\displaystyle[\tau(X),[Y,\tau^{-1}(Z)]]$ $\displaystyle=$
$\displaystyle[[X,Y],Z]-[[X,\tau^{-1}(Z)],\tau(Y)]=-[Z,[X,Y]]+[\tau(Y),[X,\tau^{-1}(Z)]]$
$\displaystyle=$ $\displaystyle-[Z,[X,Y]]-[\tau(Y),[\tau^{-1}(Z),X]].$

Then, by replacing $Z$ with $\tau(Z)$ in the above equality, we get

$[\tau(X),[Y,Z]]+[\tau(Y),[Z,X]]+[\tau(Z),[X,Y]]=0,$

which is the Hom-Jacobi identity. In the above theorem, items $(2)$ and $(3)$
are the quantum versions of the antisymmetry relation and the Hom-Jacobi
identity. Therefore, $(\mathcal{T}_{\Gamma},\tau)$ is called the quantum Hom-
Lie algebra of the bicovariant $(H,\alpha)$-Hom-FODC.

##  6\. Acknowledgments

The author would like to thank Professor Christian Lomp for his valuable
suggestions. This research was funded by the European Regional Development
Fund through the programme COMPETE and by the Portuguese Government through
the FCT- Fundação para a Ciência e a Tecnologia under the project
PEst-C/MAT/UI0144/2013. The author was supported by the grant
SFRH/BD/51171/2010.

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