\documentstyle[12pt]{article}
\textwidth 6.1in 
\textheight 9in 
\oddsidemargin .3in 
\evensidemargin .3in 
\begin{document}

\begin{center}
{\Large BRAIDED DIFFERENTIAL BIALGEBRAS}
\end{center}

\vspace{.5cm}

\begin{center}
{\large A.A.VLADIMIROV}
\end{center}
\begin{center}
\large{Bogolubov Laboratory of Theoretical Physics, \\
Joint Institute for Nuclear Research, \\
Dubna, Moscow region 141980, Russia}
\end{center}

\begin{center}
E-mail: \ alvladim@thsun1.jinr.dubna.su
\end{center}

\vspace{1cm}

\begin{center}
ABSTRACT
\end{center}

A possibility to introduce coproduct into $q$-deformed
differential complexes on quantum spaces is investigated and
illustrated by several examples.

\vspace{1cm}

{\bf 1.} In the present note we discuss, considering typical examples,
some algebraic structures specific to differential calculi
on quantum spaces. Our main objective is to define a coproduct
$\Delta$ on the differential complex, i.e. on the whole
$q$-deformed algebra of coordinate functions and their differentials.
This standpoint has been prompted by Brzezinski's paper~\cite{Br},
where the existence of a proper coproduct was shown to imply
bicovariance of the corresponding differential calculus~\cite{Wo,Ju,AC}.
In what follows, we adopt the existence of $\Delta$ as a guideline of
our approach to constructing (or selecting) differential calculi on
arbitrary quantum spaces. In other words, we are seeking for
differential bialgebras~\cite{Mal,Man}.

Below, we are dealing with coproducts of the two following types: 
multiplicative, \begin{equation} \Delta(u^i_j)=u^i_k\otimes u^k_j\,, 
\label{1} \end{equation} with $u$ being a matrix, and additive, 
\begin{equation}
\Delta(x)=x\otimes 1+1\otimes x\,, \label{2}
\end{equation}
which is also known as coaddition. In both cases, $\Delta$ for the
differential 1-forms is introduced as follows. If
$\Delta(a)=a_{(1)}\otimes a_{(2)}$, then~\cite{Br}
\begin{equation}
\Delta(da)=
da_{(1)}\otimes a_{(2)}+a_{(1)}\otimes da_{(2)}\,. \label{3}
\end{equation}
For the multiplicative case (\ref{1}), this choice of $\Delta(da)$
is closely connected with the undeformed (graded) nature of the
Leibnitz rule,
\begin{equation}
d\,(ab)=da\,b+(-)^ka\,db \label{4}
\end{equation}
($k$ is the degree of the form $b$), which is used, together with the 
nilpotency property $d^2=0$, in the $q$-deformed differential calculus.
A possibility to deform the Leibnitz rule as well is briefly discussed
in the last section.

Differential bialgebras (actually, Hopf algebras) of the type 
considered in this paper are readily seen to require nontrivial
braiding map $\Psi \neq P$~\cite{Ma1,Ma2,IV,Vl} and, therefore,
nontrivial multiplication rules in the tensor product of the two
independent copies of a quantum space~\cite{Ma3}:
\begin{equation}
(1\otimes a)\,(b\otimes 1)=\Psi(a\otimes b)\neq b\otimes a\,.
\label{5}
\end{equation}
Below, we describe several examples of such braided differential
bialgebras.

\vspace{.3cm}

{\bf 2.} Let us first consider the case of braided
matrix algebra $BM_q(N)$ with the generators $\{1,u_j^i\}$ (the latter
form the $N\!\times\! N$-matrix $u$) and relations 
\begin{equation}
R_{21}u_2R_{12}u_1=u_1R_{21}u_2R_{12}\,,  \label{6}
\end{equation}
where $R$ is the $GL_q(N)$ $R$-matrix~\cite{FRT,Ji}. The multiplication
rule (\ref{6}) is invariant under adjoint coaction of $GL_q(N)$,
\begin{equation}
u^i_j\rightarrow T^i_mS(T^n_j)\otimes u^m_n\,, \ \ {\rm or}\ \
u\rightarrow TuT^{-1}\,, \label{7}
\end{equation}
where $T^i_j$ obey the relations
\begin{equation}
R_{12}T_1T_2=T_2T_1R_{12} \label{8}
\end{equation}
and commute with $u^m_n$\,. Eq.(\ref{6}) (`reflection equations')
first appeared in the course of investigations of 2-dimensional
integrable models on a half-line (see~\cite{KS} and references
therein). Further it was studied by Majid~\cite{Ma1} within the
general framework of braided algebras.

Note that the quantum group $GL_q(N)$ (\ref{8}) itself is not
invariant under (\ref{7}), so $BM_q(N)$ can be viewed as its covariant
(and braided) counterpart~\cite{Ma3}. Unlike the adjoint action
of the classical Lie group upon itself, in the quantum case the
`quantum symmetry group' (quantum group $GL_q(N)$) and the object of 
symmetry transformation, `quantum manifold' ($GL_q(N)$-covariant
quantum space $BM_q(N)$), do not coincide, as if `splitted'
through the q-deformation.

From now on we prefer to use the following notation~\cite{IP,IV}:
\begin{equation}
P_{12}R_{12}\equiv\hat{R}_{12}\equiv R\,, \ \ \ \hat{R}_{23}\equiv R'\,,
\ \ \ R^{-1}\equiv\overline{R}\,, \ \ \ q^{-1}\equiv \bar{q}\,, \ \ \ 
q-\bar{q}\equiv \lambda \,, \label{9} 
\end{equation} and also, for any $a$, \begin{equation}
a_1\equiv a\,, \ \ \ a_2\equiv a'\,,\ \ \ a_3\equiv a''\,,\ \ \
a\otimes 1\equiv a\,,\ \ \ 1\otimes a\equiv \tilde{a}\,. \label{10}
\end{equation} For instance, the Yang-Baxter equation and the Hecke
condition for the $R$-matrix look now, respectively, \begin{equation}
R\,R'\,R=R'\,R\,R' \label{11}
\end{equation}
and
\begin{equation}
R-\overline{R}=\lambda \ \ \ \ {\rm or} \ \ \ \
R^2=1+\lambda R\,. \label{12}
\end{equation}

Differential complex on $BM_q(N)$ is defined by the following set
of equations~\cite{OSWZ,AKR}: 
\begin{equation}
\label{13} \left\{ \begin{array}{l}
R\,u\,R\,u=u\,R\,u\,R\,, \\
R\,u\,R\,du=du\,R\,u\,\overline{R}\,, \\
R\,du\,R\,du=-du\,R\,du\,\overline{R}\,.
\end{array}  \right.
\end{equation}

As it is found in~\cite{IV}, the map $\Delta$ of the form (\ref{1}),
(\ref{3}), i.e.,
\begin{equation}
\Delta(u)=u\,\tilde{u}\,, \ \ \ 
\Delta(du)=du\,\tilde{u}+u\,d\tilde{u}\,, \label{14} \end{equation} 
yields a proper coproduct for the whole
algebra (\ref{13}), provided one of the following two sets of the
braiding relations is used:
\begin{equation}
\label{15}
 \;\;\;\;\;  \left\{
\begin{array}{l}
 \overline{R}\,\tilde{u}\,R\,u=u\,\overline{R}\,\tilde{u}\,R\,, \\
 \overline{R}\,d\tilde{u}\,R\,u=u\,\overline{R}\,d\tilde{u}\,R\,, \\
 \overline{R}\,\tilde{u}\,R\,du=du\,\overline{R}\,\tilde{u}\,R\,,   \\
 \overline{R}\,d\tilde{u}\,R\,du=-du\,\overline{R}\,d\tilde{u}\,R\,,
\end{array}
\right.
\end{equation}
or
\begin{equation}
\label{16}
 \left\{
\begin{array}{l}
 \overline{R}\,\tilde{u}\,R\,u=u\,R\,\tilde{u}\,R\,, \\
 \overline{R}\,d\tilde{u}\,R\,u=u\,R\,d\tilde{u}\,R\,, \\
 \overline{R}\,\tilde{u}\,R\,du=du\,R\,\tilde{u}\,R\,, \\
 \overline{R}\,d\tilde{u}\,R\,du=-du\,R\,d\tilde{u}\,R\,.
\end{array}
\right.
\end{equation}
Therefore, $BM_q(N)$ becomes a differential bialgebra.

\vspace{.3cm}

{\bf 3.} In this section we briefly describe the general 
method~\cite{Vl} of introducing additive coproduct (\ref{2}) into the 
$q$-deformed differential complexes on the quantum spaces related to 
the Hecke-type $R$-matrices. Principal ideas of this method can be best 
explained by considering the well accustomed quantum hyperplane 
\begin{equation}
R_{12}\,x_1\,x_2=q\,x_2\,x_1\,.  \label{17}
\end{equation}
The corresponding differential complex~\cite{WZ} is defined by
\begin{equation} \label{18}
\left\{ \begin{array}{l}
R\,x\,x'=q\,x\,x'\,, \\
R\,dx\,x'=\bar{q}\,x\,dx'\,, \\
R\,dx\,dx'=-\bar{q}\,dx\,dx'\,.
\end{array} \right.
\end{equation}
Adding formally to this set of equations an extra one,
\begin{equation}
dx\,x'=q\,\overline{R}\,x\,dx'-\lambda \,q\,dx\,x'\,, \label{19}
\end{equation}
which trivially follows from the second line in (\ref{18}), one can
recast (\ref{18}),(\ref{19}) into the matrix form
\begin{equation}
\chi _2\,\chi _1'=Y_{12}\,\chi _1\,\chi _2'\,, \label{20}
\end{equation}
where
\begin{equation}
\chi =\left( \begin{array}{c} x \\ dx \end{array} \right)\,, \ \ \
Y_{12}=q \left( \begin{array}{cccc} \overline{R}&\cdot&\cdot&\cdot \\
\cdot&\overline{R}&-\lambda &\cdot \\ \cdot&\cdot&R&\cdot \\ \cdot&
\cdot&\cdot&-R \end{array} \right)\,, \label{21}
\end{equation}
and the dots denote zeros. 

Now we are to demonstrate that the differential complex (\ref{18}) 
admits coaddition of the form (\ref{2})  \begin{equation} \Delta(x)= 
x+\tilde{x}\,, \ \ \ \ \Delta(dx)=dx+d\tilde{x}\,, \label{22} 
\end{equation}
or, in short notation,
\begin{equation}
\Delta(\chi )=\chi +\tilde{\chi}\,. \label{23}
\end{equation}
A natural Ansatz for the braiding is
\begin{equation}
\tilde{\chi }_2\,\chi _1'=Z_{12}\,\chi _1\,\tilde{\chi }_2'\,,
\label{24} \end{equation}
where $Z$ is a $4\!\times\!4$-matrix whose elements may themselves
depend on $R$.

The first restriction on $Z$ is caused by the graded nature of the
differential complex (\ref{18}). This leads to
\begin{equation}
Z_{12}=\left( \begin{array}{cccc} \alpha &\cdot &\cdot&\cdot \\
\cdot&\gamma &\delta &\cdot \\ \cdot&\mu &\beta &\cdot \\
\cdot&\cdot&\cdot&\nu \end{array} \right)\,. \label{25}
\end{equation}
Further, the result of external differentiation of (\ref{24}) must be
consistent with (\ref{24}) itself. Taking into account $d^2=0$ and the
graded Leibnitz rule, we come to
\begin{equation}
\alpha =\beta +\delta \,, \ \ \ \gamma =\delta -\nu \,, \ \ \
\mu =\beta +\nu \,. \label{26}
\end{equation}
The next step is to ensure the key property of $\Delta$, i.e.
\begin{equation}
\Delta(\chi _2)\,\Delta(\chi _1')=Y_{12}\,\Delta(\chi _1)\,\Delta(\chi
_2')\,. \label{27}
\end{equation}
With the help of (\ref{24}), this boils down to 
\begin{equation}
\ [Y_{12}\,Z_{21}+(Y_{12}-Z_{12})P_{12}-{\bf 1}]\,\chi _2\,
\tilde{\chi }_1'=0\,, \label{28}
\end{equation}
producing new constraints:
\begin{equation}
\beta =(\delta +1)\,q\,R\,, \ \ \ \ (\nu +1)(R+\bar{q})=0\,. \label{29}
\end{equation}

At last, we must guarantee that our braiding (\ref{24}) obeys so-called
hexagon identities~\cite{Ma4} or, equivalently, that our
commutation rules for elements with and without a tilde are
associative. To do this, we perform a reordering
\begin{equation}
\tilde{\chi }_3\,\chi _2'\,\chi _1''\rightarrow \chi _1\,\chi _2'\,
\tilde{\chi }_3'' \label{30}
\end{equation}
in two different ways, using (\ref{20}), (\ref{24}) and
\begin{equation}
\chi _2'\,\chi _1''=Y'_{12}\,\chi _1'\,\chi _2''\,, \ \ \ \ \
\tilde{\chi }_2'\,\chi _1''=Z'_{12}\,\chi _1'\,\tilde{\chi }_2''\,,
\label{31} \end{equation}
where $Y'$ and $Z'$ mean that a substitution $R\rightarrow R'$ in the
corresponding elements of $Y$ and $Z$ has to be carried out. Following
this strategy, we obtain
\begin{equation}
Y'_{12}\,Z_{13}\,Z'_{23}=Z_{23}\,Z'_{13}\,Y_{12}\,. \label{32}
\end{equation}

Finally, we come to the following two possibilities for $Z$:
\begin{equation}
Z_{12}^{(1)}=\bar{q} \left( \begin{array}{cccc}
\overline{R}&\cdot &\cdot&\cdot \\ \cdot&\overline{R}&
-\lambda &\cdot \\ \cdot&\cdot &R &\cdot \\
\cdot&\cdot&\cdot&-R  \end{array} \right), \ \ \ \
Z_{12}^{(2)}=\overline{R} \left( \begin{array}{cccc}
\bar{q}&\cdot &\cdot&\cdot \\ \cdot&\bar{q}&
-\lambda &\cdot \\ \cdot&\cdot &q &\cdot \\
\cdot&\cdot&\cdot&-q  \end{array} \right)\,.  \label{39}
\end{equation}
In the explicit form this reads:
\begin{equation}
\label{40}
\left\{ \begin{array}{l}
\tilde{x}\,x'=\bar{q}\,\overline{R}\,x\,\tilde{x}'\,,  \\
d\tilde{x}\,x'=\bar{q}\,\overline{R}\,x\,d\tilde{x}'
-\lambda \,\bar{q}\,dx\,\tilde{x}'\,,  \\
\tilde{x}\,dx'=\bar{q}\,R\,dx\,\tilde{x}'\,,  \\
d\tilde{x}\,dx'=-\bar{q}\,R\,dx\,d\tilde{x}'\,;
\end{array} \right.
\end{equation}

\vspace{.3cm}

\begin{equation}
\label{41}
\left\{ \begin{array}{l}
\tilde{x}\,x'=\bar{q}\,\overline{R}\,x\,\tilde{x}'\,,  \\
d\tilde{x}\,x'=\bar{q}\,\overline{R}\,x\,d\tilde{x}'
-\lambda \,\overline{R}\,dx\,\tilde{x}'\,,  \\
\tilde{x}\,dx'=q\,\overline{R}\,dx\,\tilde{x}'\,,  \\
d\tilde{x}\,dx'=-q\,\overline{R}\,dx\,d\tilde{x}'\,.
\end{array} \right.
\end{equation}

Two other solutions, characterized by the matrices  
$\overline{Z}_{21}^{(1)}$ and $\overline{Z}_{21}^{(2)}$ instead of
(\ref{39}), evidently correspond to changing the position of a
tilde ($\tilde{\chi }\leftrightarrow \chi , \tilde{x}\leftrightarrow
x$) in (\ref{24}), (\ref{40}) and (\ref{41}), i.e., to the inverse
braiding transformation $\Psi^{-1}$. These results~\cite{IV,Vl} 
exhaust all the allowed
braiding relations within the homogeneous Ansatz (\ref{24}).

\vspace{.3cm}

{\bf 4}. Here we outline the results~\cite{IV,Vl} 
of the application of the same method to the differential complex 
(\ref{13}) of the braided matrix algebra $BM_q(N)$.

Let us rewrite(\ref{13}) in the form
\begin{equation}
\varphi _2\,R\,\varphi _1=V_{12}\,\varphi _1\,R\,\varphi _2\,R\,,
\label{42} \end{equation}
where
\begin{equation} \label{43}
\varphi =\left( \begin{array}{c} u \\ du \end{array} \right), \ \ \ \
V_{12}= \left( \begin{array}{cccc}
\overline{R}&\cdot &\cdot&\cdot \\ \cdot&R&
\cdot &\cdot \\ \cdot&-\lambda  &\overline{R} &\cdot \\
\cdot&\cdot&\cdot&-R  \end{array} \right),
\end{equation}
and assume the braiding relations to be
\begin{equation}
\tilde{\varphi} _2\,R\,\varphi _1=W_{12}\,\varphi _1\,R\,
\tilde{\varphi _2}\,R\,. \label{44}
\end{equation}
We wish that
\begin{equation}
\Delta(\varphi )=\varphi +\tilde{\varphi } \label{45}
\end{equation}
be a consistent coproduct. A solution of the key equation
\begin{equation}
(V_{12}\,W_{21}-{\bf 1})\,\varphi _2\,R\,\tilde{\varphi }_1+[\lambda
V_{12}\,W_{21}+(V_{12}-W_{12})P_{12}]\,\varphi _2\,R\,\tilde{\varphi
}_1\,R=0\,.  \label{47}
\end{equation}
is
\begin{equation}
W_{12}=\overline{V}_{21}= \left( \begin{array}{cccc}
R&\cdot &\cdot&\cdot \\ \cdot&R&
\lambda &\cdot \\ \cdot&\cdot &\overline{R} &\cdot \\
\cdot&\cdot&\cdot&-\overline{R}  \end{array} \right)\,,
\label{48}
\end{equation}
or, in the component form, 
\begin{equation} \label{56} \left\{ \begin{array}{l} 
\tilde{u}\,R\,u=R\,u\,R\,\tilde{u}\,R\,, \\
d\tilde{u}\,R\,u=R\,u\,R\,d\tilde{u}\,R+\lambda
\,du\,R\,\tilde{u}\,R\,, \\
\tilde{u}\,R\,du=\overline{R}\,du\,R\,\tilde{u}\,R\,, \\
d\tilde{u}\,R\,du=-\overline{R}\,du\,R\,d\tilde{u}\,R\,.
\end{array} \right.
\end{equation}
Other solutions are given in~\cite{IV,Vl}.

\vspace{.3cm}

{\bf 5.} In all the above examples the undeformed Leibnitz rule 
(\ref{4}) is implied. However, the problem of its consistent
$q$-deformation is highly intriguing. In~\cite{FP}, a deformed version 
of the Leibnitz rule is used to build a proper differential complex on 
$SL_q(N)$. In this section, we investigate the possibility to construct
differential bialgebras based on a deformed Leibnitz rule.

For additive $\Delta$ (\ref{2}) this can be achieved rather easily.
Let us consider, for example, $BM_q(N)$ (first eq. of (\ref{13})) and
admit the Leibnitz rule of the following type:
\begin{equation}
d(u\,R\,u)=du\,R\,u+u\,Q\,du\,, \ \ d(du\,R\,u)=-du\,Q\,du\,, \label{70}
\end{equation}
where $Q$ can be any function of $R$. A consistent set of relations
analogous to (\ref{13}),(\ref{56}) now reads:
\begin{equation}
\label{71} \left\{ \begin{array}{l}
R\,u\,R\,u=u\,R\,u\,R\,, \\
R\,u\,Q\,du=du\,R\,u\,\overline{R}\,, \\
R\,du\,Q\,du=-du\,Q\,du\,\overline{R}\,;
\end{array}  \right.
\end{equation}
\begin{equation} \label{72} \left\{ \begin{array}{l} 
\tilde{u}\,R\,u=R\,u\,R\,\tilde{u}\,R\,, \\
d\tilde{u}\,R\,u=R\,u\,Q\,d\tilde{u}\,R+\lambda
\,du\,R\,\tilde{u}\,R\,, \\
\tilde{u}\,Q\,du=\overline{R}\,du\,R\,\tilde{u}\,R\,, \\
d\tilde{u}\,Q\,du=-\overline{R}\,du\,Q\,d\tilde{u}\,R\,.
\end{array} \right.
\end{equation}

Unlike this, the multiplicative coproduct seems to be incompatible
with the deformed Leibnitz rule. At present, I have some preliminary
arguments supporting (but not a sistematic proof of) this statement.
Namely, it is hardly possible (impossible, if the $GL_q(N)$
covariance is required) to write down coassociative expression for
$\Delta(du)$ except for (\ref{14}) which, as it is, strongly
suggests the undeformed version of the Leibnitz rule. Really, when
working with (\ref{14}), it is hardly possible (impossible within
natural Ans\"{a}tze like (\ref{13})) to write down a consistent set
of commutational relations for the differential complex endowed with 
both the bialgebra structure and the deformed Leibnitz rule. However, 
the whole subject of $q$-deformations of the Leibnitz rule apparently 
deserves further investigation.

\vspace{.5cm}

I would like to thank A.Isaev (in collaboration with whom part of
these results was obtained) and P.Pyatov for fruitful discussions.

\begin{thebibliography}{99}
\bibitem{Br} T.Brzezinski, Lett.Math.Phys. 27 (1993) 287.
\bibitem{Wo} S.L.Woronowicz, Comm.Math.Phys. 122 (1989) 125.
\bibitem{Ju} B.Jur\v{c}o, Lett.Math.Phys. 22 (1991) 177.
\bibitem{AC} P.Aschieri and  L.Castellani, Int.J.Mod.Phys. A8 (1993)
1667.
\bibitem{Mal} G.Maltsiniotis, C.R.Acad.Sci.Paris 331 (1990) 831;
Comm.Math.Phys. 151 (1993) 275.
\bibitem{Man} Yu.I.Manin, Teor.Mat.Fiz. 92 (1992) 425.
\bibitem{Ma1} S.Majid, J.Math.Phys. 32 (1991) 3246;
ibid. 34 (1993) 1176.
\bibitem{Ma2} S.Majid, J.Math.Phys. 34 (1993) 2045.
\bibitem{IV} A.P.Isaev and A.A.Vladimirov, JINR preprint E2-94-32,
Dubna, 1994.
\bibitem{Vl} A.A.Vladimirov, JINR preprint E2-94-39, Dubna, 1994,
to appear in J.Phys.A.
\bibitem{Ma3} S.Majid, Beyond supersymmetry and quantum symmetry
(an introduction to braided groups and braided matrices),
in: M.-L. Ge and H.J. de Vega, editors, Quantum Groups, Integrable
Statistical Models and Knot Theory, pp. 231-282, World.Sci., 1993.
\bibitem{FRT} L.D.Faddeev, N.Yu.Reshetikhin and L.A Takhtajan,
Algebra i Analiz 1 Vol.1 (1989) 178; English transl: Leningr.Math.J.
1 (1990) 193.
\bibitem{Ji} M.Jimbo, Lett.Math.Phys. 11 (1986) 247.
\bibitem{KS} P.P.Kulish and E.K.Sklyanin, J.Phys. A 25 (1992) 5963;
P.P.Kulish and R.Sasaki, Progr.Theor.Phys. 89 (1993) 741.
\bibitem{IP} A.P.Isaev and P.N.Pyatov, JINR preprint
E2-93-416, Dubna, 1993.
\bibitem{OSWZ} O.Ogievetsky, W.B.Schmidke, J.Wess and
B.Zumino, Comm.Math.Phys. 150 (1992) 495.
\bibitem{AKR} J.A. de Azcarraga, P.P.Kulish and F.Rodenas, Valencia
University preprint FTUV 93-36, 1993; P.P.Kulish, Valencia University
preprint FTUV 93-54, 1993.
\bibitem{WZ} J.Wess and B.Zumino, Nucl.Phys. B (Proc.Suppl.) 18B
(1990) 302.
\bibitem{Ma4} S.Majid, Int.J.Mod.Phys. A 5 (1990) 1.
\bibitem{FP} L.D.Faddeev and P.N.Pyatov, 1994, to be published in the 
memorial volume of F.A.Berezin.
\end{thebibliography} \end{document}