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\begin{document}
\title[Braided Quantum Groups]{Generalized Braided Quantum Groups}
\author{micho {\Dj}UR{\Dj}EVICH}
\address{Instituto de Matematicas, UNAM, Area de la Investigacion
Cientifica, Circuito Exterior, Ciudad Universitaria, M\'exico DF,
cp 04510, MEXICO}
\email{micho@matem.unam.mx}
\begin{abstract}
A generalization of Hopf algebras (quantum groups) and braided-Hopf
algebras (braided quantum groups) in which the multiplicativity axiom
for the counit is dropped is presented.
The generalization overcomes an inherent geometrical
inhomogeneity of standard quantum groups and braided quantum groups,
in the sense of allowing completely
`pointless' objects. All braid-type equations appear as a consequence of
deeper axioms. Braided counterparts of basic algebraic relations
between fundamental entities of the standard theory are found.
\end{abstract}
\maketitle
\section{Introduction}
\renewcommand{\thepage}{}
     The aim of this study is to present basic elements
of a braided  theory  which generalizes standard quantum groups and
braided quantum groups in a non-trivial and effective way.

     The theory allows  a  possibility  of
completely  `pointless'  objects  and  includes,  besides   standard
braided quantum groups, various geometrically interesting structures which
are not braided-Hopf algebras, but which are more or less similar to them.

     Let us start with a simple  geometrical  consideration.
According to the classical Gelfand-Naimark theorem, there exists a natural
correspondence between compact topological spaces $X$ and commutative
unital $C^*$-algebras $\cal{A}$.
For a given $X$, the algebra $\cal{A}$ is consisting of complex-valued
continuous functions on $X$, endowed with the
standard algebraic operations and the maximum norm.
Conversely, if $\cal{A}$ is given then points of $X$ are recovered
as characters (non-trivial multiplicative hermitian linear functionals)
on $\cal{A}$.
In terms of this identification, topology of $X$ is induced by the *-weak
topology of the dual space.

Furthermore, in differential geometry it is possible to re-express all
properties of a smooth manifold $X$, in terms of the *-algebra of smooth
complex-valued functions on $X$.
A similar situation holds in algebraic geometry, where $\cal{A}$ is
consisting of polinomial functions on the algebraic variety $X$.

The starting idea of non-commutative differential geometry \cite{C},
consists in replacing function algebras by appropriate non-commutative
algebras $\cal{A}$, but still interpreting the  elements of $\cal{A}$
as `functions' on the qualitatively new `quantum spaces'.
In non-commutative geometry, the `existence' of such `quantum spaces'
always appears through $\cal{A}$.
In other words, we work directly with the algebra $\cal{A}$, and all
geometrical concepts and structures are
expressed exclusively in terms of the algebra $\cal{A}$.
This means that formally we {\it define} quantum spaces as ordered pairs
$X=(\cal{A},S)$, where $S$ is the appropriate additional
algebraic structure on $\cal{A}$, corresponding in the
classical (commutative) case to the appropriate geometrical structure on
the classical underlying space $X$.

However, if $\cal{A}$ is non-commutative, then the corresponding quantum
space $X$ can not be re-interpreted in classical terms, as a structuralized
collection of points.
On the other hand, it is important to notice that
the concept of a classical point is easily incorporable in the
non-commutative context.
In analogy with the classical geometry, it is natural to {\it define}
points of $X$ as characters of the algebra $\cal{A}$, assuming that $\cal{A}$
is equipped with a *-structure (if $\cal{A}$ is not equipped
with a *-structure we can simply consider
all multiplicative functionals, in analogy with complex algebraic geometry).
Generally, the space $X$ may be `completely quantum'--without points at all.

A particularly important class of quantum spaces is given by quantum groups.
Geometrically, quantum groups are quantum spaces endowed
with a group structure. Let us consider a quantum group $G=(\cal{A},S)$.
By definition, this means that $S=\{\phi,\e,\k\}$ is a Hopf algebra \cite{A}
structure on the algebra $\cal{A}$, specified by the coproduct
$\phi\colon\cal{A}\rightarrow\cal{A}\otimes
\cal{A}$, the counit $\e\colon\cal{A}\rightarrow\Bbb{C}$ and the
antipode $\k\colon\cal{A}\rightarrow\cal{A}$ map (we follow the
notation of \cite{W}).
The three maps should be mutually related in the following way.

As first, the maps $\phi$ and $\e$ determine a counital coalgebra
structure on $\cal{A}$, in other words
$$
(\id\otimes\phi)\phi=(\phi\otimes\id)\phi\qquad(\id\otimes\e)\phi
=(\e\otimes\id)\phi=\id.
$$
Secondly, we have the antipode axiom
$$
m(\k\otimes \id)\phi=m(\id\otimes \k)\phi=1\e,
$$
where $m\colon\cal{A}\otimes \cal{A}\rightarrow\cal{A}$
is the multiplication in $\cal{A}$
and $1\in\cal{A}$ is the unit element.

Finally, the map $\phi$ should  be multiplicative, in the sense that
$$
\phi(ab)=\phi(a)\phi(b),
$$
for  each  $a,b\in\cal{A}$.
In the above relation, $\cal{A}\otimes \cal{A}$ is understood as an algebra,
in a natural manner.

As a consequence of mentioned properties, it turns out  that  the
antipode $\k$ is an anti(co)multiplicative map.
The multiplicativity of the counit is another  important  consequence.
Further, if $\cal{A}$ is equipped with a *-structure and if the
coproduct is such that $\phi*=(*\otimes *)\phi$,
then the composition $*\k$ is involutive and the counit is hermitian.
\renewcommand{\thepage}{\arabic{page}}

In particular, the space $G$ always possesses  at  least  one  point,
corresponding to the counit map (the neutral element).
The quantum group structure on $G$ induces, in a  natural  manner,  a  group
structure on the set $G_{cl}$ of all classical points of $G$,
such that  $G_{cl}$ is geometrically interpretable
as a `subgroup' of $G$.
Explicitly, the product and the inverse are given by
$$
gh=(g\otimes h)\phi\qquad g^{-1}=g\k.
$$

In summary, quantum groups are, in  contrast  to  ordinary
ones, {\it inhomogeneous objects}.
The inhomogeneity  is  explicitely  visible  in  the
situations  in  which   `diffeomorphisms'   of   $G$   appear.
All `diffeomorphisms'  must  `preserve'  the classical part $G_{cl}$.
For example, in the theory of locally trivial quantum principal  bundles
over smooth manifolds \cite{D} a natural correspondence between
quantum  $G$-bundles and ordinary $G_{cl}$-bundles (over the
same manifold) holds.
Geometrically speaking, this phenomena appears because the corresponding
right-covariant `transition functions' are completely determined by
their `restrictions' on the classical part $G_{cl}$.

     On  the  other  hand,  it  is  natural  to  expect  that   in
noncommutative geometry quantum spaces with a group structure play
a similar role as Lie groups in classical  differential  geometry.
And,  among  other  things,  Lie  groups   provide   examples   of
particularly regular geometrical objects.
From this point of view,  the necessity  of the described geometrical
inhomogeneity of quantum groups seems strange.

     Such a thinking naturally leads to an idea of generalizing  a
notion of a group structure on a noncommutative space, in order to
include objects of a more elaborate geometrical nature.

There has already been introduced in \cite{Maj} one generalization of
quantum groups, in the framework of braided categories.
In this generalization, the standard transposition (figuring in the
product in $\cal{A}\otimes\cal{A}$) is replaced by the
appropriate braid operator
$\sigma\colon\cal{A}\otimes\cal{A}\rightarrow\cal{A}\otimes\cal{A}$, so
that all group entities are undersandable as morphisms in the braided
category generated by $\cal{A}$ and $\sigma$.
Such a generalization has proven useful for various applications
in non-commutative geometry,
however it does not address the above discussed `completely pointless'
question because the counit map is always multiplicative (and hermitian,
if we have the appropriate *-structure on $\cal{A}$).

In this paper, we further generalize this concept of a braided-Hopf
algebra (by not demanding that the counit is multiplicative) replacing
the standard axiom of the $\sigma$-functoriality of the coproduct $\phi$
by a more general octagonal diagram.
This opens a possibility of the
existence of `completely pointless' structures (in particular, in this
case the counit is not multiplicative).
Moreover, we shall not demand directly that $\sigma$ obeys the braid
equation, though this will be derived as a consequence of the initial axioms.

The paper is organized as follows.
The next section is devoted to the definition of braided quantum groups.
In Section 3 the most important interrelations between all  relevant
maps will be investigated.
In particular, we shall see that  besides the flip-over  operator
$\sigma$,  another  braid  operator
$\tau\colon\cal{A}\otimes\cal{A}\rightarrow\cal{A}\otimes\cal{A}$
naturally enters the game.
This  operator  is  expressible  via  $\e$, $\phi$  and $\sigma$.
Two  braid  operators  $\sigma$  and  $\tau$  will   play   a
fundamental role in the whole analysis.
In particular, it will  be shown that $\sigma$ and $\tau$ are
mutually compatible in a  `braided sense'.

The standard theory of braided quantum groups \cite{Maj} is recovered
when $\sigma=\tau$.
Interestingly, this is further equivalent to the multiplicativity of
the counit map.

A large class of examples of `completely pointless' braided quantum groups
is given by braided Clifford algebras \cite{DD}
associated to involutive braidings.
This includes classical Clifford and Weyl algebras.
Another class of interesting examples is given by quantum tori \cite{C}.
Endowed with appropriate group structures \cite{D-qtor}, quantum tori
can be viewed as braided quantum groups, in the sense of the formalism
presented in this paper.

Finally, the Appendix is devoted  to  the  main
properties of systems of braid operators, mutually compatible in a
`braided  sense'.
A  motivation  for  this  comes  from   already
mentioned  braided  compatibility  between  $\sigma$  and $\tau$.
In particular, it will be shown that $\sigma$ and $\tau$ can be naturally
included in a (generally  infinite)  `braid  system'
expressing concisely all twisting properties.

\newpage
\section{Definition of Braided Quantum Groups}

Let $\cal{A}$ be a complex associative algebra,  with  the  product
$m\colon\cal{A}\otimes \cal{A}\rightarrow\cal{A}$ and the unit element
$1\in\cal{A}$.
Let  us  assume that $\cal{A}$ is  endowed  with  a  coassociative coalgebra
structure, specified by the coproduct
$\phi\colon\cal{A}\rightarrow\cal{A}\otimes
\cal{A}$ and the counit $\e\colon\cal{A}\rightarrow\Bbb{C}$.
Further, let  us  assume  that
bijective linear maps $\k\colon\cal{A}\rightarrow\cal{A}$ and
$\sigma\colon\cal{A}\otimes
\cal{A}\rightarrow\cal{A}\otimes \cal{A}$ are given such that the  following
equalities hold
\begin{align}
\sigma(m\otimes\id)&=(\id\otimes m)(\sigma\otimes\id)(\id\otimes\sigma)
\label{21}\\
\sigma(\id\otimes m)&=(m\otimes\id)(\id\otimes\sigma)(\sigma\otimes \id)
\label{22}\\
\phi m=(m&\otimes m)(\id\otimes\sigma\otimes\id)(\phi\otimes\phi),\label{23}
\end{align}
together with the antipode axiom
\begin{equation}
1\e=m(\id\otimes\k)\phi=m(\k\otimes\id)\phi.\label{25}
\end{equation}

Finally, let us assume that the diagram
\begin{equation}\label{24}
\begin{CD}
\cal{A}\otimes\cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\id\otimes\phi\otimes\id$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\sigma^{-1}\otimes\id$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}\\
@V{\mbox{$\sigma\otimes\id^2$}}VV @. @AA{\mbox{$\id\otimes\phi$}}A\\
\cal{A}\otimes\cal{A}\otimes\cal{A}\otimes\cal{A} @. {} @.
\cal{A}\otimes\cal{A}\\
@A{\mbox{$\id^2\otimes\sigma$}}AA @. @VV{\mbox{$\phi\otimes\id$}}V\\
\cal{A}\otimes\cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\id\otimes\phi\otimes\id$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\id\otimes\sigma^{-1}$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}
\end{CD}
\end{equation}
is commutative.

\begin{defn}\label{def:21} Every pair
$G=\bigl(\cal{A},\{\phi,\e,\k,\sigma\}\bigr)$
satisfying the above requirements is called {\it a braided quantum group}.
\end{defn}

The map $\sigma$ is interpretable as the  `twisting  operator'.
In the standard theory, $\sigma$ reduces to the ordinary transposition.
Identities \eqref{21}--\eqref{23} and \eqref{24} express  mutual
compatibility  between
maps $\phi$, $m$ and $\sigma$.
It  is  important  to  mention  that  the asymmetry between
\eqref{21}--\eqref{22} and \eqref{24} implies that the theory is not
`selfdual'.
However if  we  replace \eqref{24} with `dual' counterparts of
\eqref{21}--\eqref{22} then the theory reduces to braided quantum
groups of \cite{Maj} (and in particular becomes selfdual).

 The space $\cal{A}\otimes \cal{A}$ is  an  $\cal{A}$-bimodule,  in  a
natural manner.
With the help of $\sigma$, a natural product can  be
defined on $\cal{A}\otimes \cal{A}$, by requiring
\begin{equation}
 (a\otimes b)(c\otimes d)=a\sigma(b\otimes c)d.\label{26}
\end{equation}
Identities \eqref{21}--\eqref{22} ensure that this defines
an  associative  algebra
structure on $\cal{A}\otimes \cal{A}$, such that $1\otimes 1$  is  the
unit element.
In particular,
\begin{equation}\label{27}
\sigma\bigl(1\otimes (\,)\bigr)=(\,)\otimes 1 \qquad\quad
\sigma\bigl((\,)\otimes 1\bigr)=1\otimes (\,).
\end{equation}
In the following, it will be assumed that  $\cal{A}\otimes  \cal{A}$
is endowed with this algebra structure.
Equality \eqref{23}  then
says that $\phi$ is multiplicative.

Identity \eqref{24}  expresses  the  coassociativity  of  the  map
$(\id\otimes \sigma^{-1}  \otimes \id)(\phi\otimes  \phi)$.
It turns out that the  `inverse' diagram
\begin{equation}\label{28}
\begin{CD}
\cal{A}\otimes\cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\id\otimes\phi\otimes\id$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\sigma\otimes\id$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}\\
@V{\mbox{$\id^2\otimes\sigma$}}VV @. @AA{\mbox{$\id\otimes\phi$}}A\\
\cal{A}\otimes\cal{A}\otimes\cal{A}\otimes\cal{A} @. {} @.
\cal{A}\otimes\cal{A}\\
@A{\mbox{$\sigma\otimes\id^2$}}AA @. @VV{\mbox{$\phi\otimes\id$}}V\\
\cal{A}\otimes\cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\id\otimes\phi\otimes\id$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}
@<{\mbox{$\id\otimes\sigma$}}<< \cal{A}\otimes\cal{A}\otimes\cal{A}
\end{CD}
\end{equation}
holds, too. It expresses  the  coassociativity  of
$(\id\otimes\sigma\otimes\id)(\phi\otimes \phi)$.

\section{Elementary Algebraic Properties}

Let $G=\bigl(\cal{A},\{\phi,\e,\k,\sigma\}\bigr)$ be a
braided quantum group.
As in the standard theory, the antipode is uniquely determined by \eqref{25}.
The flip-over operator $\sigma$ is expressible through
$\phi,m$ and $\k$ in the following way
\begin{equation}\label{29}
\sigma=(m\otimes m)(\k\otimes\phi m\otimes\k)(\phi\otimes \phi),
\end{equation}
as directly follows from \eqref{23} and \eqref{25}.

     It is easy to see that
\begin{equation}
\phi(1)=1\otimes 1.\label{210}
\end{equation}
Indeed, $\phi(1)$    is    the    unity in the    subalgebra
$\phi(\cal{A})\subseteq\cal{A}\otimes \cal{A}$, as  follows from \eqref{23}.
On the other hand, $\cal{A}\otimes \cal{A}$ is generated by
$\phi(\cal{A})$, as a left (right) $\cal{A}$-module.
Hence, $\phi(1)$ is the unity of $\cal{A}\otimes \cal{A}$.
From \eqref{210} we obtain
\begin{align}
                            \e(1)&=1\label{211} \\
                            \k(1)&=1.\label{212}
\end{align}
 In   further   computations   the   result   of   an $(n-1)$-fold
comultiplication of an  element  $a\in\cal{A}$  will  be  symbolically
denoted by $a^{(1)}\otimes \dots\otimes a^{(n)}$.
Clearly, this  element  of $\cal{A}$   is  independent  of
ways  in  which  the  corresponding
comultiplications are performed.
\begin{lem}\label{lem:21}
The following identities hold
\begin{align}
 (\e\otimes\id)&=(\id\otimes
\e m)(\sigma\otimes  \id)(\id\otimes \phi) \label{213} \\
(\id\otimes \e)&=(\e m\otimes\id)(\id\otimes\sigma)(\phi\otimes\id).\label{214}
\end{align}
\end{lem}
\begin{pf} According to \eqref{23},
$$
ab^{(1)}\otimes b^{(2)}=(\e\otimes\id)(m\otimes m)(\id\otimes
\sigma\otimes \id)\bigl(
a^{(1)}\otimes a^{(2)}   \otimes b^{(1)}   \otimes b^{(2)}\bigr)\otimes
b^{(3)},
$$
for each $a,b\in\cal{A}$.
Acting by $m(\id\otimes \k)$ on  this  equality,
and using \eqref{25} we obtain
$$
a\e(b)=(\e\otimes \id)\bigl(a^{(1)}\sigma(a^{(2)}\otimes b)\bigr).
$$
Similarly, acting by $m(\k\otimes \id)$ on the identity
$$
a^{(1)}\otimes a^{(2)}b=a^{(1)}   \otimes (\id\otimes \e)
(m\otimes m)(\id\otimes \sigma\otimes \id)(a^{(2)}\otimes
a^{(3)}\otimes b^{(1)}   \otimes b^{(2)}   )
$$
we obtain
$$
\e(a)b=(\id\otimes \e)\bigl(\sigma(a\otimes b^{(1)})b^{(2)}\bigr).\qed
$$
\renewcommand{\qed}{}
\end{pf}

A `secondary' flip-over operator $\tau$ will be now introduced in the game.
From \eqref{24} we obtain
\begin{equation}
(\id^2 \otimes \e)(\id\otimes \sigma^{-1}  )(\phi\otimes \id)=
(\e\otimes \id^2 )(\sigma^{-1}\otimes \id)(\id\otimes \phi). \label{215}
\end{equation}
     Let $\tau\colon\cal{A}\otimes \cal{A}\rightarrow\cal{A}\otimes \cal{A}$
be a linear map defined by
\begin{equation}
\tau=(\id^2 \otimes \e)(\id\otimes\sigma^{-1})(\phi\otimes
\id)\sigma=(\e\otimes \id^2)(
\sigma^{-1}\otimes \id)(\id\otimes\phi)\sigma. \label{216}
\end{equation}
\begin{lem}\label{lem:22}
The map $\tau$ is bijective and
\begin{equation}
\tau^{-1}\sigma=(\id^2 \otimes \e)(\id\otimes \sigma)(\phi\otimes
\id)=(\e\otimes\id^2)
(\sigma\otimes\id)(\id\otimes \phi).\label{217}
\end{equation}
\end{lem}
\begin{pf}
The second equality in \eqref{217} follows from \eqref{28}.
Let $\tau'\sigma$ be the map given by the second term in \eqref{217}.
A direct computation gives
\begin{equation*}
\begin{split}
{}&\tau\tau'\sigma=(\e\otimes \id^2 \otimes \e)(\sigma^{-1}\otimes \id^2)
(\id\otimes \phi
\otimes\id)(\sigma\otimes\id)(\id\otimes\sigma)(\phi\otimes\id)\\
=&(\e\otimes    \id^2 \otimes     \e)(\sigma^{-1}  \otimes\id^2 )
(\id\otimes \phi\otimes \id)(\sigma\otimes \e\otimes \id)
(\id\otimes \phi\otimes \id)(\id\otimes \sigma)(\phi\otimes\id)\\
=&(\e\otimes\id^2 \otimes   \e\otimes   \e)(\sigma^{-1}  \otimes   \id
\otimes \sigma)(\id\otimes\phi\otimes\id^2)(\id\otimes\phi\otimes
\id)(\sigma\otimes \id)(\id\otimes\phi)\\
=&(\e\otimes \id^2 \otimes  \e\otimes  \e)(\id^3 \otimes\sigma)(\id^2
\otimes \phi\otimes\id)(\id^2 \otimes\sigma^{-1})(\id\otimes\phi\otimes
\id)(\id\otimes \sigma)(\phi\otimes \id)\\
=&(\id^2 \otimes \e\otimes \e)(\id^2 \otimes \sigma)(\id\otimes  \phi\otimes
\id)(\id\otimes \sigma^{-1}  )(\phi\otimes \id)\sigma\\
=&(\id^2 \otimes \e\otimes \e)(\sigma\otimes\id^2 )(\id\otimes  \phi\otimes
\id)(\sigma^{-1}  \otimes \id)(\id\otimes \phi)\sigma=\sigma.
\end{split}
\end{equation*}

 Similarly,  interchanging  $\sigma$  and  $\sigma^{-1}$    in  the   above
computations we conclude that $\tau'$ is a left inverse for $\tau$.
Hence,  $\tau$  is bijective and $\tau^{-1}=\tau'$.
\end{pf}

 Let us write down some important  algebraic  relations  including
the map $\tau$.
As first, let us observe that
\begin{gather}
(\e\otimes \id)\tau=\id\otimes \e \quad\qquad
(\id\otimes \e)\tau=\e\otimes \id  \label{218}\\
\tau\bigl(1\otimes (\,)\bigr)=(\,)\otimes 1\quad\qquad
\tau\bigl((\,)\otimes 1\bigr)=1\otimes (\,). \label{219}
\end{gather}
This is a direct consequence of the definition of $\tau$, and  property
\eqref{27}.
Further, coassociativity of $\phi$ and relations
\eqref{216}--\eqref{217} imply
\begin{align}
(\phi\otimes\id)\tau^{-1}\sigma&
=(\id\otimes\tau^{-1}\sigma)(\phi\otimes\id)\label{220}\\
(\id\otimes\phi)\tau^{-1}\sigma&=(\tau^{-1}\sigma\otimes\id)(\id\otimes\phi)
\label{221}\\
(\phi\otimes\id)\tau\sigma^{-1}&=(\id\otimes\tau\sigma^{-1})(\phi\otimes\id)
\label{222}\\
(\id\otimes \phi)\tau\sigma^{-1}&=(\tau\sigma^{-1}\otimes \id)
(\id\otimes \phi).\label{223}
\end{align}
In  other  words,   maps   $\sigma\tau^{-1}$     and
$\sigma^{-1}\tau$   are
automorphisms of  the  $\cal{A}$-bicomodule  $\cal{A}\otimes  \cal{A}$
(with the left and the right $\cal{A}$-comodule structures given  by
$\phi\otimes \id$  and  $\id\otimes  \phi$  respectively).
Moreover, the following commutation relations hold
\begin{align}
(\sigma\tau^{-1}\otimes \id)(\id\otimes \sigma\tau^{-1})&
=(\id\otimes\sigma\tau^{-1})(\sigma\tau^{-1}\otimes \id)\label{224}\\
(\sigma\tau^{-1}\otimes \id)(\id\otimes \sigma^{-1}\tau)&
=(\id\otimes \sigma^{-1}\tau)(\sigma\tau^{-1}\otimes \id) \label{225} \\
(\sigma^{-1}\tau\otimes \id)(\id\otimes \sigma\tau^{-1})&
=(\id\otimes \sigma\tau^{-1})(\sigma^{-1}\tau\otimes \id)\label{226}\\
(\sigma^{-1}\tau\otimes \id)(\id\otimes \sigma^{-1}\tau)&
=(\id\otimes \sigma^{-1}\tau)(\sigma^{-1}\tau\otimes \id).\label{227}
\end{align}

The above equalities follow from \eqref{220}--\eqref{223} and
\eqref{216}--\eqref{217}.
As a direct consequence of Lemma~\ref{lem:21} and
\eqref{217} we find
\begin{equation}
\e m=(\e\otimes \e)\sigma^{-1}\tau. \label{228}
\end{equation}
This generalizes the standard multiplicativity law for the counit.

Identities \eqref{24} and \eqref{28} can be rewritten in a simpler
`pentagonal form', including the  operator
$\tau$  and  explicitly expressing  twisting properties of the coproduct map.
\begin{pro}\label{pro:23}
The following identities hold
\begin{align}
(\phi\otimes\id)\sigma&=(\id\otimes\tau)(\sigma\otimes\id)(\id\otimes\phi)
\label{229}\\
(\id\otimes\phi)\sigma&=(\tau\otimes\id)(\id\otimes\sigma)(\phi\otimes\id)
\label{230}\\
(\phi\otimes\id)\sigma&=(\id\otimes\sigma)(\tau\otimes\id)(\id\otimes\phi)
\label{231}\\
(\id\otimes\phi)\sigma&=(\sigma\otimes\id)(\id\otimes\tau)(\phi\otimes\id).
\label{232}
\end{align}
\end{pro}
\begin{pf}
Using \eqref{24} and \eqref{217} we obtain
\begin{multline*}
(\e\otimes \id^3 )(\sigma\otimes \id^2 )(\id\otimes \phi\otimes \id)
(\sigma^{-1}\otimes\id)(\id\otimes\phi)=(\tau^{-1}\otimes\id)(\id\otimes\phi)\\
=(\e\otimes \id\otimes \sigma)(\id\otimes \phi\otimes \id)
(\id\otimes \sigma^{-1})(\phi\otimes \id)=(\id\otimes \sigma)(\phi\otimes \id)
\sigma^{-1}.
\end{multline*}
Similarly,
\begin{multline*}
(\id^3 \otimes    \e)(\id^2 \otimes\sigma)(\id\otimes\phi\otimes
\id)(\id\otimes \sigma^{-1})(\phi\otimes \id)=(\id\otimes \tau^{-1})
(\phi\otimes \id)\\
=(\sigma\otimes\id\otimes\e)(\id\otimes\phi\otimes\id)(\sigma^{-1}\otimes
\id)(\id\otimes\phi)=(\sigma\otimes\id)(\id\otimes \phi)\sigma^{-1}.
\end{multline*}
Hence, \eqref{229}--\eqref{230}  hold.
Starting  from equalities \eqref{28}  and \eqref{216} and applying the same
computation we obtain \eqref{231}--\eqref{232}.
\end{pf}

 In the next proposition `pentagonal'  twisting  relations  including
only $\tau$ are collected.
\begin{pro}\label{pro:24} We have
\begin{align}
(\phi\otimes\id)\tau&=(\id\otimes\tau)(\tau\otimes\id)(\id\otimes\phi)
\label{233}\\
(\id\otimes\phi)\tau&=(\tau\otimes\id)(\id\otimes\tau)(\phi\otimes\id)
\label{234}\\
\tau(m\otimes\id)&=(\id\otimes m)(\tau\otimes\id)(\id\otimes \tau)
\label{235}\\
\tau(\id\otimes m)&=(m\otimes\id)(\id\otimes\tau)(\tau\otimes \id).
\label{236}
\end{align}
\end{pro}
\begin{pf} Direct transformations give
$$
(\id\otimes\tau)(\tau\otimes\id)(\id\otimes\phi)=(\id\otimes
\tau\sigma^{-1})(\phi\otimes \id)\sigma=(\phi\otimes \id)\tau.
$$
Similarly,
$$
(\tau\otimes\id)(\id\otimes\tau)(\phi\otimes
\id)=(\tau\sigma^{-1}\otimes\id)(\id\otimes\phi)\sigma=(\id\otimes
\phi)\tau.
$$
     Applying \eqref{216}, \eqref{231} and \eqref{21} we obtain
\begin{equation*}
\begin{split}
(\id\otimes m)&(\tau\otimes \id)(\id\otimes \tau)=(\id\otimes  m\otimes
\e)(\tau\otimes\sigma^{-1})(\id\otimes\phi\otimes\id)(\id\otimes
\sigma)\\
&=(\id\otimes m\otimes\e)(\id^2\otimes\sigma^{-1})(\id\otimes
\sigma^{-1}\otimes\id)(\phi\otimes  \id^2 )(\sigma\otimes  \id)(\id\otimes
\sigma)\\
&=(\id^2 \otimes \e)(\id\otimes\sigma^{-1})(\phi\otimes  m)(\sigma\otimes
\id)(\id\otimes\sigma)\\
&=(\id^2\otimes\e)(\id\otimes
\sigma^{-1})(\phi\otimes \id)\sigma(m\otimes \id)=
\tau(m\otimes \id).
\end{split}
\end{equation*}
Similarly,
\begin{equation*}
\begin{split}
(m\otimes \id)&(\id\otimes \tau)(\tau\otimes\id)=(\e\otimes  m\otimes
\id)(\sigma^{-1}\otimes \tau)(\id\otimes \phi\otimes  \id)(\sigma\otimes
\id)\\
&=(\e\otimes m\otimes\id)(\sigma^{-1}\otimes\id^2)(\id\otimes
\sigma^{-1}\otimes\id)(\id^2 \otimes\phi)(\id\otimes
\sigma)(\sigma\otimes \id)\\
&=(\e\otimes   \id^2 )(\sigma^{-1}\otimes\id)(m\otimes\phi)(\id\otimes
\sigma)(\sigma\otimes \id)=\tau(\id\otimes m).\qed
\end{split}
\end{equation*}
\renewcommand{\qed}{}
\end{pf}
     We pass to the study of  algebraic  relations  including  the
antipode map.
In the standard theory, the antipode is an anti(co)-multiplicative map.
The next proposition  gives  a  braided counterpart of this property.
\begin{pro}\label{pro:25}
We have
\begin{align}
\phi\k&=\sigma(\k\otimes \k)\phi  \label{237}\\
\k m&=m(\k\otimes \k)\tau\sigma^{-1}\tau\sigma^{-1}\tau. \label{238}
\end{align}
\end{pro}
\begin{pf}
Let us start from the identity
$$
\k(a^{(1)})a^{(2)}\otimes a^{(3)}=1\otimes a.
$$
Acting by $\phi\otimes \phi$ on both sides, and using \eqref{23} and
\eqref{210} we obtain
$$
\bigl(\phi\k(a^{(1)})\bigr)\bigl(a^{(2)}\otimes a^{(3)}\bigr)
\otimes a^{(4)}\otimes a^{(5)}=1\otimes 1\otimes a^{(1)}\otimes a^{(2)}.
$$
After the action of $(\id\otimes m\otimes \id)(\id^2\otimes \k\otimes \id)$
on both sides the above equality becomes
$$
\bigl(\phi\k(a^{(1)})\bigr)\bigl(a^{(2)}\otimes
1\bigr)\otimes a^{(3)}=1\otimes
\k(a^{(1)})\otimes a^{(2)}.
$$
Hence
$$
\bigl(\phi\k(a^{(1)})\bigr)\bigl(a^{(2)}\k(a^{(3)})\otimes 1\bigr)
=\bigl(1\otimes \k(a^{(1)})\bigr)\bigl(\k(a^{(2)})\otimes 1\bigr).
$$
Applying \eqref{25} and \eqref{26} we obtain
$$
\phi\k(a)=\sigma\bigl(\k(a^{(1)})\otimes \k(a^{(2)})\bigr).
$$
This proves \eqref{237}.
To  prove  \eqref{238},  let  us  start  from
$m(\k\otimes m)(\phi\otimes \id)=\e\otimes \id$, act by  it  on
$m\otimes m$, and apply \eqref{23} and \eqref{228}.
We find
$$
m(\k\otimes m)(m\otimes m\otimes m)(\id\otimes \sigma\otimes \id^3 )(\phi
\otimes\phi\otimes \id^2 )=(\e\otimes \e)\sigma^{-1}\tau\otimes m.
$$
Acting by this  equality  on  $(\id^2 \otimes  \k\otimes  \id)(\id\otimes
\phi\otimes \id)$ and simplifying the expression we find
$$
m(\k m\otimes m)(\id\otimes     \sigma\otimes     \id)(\phi\otimes
\id^2)=(\e\otimes m)(\id\otimes \k\otimes \id)(\sigma^{-1}\tau\otimes \id).
$$
Acting by this on $(\id^2\otimes
\k)(\id\otimes \sigma)(\phi\otimes \id)$ we obtain
\begin{multline*}
m(\k m\otimes m)(\id\otimes\sigma\otimes\k)(\id^2 \otimes
\sigma)(\id\otimes \phi\otimes \id)(\phi\otimes \id)\\
=(\e\otimes m)(\id\otimes \k\otimes \k)(\sigma^{-1}\tau\otimes
\id)(\id\otimes \sigma)(\phi\otimes \id).
\end{multline*}
After simple twisting transformations the left-hand  side  of  the
above equality becomes
\begin{multline*}
m(\k m\otimes m)(\id^3 \otimes \k)(\id^2 \otimes \phi)(\id\otimes
\sigma\tau^{-1}  \sigma
)(\phi\otimes \id)\\
=(\k m\otimes \e)(\id\otimes \sigma\tau^{-1}\sigma)(\phi\otimes \id)
=\k m\tau^{-1}\sigma\tau^{-1}\sigma.
\end{multline*}
The right-hand side of the mentioned equality reduces to
$$
m(\e\otimes\k\otimes\k)(\sigma^{-1}\otimes\id)(\id\otimes
\phi)\sigma=m(\k\otimes\k)\tau.
$$
Consequently, \eqref{238} holds.
\end{pf}

Twisting properties of the antipode will be now analyzed.
As first, a technical lemma
\begin{lem}\label{lem:26} We have
\begin{gather}
\bigl[\sigma(\k\otimes \id)\tau^{-1}\sigma\tau^{-1}(a\otimes b^{(1)})\bigr]
b^{(2)}=a\otimes 1\e(b)\\
a^{(1)}\bigl[\sigma(\id\otimes \k)\tau^{-1}\sigma\tau^{-1}(a^{(2)}\otimes b)
\bigr]=\e(a)1\otimes b,
\end{gather}
for each $a,b\in\cal{A}$.
\end{lem}
\begin{pf}
We compute
\begin{equation*}
\begin{split}
&(\id\otimes m)(\sigma\otimes\id)(\k\otimes
\id^2)(\tau^{-1}\sigma\tau^{-1}\otimes\id)(\id\otimes \phi)\\
=&(\id\otimes   m)(\sigma\otimes\id)(\k\otimes   \id^2 )(\tau^{-1}\otimes
\id)(\id\otimes\phi)\sigma\tau^{-1}\\
=&(\id\otimes  m)(\sigma\otimes   \id)(\k\otimes   \sigma)(\phi\otimes
\id)\tau^{-1}=\sigma(m\otimes \id)(\k\otimes \id^2)(\phi\otimes \id)\tau^{-1}\\
=&\sigma(1\e\otimes \id)\tau^{-1}=\id\otimes 1\e.
\end{split}
\end{equation*}
Similarly,
\begin{equation*}
\begin{split}
&(m\otimes \id)(\id\otimes \sigma)(\id^2 \otimes\k)(\id\otimes \tau^{-1}
\sigma\tau^{-1})(\phi\otimes \id)\\
=&(m\otimes    \id)(\id\otimes    \sigma)(\id^{2}\otimes\k)(\id\otimes
\tau^{-1})(\phi\otimes\id)\sigma\tau^{-1}\\
=&\sigma(\id\otimes m)(\id \otimes\k)(\id\otimes
\phi)\tau^{-1}=\sigma(\id\otimes 1\e)\tau^{-1}=1\e\otimes\id.\qed
\end{split}
\end{equation*}
\renewcommand{\qed}{}
\end{pf}
\begin{pro}\label{pro:27}
The following identities hold
\begin{gather}
\sigma(\k\otimes \id)=(\id\otimes \k)\tau\sigma^{-1}\tau\label{242}\\
\tau(\id\otimes \k)=(\k\otimes \id)\tau\label{243}\\
\tau(\k\otimes \id)=(\id\otimes \k)\tau\label{244}\\
\sigma(\id\otimes\k)=(\k\otimes \id)\tau\sigma^{-1}\tau.\label{241}
\end{gather}
\end{pro}
\begin{pf}
Applying Lemma~\ref{lem:26} and property \eqref{25} we obtain
$$
\sigma(\k\otimes  \id)\tau^{-1}\sigma\tau^{-1}(a\otimes b)=
\bigl[\sigma(\k\otimes \id)\tau^{-1}\sigma\tau^{-1}(a\otimes b^{(1)})\bigr]
b^{(2)}\k(b^{(3)})=a\otimes \k(b).
$$
Similarly,
$$
(\id\otimes \k)\tau^{-1}\sigma\tau^{-1}(a\otimes b)
=\k(a^{(1)})a^{(2)}\bigl[\sigma(\k\otimes    \id)\tau^{-1}\sigma\tau^{-1}
(a^{(3)}   \otimes b)\bigr]=\k(a)\otimes b.
$$
This shows \eqref{242} and \eqref{241}.
Using properties \eqref{216}, \eqref{242}, \eqref{241},
\eqref{222}--\eqref{223}  and  \eqref{233}--\eqref{234}
we obtain
\begin{equation*}
\begin{split}
\tau(\id\otimes \k)&=(\e\otimes \id^2)(\sigma^{-1}\otimes \id)
(\id\otimes \phi)\sigma(\id\otimes \k)\\
&=(\e\otimes \k\otimes \id)(\tau^{-1}\sigma\tau^{-1}\otimes \id)
(\id\otimes \phi)\tau\sigma^{-1}\tau\\
&=(\e\otimes\k\otimes \id)(\tau^{-1}\otimes \id)(\id\otimes \phi)\tau=(\k
\otimes\id)\tau.
\end{split}
\end{equation*}
Similarly,
\begin{equation*}
\begin{split}
\tau(\k\otimes \id)&=(\id^2 \otimes \e)(\id\otimes \sigma^{-1})
(\phi\otimes \id)\sigma(\k\otimes \id)\\
&=(\id\otimes\k\otimes\e)(\id\otimes \tau^{-1}\sigma\tau^{-1})
(\phi\otimes \id)\tau\sigma^{-1}\tau\\
&=(\id\otimes\k\otimes \e)(\id\otimes \tau^{-1})(\phi\otimes \id)\tau=
(\id\otimes\k)\tau. \qed
\end{split}
\end{equation*}
\renewcommand{\qed}{}
\end{pf}

     As a direct consequence of the previous proposition we find
\begin{align}
(\k\otimes\k)\tau&=\tau(\k\otimes \k)\label{245}\\
(\k\otimes\k)\sigma&=\sigma(\k\otimes\k).\label{246}
\end{align}
For the end of this section, we shall prove that $\sigma$ and  $\tau$
satisfy a system of braid equations.
\begin{pro}\label{pro:28}
The following identities hold
\begin{align}
(\sigma\otimes \id)(\id\otimes \sigma)(\sigma\otimes \id)&=(\id\otimes
\sigma)(\sigma\otimes \id)(\id\otimes \sigma)\label{247}\\
(\tau\otimes \id)(\id\otimes \sigma)(\sigma\otimes \id)&=(\id\otimes \sigma)(
\sigma\otimes \id)(\id\otimes \tau)\label{248}\\
(\sigma\otimes \id)(\id\otimes \tau)(\sigma\otimes \id)&=(\id\otimes \sigma
)(\tau\otimes \id)(\id\otimes \sigma)\label{249} \\
(\sigma\otimes \id)(\id\otimes \sigma)(\tau\otimes \id)&=(\id\otimes \tau)(
\sigma\otimes \id)(\id\otimes \sigma)\label{250}\\
(\tau\otimes \id)(\id\otimes \tau)(\sigma\otimes\id)
&=(\id\otimes \sigma)(\tau\otimes \id)(\id\otimes \tau)\label{251}\\
(\tau\otimes \id)(\id\otimes \sigma)(\tau\otimes
\id)&=(\id\otimes \tau)(\sigma\otimes \id)(\id\otimes \tau)\label{252}\\
(\sigma\otimes \id)(\id\otimes \tau)(\tau\otimes \id)&=(\id\otimes \tau)(
\tau\otimes \id)(\id\otimes \sigma)\label{253}\\
(\tau\otimes \id)(\id\otimes \tau)(\tau\otimes \id)&=(\id\otimes
\tau)(\tau\otimes \id)(\id\otimes\tau).\label{254}
\end{align}
\end{pro}
\begin{pf} We shall first prove \eqref{248}--\eqref{251}  and  \eqref{253},
secondly
\eqref{254}, thirdly \eqref{252} and finally \eqref{247}.
A direct computation gives
\begin{multline*}
(\tau\otimes \id)(\id\otimes \sigma)(\sigma\otimes  \id)=(\tau\otimes
\id)(\id\otimes \sigma)(m\otimes m\otimes \id)\\
\hfill(\k\otimes\phi m\otimes
\k\otimes \id)(\phi\otimes \phi\otimes \id)\\
=(\id\otimes  m\otimes  m)(\tau\otimes  \id^3)(\id\otimes\tau\otimes
\id^2)(\id^2\otimes\sigma\otimes\id)(\id^3\otimes\sigma)\hfill\\
\hfill(\k\otimes
\phi m\otimes \k\otimes\id)(\phi\otimes \phi\otimes \id)\\
=(\id\otimes    m\otimes    m)(\id\otimes    \k\otimes    \phi\otimes
\k)(\tau\otimes \id^2)(\id\otimes \sigma\otimes \id)(\id\otimes m\otimes
\tau\sigma^{-1}\tau)(\phi\otimes \phi\otimes \id)\hfill\\
=(\id\otimes m\otimes    m)(\id\otimes\k\otimes\phi m\otimes
\k)(\tau\otimes \id^3 )(\id\otimes\sigma\otimes\id^2)(\phi\otimes
\id\otimes \phi)(\id\otimes \tau)\hfill\\
=(\id\otimes   m\otimes    m)(\id\otimes    \k\otimes    \phi m\otimes
\k)(\id\otimes   \phi\otimes    \phi)(\sigma\otimes    \id)(\id\otimes
\tau)\hfill\\
=(\id\otimes \sigma)(\sigma\otimes \id)(\id\otimes \tau).
\end{multline*}
Similarly,
\begin{multline*}
(\id\otimes \tau)(\sigma\otimes \id)(\id\otimes \sigma)=(\id\otimes \tau)(
\sigma\otimes   \id)(\id\otimes   m\otimes   m)\\
\hfill(\id\otimes   \k\otimes
\phi m\otimes \k)(\id\otimes \phi\otimes \phi)\\
=(m\otimes m\otimes \id)(\id^3 \otimes \tau)(\id^2 \otimes \tau\otimes \id)
(\id\otimes\sigma\otimes \id^2)(\sigma\otimes \id^3)\hfill\\
\hfill(\id\otimes \k
\otimes \phi m\otimes\k)(\id\otimes \phi\otimes \phi)\\
=(m\otimes m\otimes \id)(\k\otimes\phi\otimes\k\otimes
\id)(\id^2 \otimes\tau)(\id\otimes\sigma\otimes
\id)(\tau\sigma^{-1}\tau\otimes  m\otimes  \id)(\id\otimes   \phi\otimes
\phi)\hfill\\
=(m\otimes m\otimes \id)(\k\otimes \phi m\otimes \k\otimes \id)
(\id^3 \otimes \tau)(\id^2 \otimes\sigma\otimes\id)(\phi\otimes\id\otimes
\phi)(\tau\otimes \id)\hfill\\
=(m\otimes m\otimes\id)(\k\otimes \phi m\otimes\k\otimes
\id)(\phi\otimes  \phi\otimes   \id)(\id\otimes   \sigma)(\tau\otimes
\id)\hfill\\
=(\sigma\otimes \id)(\id\otimes \sigma)(\tau\otimes \id).
\end{multline*}
Essentially  the  same  transformations  lead  to  identities
\eqref{249}, \eqref{251} and \eqref{253}. Let us prove \eqref{254}.
We have
\begin{equation*}
\begin{split}
&(\id\otimes   \tau)(\tau\otimes   \id)(\id\otimes    \tau)=(\id\otimes
\tau\otimes\e)(\tau\otimes     \sigma^{-1})(\id\otimes\phi\otimes
\id)(\id\otimes \sigma)\\
=&(\id^2 \otimes \e\otimes \id)(\id^2 \otimes  \tau)(\id\otimes  \tau\otimes
\id)(\tau\otimes  \sigma^{-1})(\id\otimes   \phi\otimes   \id)(\id\otimes
\sigma)\\
=&(\id^2 \otimes \e\otimes \id)(\id\otimes \sigma^{-1}\otimes \id)
(\id^2 \otimes \tau)(\id\otimes \tau\otimes \id)(\tau\otimes \id^2)
(\id\otimes \phi\otimes \id)(\id\otimes\sigma)\\
=&(\id^2 \otimes\e\otimes\id)(\id\otimes \sigma^{-1}\otimes
\id)(\phi\otimes \tau)(\tau\otimes \id)(\id\otimes \sigma)\\
=&(\tau\sigma^{-1}\otimes \id)(\id\otimes \tau)(\tau\otimes \id)(\id
\otimes\sigma)
=(\tau\otimes \id)(\id\otimes \tau)(\tau\otimes \id).
\end{split}
\end{equation*}
Identities \eqref{225}, \eqref{248}, \eqref{251} and \eqref{254} imply
\begin{equation*}
\begin{split}
(\id\otimes\tau)&(\sigma\otimes\id)(\id\otimes\tau)=(\id\otimes
\tau)(\sigma\tau^{-1}\otimes \id)(\tau\otimes \id)(\id\otimes\tau)\\
=&(\id\otimes\sigma)(\sigma\tau^{-1}\otimes\id)(\id\otimes
\sigma^{-1}\tau)(\tau\otimes \id)(\id\otimes \tau)\\
=&(\id\otimes\sigma)(\sigma\tau^{-1}\otimes\id)(\id\otimes
\sigma^{-1})(\tau\otimes \id)(\id\otimes \tau)(\tau\otimes \id)\\
=&(\id\otimes\sigma)(\sigma\otimes\id)(\id\otimes
\tau)(\sigma^{-1}\tau\otimes\id)=(\tau\otimes\id)(\id\otimes
\sigma)(\tau\otimes \id).
\end{split}
\end{equation*}
     Finally, \eqref{224}, \eqref{248}, \eqref{250} and \eqref{252}
imply
\begin{equation*}
\begin{split}
(\id\otimes \sigma)&(\sigma\otimes \id)(\id\otimes\sigma)=(\id\otimes
\sigma\tau^{-1})(\sigma\otimes \id)(\id\otimes \sigma)(\tau\otimes \id)\\
&=(\id\otimes \sigma\tau^{-1})(\sigma\tau^{-1}\otimes \id)(\id\otimes
\tau)(\sigma\otimes \id)(\id\otimes \tau)\\
&=(\sigma\tau^{-1}\otimes \id)(\id\otimes \sigma)(\sigma\otimes \id)
(\id\otimes\tau)
=(\sigma\otimes \id)(\id\otimes \sigma)(\sigma\otimes \id).\qed
\end{split}
\end{equation*}
\renewcommand{\qed}{}
\end{pf}



\appendix
\section{Braid Systems}

The presence of two different braid operators $\sigma$ and $\tau$ in the
twisting properties of $\phi$ and $\k$ implies that, in contrast to the
standard formalism \cite{Maj}, the theory is not includable in the
conceptual framework of braided categories.
In this appendix we shall prove that $\sigma$ and $\tau$ can be
included in a generally infinite system of braid operators indexed
by integers, expressing all twisting
properties in a concise and elegant way.
Finally, we give a characterization of the standard theory, in terms of the
multiplicativity of the counit map.

Let us consider a complex associative algebra $\cal{A}$ with the unit element
$1\in\cal{A}$ and the product
$m\colon\cal{A}\otimes\cal{A}\rightarrow\cal{A}$.
\begin{defn}\label{def:A1} A {\it braid system} over $\cal{A}$ is a collection $\cal{F}$
of bijective linear maps acting in $\cal{A}\otimes \cal{A}$ and satisfying
\begin{align}
(\alpha\otimes\id)(\id\otimes\beta)(\gamma\otimes\id)&=
(\id\otimes\gamma)(\beta\otimes\id)(\id\otimes\alpha)\label{A1}\\
\alpha(\id\otimes m)&=(m\otimes\id)(\id\otimes\alpha)
(\alpha\otimes\id)\label{A2}\\
\alpha(m\otimes\id)&=(\id\otimes m)(\alpha\otimes\id)
(\id\otimes\alpha)\label{A3}
\end{align}
for each $\alpha,\beta,\gamma\in\cal{F}$.
\end{defn}
\begin{defn}\label{def:A2}
A braid system $\cal{F}$ is called {\it complete} iff it is closed under the
operation $(\alpha,\beta,\gamma)\mapsto\alpha\beta^{-1}\gamma$.
\end{defn}
Let $\cal{F}$ be a braid system over $\cal{A}$.
Then
$$
\alpha\bigl(1\otimes (\,)\bigr)=(\,)\otimes 1\quad\qquad
\alpha\bigl((\,)\otimes 1\bigr)=1\otimes (\,)
$$
for each $\alpha\in\cal{F}$, as follows from \eqref{A2}--\eqref{A3}.
Further, every $\alpha\in\cal{F}$ naturally determines an associative algebra
structure on $\cal{A}\otimes\cal{A}$, with the unit element $1\otimes 1$.
The corresponding product is given by
$(m\otimes m)(\id\otimes\alpha\otimes\id)$.

We are going to prove that there exists the {\it minimal} complete
braid system $\cal{F}^*$ which extends $\cal{F}$.
Starting from the system $\cal{F}$ we can inductively
construct an increasing chain of
braid systems $\cal{F}_n$, where $n\geq 0$ and
$\cal{F}_0=\cal{F}$, while $\cal{F}_{n+1}$ is consisting of maps of the form
$\delta=\alpha\beta^{-1}\gamma$, where $\alpha,\beta,\gamma\in\cal{F}_n$.
The fact
that all $\cal{F}_n$ are braid systems easily follows by
induction, applying the definition of braid systems and the identity
\begin{equation}
(\alpha\beta^{-1}\otimes\id)(\id\otimes\gamma\delta^{-1})
=(\id\otimes\gamma\delta^{-1})(\alpha\beta^{-1}\otimes\id)\label{A4}
\end{equation}
(which holds in an arbitrary braid system).

Let $\cal{F}^*$
be the union of systems $\cal{F}_n$.
By construction, $\cal{F}^*$ is a complete
braid system.
Moreover, $\cal{F}^*$ is the minimal braid system containing
$\cal{F}$.

Let $G=(\cal{A}, \{\phi,\e,\k,\sigma\})$ be a braided quantum group.
According
to \eqref{21}--\eqref{22}, \eqref{235}--\eqref{236} and
Proposition~\ref{pro:28}
operators $\{\sigma,\tau\}$ form a braid system over the algebra $\cal{A}$.
The corresponding completion $\cal{F}=\{\sigma,\tau\}^*$ consists of maps
$\sigma_n\colon\cal{A}\otimes\cal{A}\rightarrow\cal{A}\otimes\cal{A}$ of the
form
\begin{equation}\label{A6}
\sigma_n=(\sigma\tau^{-1})^{n-1}\sigma=\sigma(\tau^{-1}\sigma)^{n-1}
\end{equation}
where $n\in\Bbb{Z}$.
\begin{pro}\label{pro:A2} The following identities hold
\begin{align}
(\phi\otimes\id)\sigma_{n+k}&=(\id\otimes\sigma_k)(\sigma_n\otimes\id)(
\id\otimes\phi)\label{A10}\\
\sigma_n(\id\otimes\k)&=(\k\otimes\id)\sigma_{-n}\label{A8}\\
\sigma_n(\k\otimes\id)&=(\id\otimes\k)\sigma_{-n}\label{A9}\\
(\id\otimes\phi)\sigma_{n+k}&=(\sigma_k\otimes\id)(\id\otimes\sigma_n)(
\phi\otimes\id).\label{A11}
\end{align}
\end{pro}
\begin{pf} Applying Proposition~\ref{pro:27} and \eqref{A6} we obtain
\begin{multline*}
\sigma_n(\id\otimes\k)=(\sigma\tau^{-1})^{n-1}\sigma(\id\otimes\k)=
(\k\otimes\id)(\tau\sigma^{-1})^{n-1}\tau\sigma^{-1}\tau\\
=(\k\otimes\id)(\sigma\tau^{-1})^{-n-1}\sigma=(\k\otimes\id)\sigma_{-n}.
\end{multline*}
Similarly,
$$
\sigma_n(\k\otimes\id)=(\id\otimes\k)(\tau\sigma^{-1})^{n-1}\tau\sigma^{-1}
\tau=(\id\otimes\k)\sigma_{-n}.
$$
Equalities \eqref{A10} and \eqref{A11} directly follow from
\eqref{220}--\eqref{223} and \eqref{229}--\eqref{230}.
Indeed,
\begin{equation*}
\begin{split}
(\sigma_k\otimes\id)(\id\otimes\sigma_n)(\phi\otimes\id)&=
\bigl((\sigma\tau^{-1})^k\tau\otimes\id\bigr)\bigl(\id\otimes\sigma(\tau^{-1}
\sigma)^{n-1}\bigr)(\phi\otimes\id)\\
&=\bigl((\sigma\tau^{-1})^k\tau\otimes\id\bigr)(\id\otimes\sigma)
(\phi\otimes\id)(\tau^{-1}\sigma)^{n-1}\\
&=\bigl((\sigma\tau^{-1})^k\otimes\id\bigr)(\id\otimes\phi)\sigma
(\tau^{-1}\sigma)^{n-1}\\
&=(\id\otimes\phi)(\sigma\tau^{-1})^{n+k-1}\sigma=
(\id\otimes\phi)\sigma_{n+k}.
\end{split}
\end{equation*}
Similarly,
\begin{multline*}
(\id\otimes\sigma_n)(\sigma_k\otimes\id)(\id\otimes\phi)=
\bigl(\id\otimes(\sigma\tau^{-1})^n\tau\bigr)\bigl(\sigma(\tau^{-1}\sigma
)^{k-1}\otimes\id\bigr)(\id\otimes\phi)\\
=(\phi\otimes\id)(\sigma\tau^{-1})^{n+k-1}\sigma=(\phi\otimes
\id)\sigma_{n+k}.\qed
\end{multline*}
\renewcommand{\qed}{}
\end{pf}
As we shall now see, an arbitrary $\sigma_n\in\cal{F}$ is interpretable as
the flip-over operator corresponding to a modified braided
quantum group structure.

For each $n\in\Bbb{Z}$, let $m_n\colon\cal{A}\otimes\cal{A}\rightarrow
\cal{A}$ and $\k_n\colon\cal{A}\rightarrow\cal{A}$ be the maps given by
\begin{align}
m_n&=m\sigma_n^{-1}\sigma\label{A12}\\
\k_n&=(\e\otimes\k)\sigma_n^{-1}\sigma\phi=(\k\otimes\e)
\sigma_n^{-1}\sigma\phi \label{A13}
\end{align}
(the second equality in \eqref{A13} will be justified in the proof
of the proposition below).
It is easy to see that each $m_n$, interpreted as
a product, determines a structure of an associative algebra on the space
$\cal{A}$.
Indeed,
\begin{equation*}
\begin{split}
m_n(m_n\otimes\id)&=m\sigma_n^{-1}\sigma(m\sigma_n^{-1}\sigma\otimes\id)
=m\sigma_n^{-1}(\id\otimes m)(\sigma\otimes\id)(\id\otimes\sigma)(
\sigma_n^{-1}\sigma\otimes\id)\\
&=m(m\otimes\id)(\id\otimes\sigma_n^{-1})(\sigma_n^{-1}\sigma\otimes\id)
(\id\otimes\sigma)(\sigma_n^{-1}\sigma\otimes\id)\\
&=m(m\otimes\id)(\id\otimes\sigma_n^{-1})(\sigma_n^{-1}\otimes\id)
(\id\otimes\sigma^{-1}_n)(\sigma\otimes\id)(\id\otimes\sigma)(\sigma\otimes
\id)\\
&=m(\id\otimes m)(\sigma_n^{-1}\otimes\id)(\id\otimes\sigma_n^{-1})
(\sigma_n^{-1}\otimes\id)(\id\otimes\sigma)(\sigma\otimes\id)
(\id\otimes\sigma)\\
&=m(\id\otimes m)(\sigma_n^{-1}\otimes\id)(\id\otimes\sigma_n^{-1}\sigma)
(\sigma\otimes\id)(\id\otimes\sigma_n^{-1}\sigma)\\
&=m\sigma_n^{-1}(m\otimes\id)(\id\otimes\sigma)(\sigma\otimes\id)
(\id\otimes\sigma_n^{-1}\sigma)\\
&=m\sigma_n^{-1}\sigma(\id\otimes m\sigma_n^{-1}\sigma)=m_n(\id\otimes m_n).
\end{split}
\end{equation*}

For each $n\in\Bbb{Z}$, let us denote by $\cal{A}_n$ the vector space
$\cal{A}$ endowed with the product $m_n$.
Evidently, $1\in\cal{A}_n$ is
the unit in this algebra, too.
\begin{pro}\label{pro:A3} The pair $G_n=\bigl(\cal{A}_n,\{\phi,\e,
\k_n,\sigma_n\}\bigr)$ is a braided quantum group.
\end{pro}
\begin{pf} We have to check the last three axioms in
Definition~\ref{def:21}.
The compatibility condition between
$\phi$ and $\sigma_n$ easily follows from \eqref{A10} and \eqref{A11}.
Further, a direct computation gives
\begin{equation*}
\begin{split}
\phi m_n&=(m\otimes m)(\id\otimes\sigma\otimes\id)(\phi\otimes
\phi)\sigma_n^{-1}\sigma=(m\otimes m)(\id\otimes\sigma\sigma_n^{-1}\sigma
\otimes\id)(\phi\otimes\phi)\\
&=(m\otimes m)(\id\otimes\sigma_{2-n}\otimes\id)(\phi\otimes\phi)\\
&=(m\sigma_n^{-1}\otimes m)(\id\otimes\phi\otimes\id)(\sigma_2\otimes\id)
(\id\otimes\phi)\\
&=(m\sigma_n^{-1}\sigma\otimes m)(\id\otimes\sigma\otimes\id)(\phi\otimes
\phi)\\
&=(m\sigma_n^{-1}\sigma\otimes m\sigma_n^{-1})(\id\otimes\phi\otimes\id)
(\id\otimes\sigma_{n+1})(\phi\otimes\id)\\
&=(m_n\otimes m_n)(\id\otimes\sigma_n\otimes\id)(\phi\otimes\phi).
\end{split}
\end{equation*}
Finally, we have to check that $k_n$ satisfies the antipode axiom.
Let us consider maps $k_n^\pm\colon\cal{A}\rightarrow\cal{A}$ given by
$$
k_n^-=(\k\otimes\e)\sigma_n^{-1}\sigma\phi\quad\qquad
k_n^+=(\e\otimes\k)\sigma_n^{-1}\sigma\phi.
$$
We have
\begin{equation*}
\begin{split}
m_n(k_n^-\otimes\id)\phi&=m\sigma_n^{-1}\sigma(\e\otimes\k\otimes\id)
(\tau\sigma_n^{-1}\sigma\otimes\id)(\phi\otimes\id)\phi\\
&=m(\e\otimes\k\otimes\id)(\id\otimes\sigma_{-n}^{-1}\sigma_{-1})
(\sigma_{1-n}\otimes\id)(\id\otimes\phi)\phi\\
&=m(\e\otimes\k\otimes\id)(\id\otimes\sigma_{-n}^{-1})(\phi\otimes
\id)\sigma_{-n}\phi\\
&=m(\e\otimes\k\otimes\id)(\tau\otimes\id)(\id\otimes\phi)\phi=
m(\k\otimes\id)\phi=1\e.
\end{split}
\end{equation*}
Similarly, it follows that
$m_n(\id\otimes\k_n^+)\phi=1\e. $
To complete the proof, let us observe that
\begin{multline*}
\k_n^+=(\e\otimes\k_n^+)\phi=m_n(m_n\otimes\id)(\k_n^-\otimes\id\otimes\k_n^+)
(\phi\otimes\id)\phi\\
=m_n(\id\otimes m_n)(\k_n^-\otimes\id\otimes\k_n^+)(\id\otimes\phi)\phi
=m_n(\k_n^-\otimes 1\e)\phi=k_n^-.
\end{multline*}
The map $k_n=k_n^\pm$ is bijective.
Its inverse is given by
$$
\k_n^{-1}\k=(\e\otimes\id)\sigma^{-1}\sigma_n\phi=(\id\otimes\e)\sigma^{-1}
\sigma_n\phi.\qed
$$
\renewcommand{\qed}{}
\end{pf}

From the point of view of this analysis, the group $G_0$ is particularly
interesting.
For example, left-covariant first-order differential structures
over $G$ (braided counterparts of structures considered in
\cite{WW}) are in a natural bijection with certain
right $\cal{A}_0$-ideals $\cal{R}
\subseteq\ker(\e)$.
Informally speaking, $G_0$ is interpretable
as a `maximal braided simplification' of $G$, with the same
coalgebra structure.
It is a standard braided-Hopf algebra.

If $G$ is a standard braided-Hopf algebra then the counit is
multiplicative.
Interestingly, the converse is also true.
\begin{lem}
The following properties are equivalent
\begin{align}
\e m&=\e\otimes\e\label{M1}\\
(\e\otimes\id)\sigma&=\id\otimes\e\label{ML}\\
(\id\otimes\e)\sigma&=\e\otimes\id\label{MR}\\
\sigma&=\tau.\label{M3}
\end{align}
\end{lem}
\begin{pf}
Equality \eqref{M3} implies \eqref{M1},
according to \eqref{228}.
If \eqref{M1} holds then \eqref{213}--\eqref{214}
imply \eqref{ML}--\eqref{MR}.
Finally, if \eqref{ML} (or
\eqref{MR}) holds \eqref{216} implies that
two flip-over operators coincide.
\end{pf}

In other words, the above listed conditions characterize the theory of
\cite{Maj}.
Indeed, $\sigma=\tau$ implies that the whole system $\cal{F}$
reduces to a single braiding $\sigma$, and all maps
appearing in the game are understandable as morphisms in
a braided category generated by $\cal{A}$ and $\sigma$.

In the standard theory \cite{Maj} all computations can be performed
diagramatically, drawing braid and tangle diagrams.
A similar situation holds here, in the general multi-braided framework.
The only difference is that diagrams should be appropriately refined,
by identifying each separate
braiding (labeling them by integers), and properly expressing
all the derived algebraic properties.

\newpage
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\end{document}