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\begin{document}
\title{Left-covariant Differential Calculi\\
on $SL_q(N)$}
\date{}\author{}
\maketitle
{\large
 \begin{center}
K.\,Schm\"udgen and A.\,Sch\"uler,\\
Fakult\"at f\"ur Mathematik und Informatik,\\
Universit\"at Leipzig, Augustusplatz 10,\\
04109 Leipzig, Federal Republic of Germany
\footnote{e-mail: schmuedgen@mathematik.uni-leipzig.de, schueler@rz.uni-leipzig.de}
\end{center}
}
${}$\medskip\\
\begin{abstract}
\noindent
We study $N^2-1$ dimensional left-covariant differential calculi on
the quantum group $SL_q(N)$. In this way we obtain four classes
of differential calculi which are algebraically much simpler as the bicovariant calculi.
The algebra generated by the left-invariant vector fields has 
only quadratic-linear  relations and posesses
a Poincar\'e-Birkhoff-Witt basis.\\
We use the concept of universal (higher order) differential calculus associated
with a given left-covariant first order differential calculus. It turns out that the space
of left-invariant $k$-forms has the dimension $N^2-1\choose k$ as in the case of
the corresponding classical Lie group $SL(N)$.
 \medskip\\ \noindent
{\bf Keywords:} 
noncommutative differential calculus, quantum Lie
algebra\\
{\bf MSC (1991):} 17\,B\,37, 46\,L\,87, 81\,R\,50.
\end{abstract}
\vskip 50pt
\thispagestyle{empty}
\newpage
\section{Introduction}
After the seminal work \cite{woro122} of S.\,L.\,Woronowicz, bicovariant
differential calculi on quantum groups (Hopf algebras) have been
extensively
studied in the literature. There is a well developed general theory of such
calculi. Bicovariant differential calculi on the quantum group $SL_q(N),
N\ge3$,
have been recently classified in \cite{ss1}.  All calculi occuring in this classification
have dimension $N^2$, i.\,e. their dimension does not coincide with the
dimension $N^2-1$ of the corresponding classical Lie group. On the other
hand, the first example of a non-commutative differential calculus on a
quantum group was Woronowicz' $3D$-calculus on $SU_q(2)$ \cite{woro}.
This is a three
dimensional left-covariant calculus which is not bicovariant. The
$3D$-calculus is algebraically much simpler and in many respects nearer
to the
classical differential calculus on $SU_q(2)$ than the four dimensional
bicovariant calculi on  $SU_q(2)$. This motivates to look for $N^2-1$
dimensional left-covariant differential calculi on $SL_q(N)$.
The main aim
of our approach is to follow the classical situation as close as possible.
Let us brief\/ly explain the basic idea of the approach given in this paper.
As in \cite{ss1}, we assume
that the differentials $\d u^i_j$ of the matrix entries $u^i_j$ generate the
left module of 1-forms. Hence the differential $\d$ can be expressed as
$\d a=\sum (X_{ij}\ast a)\omega (u^i_j)$ for $a\in SL_q(N)$, where
$\omega (u^i_j):= \sum_n\kappa (u^i_n) \d u^n_j$ are the left-invariant
Maurer-Cartan
forms and $X_{ij}$ are linear functionals on $SL_q(N)$ such that $X_{ij}
(1)=0$. In our approach
the functionals $X_{ij}$ will be chosen from the algebra of regular 
functionals $\UU_{R}$ as defined in \cite{frt}, 2.\,2.
We assume that the vector space of left-invariant 1-forms
has
dimension $N^2-1$ and that $X_{ij} (u^r_s)=\delta_{ir}\delta_{js}$ for
$i\ne j$. The main property of these calculi is that there are 
only quadratic-linear relations between the left-invariant vector fields.
The corresponding formulas show that these  calculi are
very close to the classical differential calculi on $SL(N)$ in many
respects.
Their
enveloping algebras of the Lie algebra have Poincar\'e-Birkhoff-Witt property.
The dimensions of the vector spaces of left-invariant
$k$-forms $\Gamm_{\rm inv}^{\wedge k}$
is equal to the corresponding dimension $N^2-1\choose k$ in the classical case.
Moreover, the associated higher order calculi $(\Gamm,\d)$ give the
ordinary calculus on $SL(N)$ in the limit $q\to 1$.\\
This paper is organized as follows. Section 2 contains some general
results
about left-covariant differential calculi on quantum groups which will
be needed later. 
In Section 3 we develop four classes of left-covariant first order differential calculi
$(\Gamm_r,\d)$, $r=1,\dots,4$. We start with the choice of left-invariant
functionals $X_\ig$ and compute all  relations between them.
% Moreover, there is a
%braiding matrix $\si$ has only eigenvalues $-1$ and $1$  with
%multiplicities $n\choose 2$ and $n+1\choose 2$, $n=N^2-1$, respectively.\\
%The structure constant $C=(C^\ig_{\jg\kg}) are antisymmetric with respect to
%$\si$ but there is no Jacobi identity.\\
In Sections 4 we  brief\/ly describe the construction of the
universal
higher order differential calculus associated with a given left-covariant
first order calculus on a quantum group.
%It is shown that the vector space of left-invariant $k$-forms has
%the classical dimension $N^2-1\choose k$.
%% then
%%the 2-form $\omega_{31}\wedge \omega_{13}$ vanishes and hence
%%the space
%%of 2-forms does not yield the corresponding space for the classical
%%differential calculus on $SL(3)$ when $q\rightarrow 1$.
We apply this construction to the calculi 
$(\Gamm_r,\d)$, $r=1,\dots,4$.\\
Throughout this paper $q$ is a nonzero complex number such that
$q^2\ne 1$ and we abbreviate $\lambda:=q-q^{-1}$.
% and $\lambda_+:=q+q^{-1}$.


\section{ Left-covariant Differential Calculi on Quantum Groups}
Our basic reference concerning differential calculi on quantum groups is 
\cite{woro122}.
Let $\Aa$ be a fixed Hopf algebra with comultiplication $\Delta$,
counit
$\ve$, antipode $\kappa$ and unit element\,$1$. Sometimes we use
Sweedler's notation 
$\Delta^{(n)} (a) = a_{(1)} \otimes a_{(2)}\otimes\cdots\otimes 
a_{(n+1)}.$
A {\it first order
differential calculus} (brief\/ly, a FODC) over ${\cal A}$ is a pair
$(\Gamm$,d)
of an ${\cal A}$-bimodule $\Gamm$ and a linear mapping 
$\d\colon\Aa\to\Gamm$
such that $\d(ab)=\d a{\cdot}b + a{\cdot}\d b$ for
$a, b\in\Aa$ and
$\Gamm ={\rm lin} \{ a\d b: a, b\in\Aa\}.$
A FODC $(\Gamm, \d)$ is called {\it left covariant} if there is a
linear mapping $\dl\colon\Gamm\to \Aa{\otimes} \Gamm$ for which
$\Delta_L (a {\rm d}b) = \Delta (a) ({\rm id} \otimes {\rm d}) 
\Delta (b), a, b\in{\cal A}$.\\
Suppose that $(\Gamm, {\rm d})$ is a left-covariant FODC over 
${\cal A}$. Recall that the canonical projection of $\Gamm$ into
$\gi=\{ \omega\in\Gamm:\dl (\omega) =
1\otimes\omega\}$ is defined by $\pin (\d a) = \kappa
(a_{(1)}) {\rm d}a_{(2)}$, cf. \cite{woro}. We abbreviate
$\omega(a) = \pin ({\rm d}a).$ Then ${\cal R} :=
\left\{x\in~{\rm ker}~\varepsilon :\omega (x) = 0\right\}$ is the
{\it right ideal} of $\ker\ve$ {\it associated with}
$(\Gamm, {\rm d})$.
Recall that $\Aa^\circ$ denotes the Hopf algebra of representative linear 
functionals on $\Aa$. If $\Aa$ is Hopf $^\ast$-algebra then $\Aa^\circ$ becomes  a
Hopf $^\ast$-algebra via $f^\ast(a)=\overline{f(\kappa(a)^\ast)}$,
$f\in\Aa^\circ$, $a\in\Aa$.
The vector space
${\cal X} := \{ X\in{\cal A}^\circ : X(1) = 0\mbox{ and } X(a) = 0
 \mbox{ for } a\in\RR \}$ is called the {\it quantum Lie algebra} of the
FODC  $(\Gamm, {\rm d}).$
%%%%%%%
\begin{lemma}\label{l1}
A vector space $\XX$ of linear functionals
on ${\cal A}$ is the quantum Lie algebra of a left-covariant FODC
$(\Gamm, {\rm d})$ if and only if $X(1) = 0$ and $\Delta X -
1{\otimes} X \in\XX{\otimes }\Aa^\circ$ for all $X\in{\cal X}$.
\end{lemma}
\begin{proof}
The necessity of the condition $\Delta X -1{\otimes}
X\in{\cal X}{\otimes} {\cal A}^\circ$ follows at once from formula (5.20)
 in \cite{woro122}: $\Delta(X_i)=\sum_jX_j{\otimes}f^j_i+1{\otimes}X_i$.
To prove the sufficiency part, let us note that the above conditions 
imply
that ${\cal R} := \{ a\in\ker\ve : X(a) = 0 \mbox{  for  }
X\in{\cal X}\}$ is a right ideal of $\ker\ve$. ${}$From the
general
theory (cf. Theorem 1.5 in \cite{woro122}) we conclude easily that ${\cal R}$ is
the
right ideal associated with some left-covariant FODC $(\Gamm,\d)$ over ${\cal A}$. 
Since $\XX$ is finite
dimensional, $\XX^{\perp\perp}=\XX$. Hence, $\XX$ is the quantum Lie algebra
of $(\Gamm,\d)$.
\end{proof}
Obviously, if $\XX$ is a quantum Lie algebra, so is $\XX^\ast$.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Left-covariant first order Differential Calculi on $SL_q(N)$}

Let $\ell^+=({\ell^+}^i_j)$ and $\ell^-=({\ell^-}^i_j)$ be the $N\times N$ matrices of
linear functionals ${\ell^+}^i_j$ and ${\ell^-}^i_j$ on ${\cal A} = SL_q(N)$ as defined
in \cite{frt}. Recall that $\ell^\pm$ is uniquely determined by the properties that
$\ell^\pm :{\cal A}\rightarrow M_N(\bbbc)$ is a unital algebra homomorphism and
${\ell^\pm}^i_j (u^n_m)=p^{\mp 1}(\hat{R}^{\pm 1})^{in}_{mj}$ for $i,j,n,m{=}1,{\dots},
N$, where $p$ is an $N$-th root of $q$, $\hat{R}$ is the $R$-matrix
of the quantum group $SL_q(N)$, and $u=(u^n_m)$ is the fundamental matrix
corepresentation of $SL_q(N)$.
In this section we define $N^2-1$ dimensional left-covariant FODC
$(\Gamm_r,\d)$, $r=1,\dots,4$, over ${\cal A}=SL_q(N)$.
We set
\bea\label{xxx}
X_{ij}&=&-\lam{\ell^+}^j_j{\ell^-}^j_i,\quad X_{ji}=
q^{2\delta_{jN}}\lam g_i{\ell^-}^i_i{\ell^+}^i_j,\,i<j,\\
X_i&=&q^{-1}\lam(g_i-1), i=1,\dots,N-1,\,g_i=(\lp^i_i\lm^N_N)^2
\quad \mbox{ for } r=1,\\
X_{ij}&=&-q^{-2\delta_{i1}}\lam h_j\lp^j_j\lm^j_i,\quad X_{ji}=
\lam\lm^i_i\lp^i_j,\,i<j,\\
X_j&=&q\lam(1-h_j), j=2,\dots,N,\, h_j=(\lp^1_1\lm^j_j)^2\quad\mbox{ for } r=2,\\
X_{ij}&=&q^{2\delta_{jN}}\lam\kappa(\lm^j_i)\lm^i_ig_i\quad X_{ji}=-\lam\kappa(\lp^i_j)\lp^j_j,\,i<j,\\
X_i&=&q^{-1}\lam(g_i-1), i=1,\dots,N-1,\quad \mbox{ for } r=3,\\
X_{ij}&=&\lam\kappa(\lm^j_i)\lm^i_i,\quad X_{ji}=-q^{-2\delta_{i1}}\lam \kappa(\lp^i_j)\lp^j_jh_j,\, i<j,\\
X_j&=&q\lam(1-h_j), j=2,\dots,N,\quad\mbox{ for } r=4.
\eea
Let $\XX_r$, $r=1,\dots,4$, denote the linear span of the functionals
$X_{ij},\,i\ne j,\,i,j=1,\dots, N$, and $X_n$, $n=1,\dots,N-1\,(2,\dots,N)$, for $r=1,3$ $(r=2,4)$.
\\
Computing the coproducts of these functionals we obtain for $r=1$ and $i<j$:
\bea\label{cop1}
\Delta X_{ij}&=&1\ot X_{ij}+\sum_{m=i}^{j-1}X_{mj}\ot\lp^j_j\lm^m_i,\\
\Delta X_{ji}&=&g_i\ot X_{ji}+\sum_{m=i+1}^j X_{mi}\ot g_i\lm^i_i\lp^m_j,\\
\Delta X_i&=&1\ot X_i+X_i\ot g_i,\, i=1,\dots, N-1,
\eea
for $r=2$ and $i<j$:
\bea\label{cop2}
\Delta X_{ij}&=&h_j\ot X_{ij}+\sum_{m=i}^{j-1}X_{mj}\ot h_j\lp^j_j\lm^m_i,\\
\Delta X_{ji}&=&1\ot X_{ji}+\sum_{m=i+1}^{j}X_{mi}\ot \lm^i_i\lp^m_j,\\
\Delta X_j&=&1\ot X_j+X_j\ot h_j,\, j=2,\dots,N.
\eea
These formulas show that $\Delta X-1\ott X\in\XX_r\ott \Aa^\circ$ for all $X\in\XX_r$, $r=1,2$.
The computations for $r=3,4$ are quite similar.
Therefore, by Lemma\,\ref{l1} the vector space $\XX_r$, $r=1,\dots,4$, defines
a left-covariant FODC over $\Aa$.\\
Using the pairing between $SL_q(N)$ and $\UU_R$ we verify 
that $X_{ij}(u^k_l)=\delta_{ik}\delta_{jl}$ for $i\ne j$ and 
$X_i(u^k_l)=\delta_{kl}(\delta_{ik}-q^{-2}\delta_{kN})$, $i=1,\dots,N-1$, $r=1,3$, and
$X_j(u^k_l)=\delta_{kl}(\delta_{jk}-q^2\delta_{1k})$, $j=2,\dots,N$, $r=2,4$.
\medskip\\
Now we will consider the $\ast$-structures. In case $SL_q(N,\bbbr)$, $| q |=1$, we
get $\XX_r^\ast=\XX_r$, $r=1,\dots,4$. Hence, all four calculi are $\ast$-calculi.
In case $SU_q(N)$, $q\in\bbbr$, we get $\XX_1^\ast=\XX_3$ and $\XX_2^\ast=\XX_4$.
Hence, none of them are $\ast$-calculi with respect to $SU_q(N)$.
\medskip\\
We shall study the calculus $\XX_3$ in more detail. Let
$\{\omega_{ij},\omega_n: i\ne j,\,n=1,\dots,N-1\}$ (resp. $\{u^i_j,u^n_n-1:
i\ne j, n=1,\dots,N-1\}$) be the dual
basis of $\gi$ (resp. of $\ker\ve/\RR)$ to the basis $\{X_{ij},X_n\}$ of $\XX$.
 \\
Let $\vt$ denote the Heaviside symbol, i.\,e. $\vt(m)=1$ if $m>0$ and 
$\vt(m)=0$ if $m\le 0$. The commutation rules between matrix entries and 
1-forms are as follows:
\bea\label{commxu}
\omega_{ij}u^l_k&=&q^{\delta_{ik}+\delta_{jk}-2\delta_{Nk}}u^l_k
\omega_{ij}+\la\vt(k-j)u^l_j\omega_{ik},\\
\omega_{ji}u^l_k&=&q^{-\delta_{ik}+\delta_{jk}}u^l_k\omega_{ji}-
\la\vt(i-k)u^l_i\omega_{jk},\\
\omega_iu^l_k&=&q^{2\delta_{ik}-2\delta_{kN}}u^l_k\omega_i+q\la\vt(k-i)u^l_i\omega_{ik}.
\eea
Note that the commutation rules are closed in the upper $(i<j)$
 and lower $(i>j)$
triangle of left invariant 1-forms, respectively, while the commutation
rules on the diagonal $\omega_i$ are {\it not} closed.\\
The associated right ideal $\RR_3$ of $\ker\ve$ is generated by the 
following set of quadratic-linear terms:
\bea\label{ideal}
&&u^i_ju^k_l\mbox{ for } i\ne j,\,i\ne l, \,k\ne j,\, k\ne l;\quad u^i_ju^j_i\mbox{ for }i\ne j;\\
&&u^i_au^a_j\mbox{ for } i>a,\,j>a \mbox{ and for }i<a,\,j<a;\\
&&u^i_au^a_j-\la u^i_j\mbox{ for }i<a<j \mbox{ and for } i>a>j;\\
&&u^a_au^i_j-q^{2\delta_{ia}-2\delta_{aN}}u^i_j\mbox{ for }i<j,
\quad u^a_au^i_j-u^i_j\mbox{ for }i>j;\\
&&(u^i_i-q^{-2\delta_{iN}})(u^i_i-q^{2-2\delta_{iN}}),\,i=1,\dots,N;\\
&&(u^i_i-1)(u^j_j-q^{-2\delta_{jN}})\mbox{ for }i<j,\,\VV-\ve(\VV).
\eea
Note that $\VV:=\sum_{i=1}^Nq^{2\delta_{iN}}u^i_i$ is the only linear 
element of $\Aa$
that is annihilated by $\XX_3$.\\
We abbreviate
$q_j:=q^{2\delta_{jN}}$ and $\quad q_{nij}:=q^{2(\delta_{ni}-\delta_{nj}
+\delta_{jN})}$. Throughout let $i<j<k<l$. Then the
commutation rules between the $\{X_\ig\}_{\ig\in I}$,
$I=\{(i,j),\,n\colon i\ne j,\,n=1,\dots,N-1\}$ are as follows:
\bea
&&X_{ij}X_{il}-qq_lX_{il}X_{ij}=0 ,\\
&&X_{il}X_{jl}-qX_{jl}X_{il}=0,\\
&&X_{ij}X_{jl}-qq_lX_{jl}X_{ij}-qq_l\la X_{il}X_j=q_lX_{il},\\
&&X_{ij}X_{kl}-q_lX_{kl}X_{ij}=0,\\
&&X_{jk}X_{il}-q_lX_{il}X_{jk}=0,\\
&&X_{ik}X_{jl}-q_lX_{jl}X_{ik}-q_l\la X_{il}X_{jk}=0,\\
\\
&&X_{ji}X_{li}-qX_{li}X_{ji}=0,\\
&&X_{li}X_{lj}-qX_{lj}X_{li}=0,\\
&&X_{ji}X_{lj}-\qm X_{lj}X_{ji}=-\qm X_{li},\\
&&X_{ji}X_{lk}-X_{lk}X_{ji}=0,\\
&&X_{kj}X_{li}-X_{li}X_{kj}=0,\\
&&X_{ki}X_{lj}-X_{lj}X_{ki}-\la X_{li}X_{kj}=0,\\
\\
\\
&&X_{ij}X_{li}-\qm\qlm X_{li}X_{ij}+\qlm\la X_{lj}X_i=-\qm\qlm X_{lj},\\
&&X_{il}X_{lj}-\qm\qlm X_{lj}X_{il}=X_{ij},\\
&&X_{ij}X_{lj}-\qm\qlm X_{lj}X_{ij}=0,\\
&&X_{ij}X_{lk}-\qlm X_{lk}X_{ij}=0,\\
&&X_{il}X_{kj}- X_{kj}X_{il}=0,\\
&&X_{ik}X_{lj}-\qlm X_{lj}X_{ik}+\qlm\la X_{lk}X_{ij}=0,\\
\\
&&X_{ji}X_{il}-qX_{il}X_{ji}=X_{jl},\\
&&X_{li}X_{jl}-\qm q_lX_{jl}X_{li}+q_l\la X_{ji}X_l=-\qm q_l X_{ji},\\
&&X_{ji}X_{jl}-\qm X_{jl}X_{ji}=0,\\
&&X_{ji}X_{kl}-X_{kl}X_{ji}=0,\\
&&X_{li}X_{jk}-q_lX_{jk}X_{li}=0,\\
&&X_{ki}X_{jl}-X_{jl}X_{ki}+\la X_{kl}X_{ji}=0,\\
\\
&&X_{il}X_{li}-\qlm X_{li}X_{il}=X_i-X_l,\\
\\
&&X_nX_m-X_mX_n=0,\\
&&X_nX_{ij}-q_{nij}X_{ij}X_n=\qm\lam(q_{nij}-1)X_{ij},\\
&&X_nX_{ji}-q_{nij}^{-1}X_{ji}X_n=\qm\lam(q_{nij}^{-1}-1)X_{ji}.
\eea
%%%%%%%
%%%%%%
Summarizing the preceding we obtain
\begin{enumerate}
\item $\XX_r$, $r=1,\dots,4$, is  an $N^2-1$ dimensional left-covariant FODC on
$SL_q(N)$.

\item The relations of the algebra $\UU_r$ generated by  $\XX_r$ are quadratic-linear. 
To be more precise,
there are complex matrices $\si=(\si^{\ig\jg}_{\kg\lg})$
and $C=(C^\kg_{\ig\jg})$,
$\ig,\jg,\kg,\lg\in I$,
% with $\si^{\ig\jg}_{\kg\lg}=0$ for $\ig\le\jg,\,\kg<\lg$ and for
%$\ig>\jg,\,\kg\ge\lg$ 
such that
\be\label{liealgebra}
X_\ig X_\jg-\sum_{\kg,\lg}\si^{\ig\jg}_{\kg\lg}X_\kg X_\lg=\sum_{\mg}C_{\ig\jg}^\mg X_\mg,\,\,\ig,\jg\in I.
\ee
The matrix $\si$ is diagonalizable and has eigenvalues $-1$ and $1$ with
multiplicities $N^2-1\choose 2$ and $N^2\choose 2$, respectively. The structure constants
$C$ are antisymmetric, i.\,e. $C\si^{\scriptscriptstyle{\sf t}}=-C$.

\item 
The algebra $\UU_r$ has a linear basis
\be\label{PBW}
\{X_{\ig_1}^{m_1}\dots X_{\ig_n}^{m_n}:
\ig_1<\ig_2<\dots<\ig_n,\,\ig_k\in I,m_k\in \bbbn_0,
\,k=1,\dots, n\},\quad n=N^2-1,
\ee
with respect to an arbitrary order of the index set $I$.
\end{enumerate}

%
%Note that
%there are only quadratic-linear relations between $X_\ig$ and $X_\jg$.
%Each relation contains at most three quadratic monomials. Fix an arbitrary
%linear order of $I$. We summarize the properties of the quantum Lie algebra
%in the following
%\begin{theo}\label{lie}
%Let $\XX_r$, $r=1,\dots,4$, one of the above quantum Lie algebras
%of $SL_q(N)$. Then there exist complex matrices $\si=(\si^{\ig\jg}_{\kg\lg})$
%and $C=(C^\kg_{\ig\jg})$,
%$\ig,\jg,\kg,\lg\in I$, with $\si^{\ig\jg}_{\kg\lg}=0$ for $\ig\le\jg,\,\kg<\lg$ and for
%$\ig>\jg,\,\kg\ge\lg$ such that
%\be\label{liealgebra}
%X_\ig X_\jg-\si^{\ig\jg}_{\kg\lg}X_\kg X_\lg=C_{\ig\jg}^\mg X_\mg,\,\,\ig,\jg\in I.
%\ee
%The matrix $\si$ is diagonalizable and has eigenvalues $-1$ and $1$ with
%multiplicities $N^2-1\choose 2$ and $N^2\choose 2$, respectively. The
%subalgebra of $\Aa^\circ$ generated by all $X_\rg$ has a linear basis
%\be\label{PBW}
%\{X_{\ig_1}^{m_1}X_{\ig_2}^{m_2}\dots X_{\ig_n}^{m_n}:
%\ig_1<\ig_2<\dots<\ig_n,\,
%(m_1\dots,m_n)\in\bbbn_0^n\},\quad n=N^2-1
%\nn\ee ${}$
%\end{theo}
%%%%
%Different order of index set $I$ yield to different matrices $\si$. But
%$\im (\si +\id)$ is allways the same. The matrix $\si$ does not fulfill the
%braid equation.
%%%%%%%%%%%%%%%%%%
\section{Higher Order Calculi}
Given a left-covariant FODC $(\Gamm,\d)$ on $\Aa$. 
%We brief\/ly describe
%the construction of an universal higher order differential calculus associated
%to $(\Gamm,\d)$. 
Let $\RR$ be the associated right ideal of $(\Gamm,\d)$ and $\SS\colon
\RR\to\gi\ott\gi$ the linear mapping defined by 
$\SS(r)=\omega(r_{(1)})\ot\omega(r_{(2)})$.
Let $S_2$ be the subbimodule of $\Gamm\ot_{\Aa}\Gamm$ generated 
by $\SS(\RR)$. Furthermore, let $S$ be the ideal of the tensor algebra 
$\gten=\sum_n\gtenn$
generated by $S_2$.\\
Then $\Gamm^\wedge=\gten/S$ is called {\it universal exterior algebra 
associated with $(\Gamm,\d)$}.
By construction, $\Gamm^\wedge$ is a left-covariant bimodule and a 
$\bzb$-graded algebra.
\begin{prop}\label{ext}
Let $(\Gamm,\d)$ be a left-covariant FODC over $\Aa$ and let $\gd$ be the 
universal exterior  algebra built over $(\Gamm,\d)$. Then there exists one
 and only one linear mapping
$ \d\colon\gd\to\gd$
such that
\begin{enumerate}
\rm\item\it $\d$ increases the grade by one.
\rm\item\it $\d$ extends $\d\colon\Aa\to \Gamm$.
\rm\item\it $\d$ fulfills the graded Leibniz rule.
\rm\item\it $\d^2=0$.
\rm\item\it $\d$ is left-covariant, i.\,e.
$\dl(\d\theta)=(\id\ot\d)\dl\theta$, $\theta\in\gd$.
\end{enumerate}${}$
\end{prop}
$(\gd,\d)$ is called {\it universal differential calculus associated with 
the left-covariant FODC $(\Gamm,\d)$}.
Woronowicz' construction of higher order calculi for {\it bicovariant} FODC
can be obtained by
taking a quotient of the universal differential calculus $(\gd,\d)$.\\
The mapping $\d{|_\gi}$ is  given  by $\d\omega(a)=
-\omega(a_{(1)})\wedge\omega(a_{(2)})$.\\
We apply the above construction to the left-covariant FODC
 $\XX_r$, $r=1,\dots,4$. Fix an arbitrary linear order of $I$.
%%%%%%%%%
\begin{theo}\label{high}
Let $(\gd,\d)$ be the universal left-covariant differential calculus
associated with one of the FODC $(\Gamm_r,\d)$, $r=1,\dots,4$. Then
\be\label{uni}
\dim( \Gamm^{\land k}_{\rm inv})={N^2-1\choose k},\quad k\ge1.\\
\ee
More precisely, the set $\{\omega_{\ig_1}\land\omega_{\ig_2}\land
\dots\land\omega_{\ig_k}:\ig_1<\dots<\ig_k,\,\ig_j\in 
I,\,j=1,\dots,k\}$
is a basis of the vector space $ \Gamm^{\land k}_{\rm inv}$.
\end{theo}

In particular, for $k=2$ Theorem\,\ref{high} states that the space
of left-invariant  2-forms is equal to $\ker\,(\si+\id)$ (where  $\si$ is the
linear mapping
$\gi\ot\gi\to\gi\ot\gi$ defined by $\omega_\ig\ott\omega_\jg
\mapsto\sum_{\kg,\lg}\si_{\ig\jg}^{\kg\lg}\omega_\kg\ott\omega_\lg$.)
In contrast, there are examples of three dimensional quadratically closed FODC 
on $SU_q(2)$  whose space of left-invariant 2-forms is only two 
dimensional.
%We conclude, the close relation of  the  constructed
%left-covariant differential calculi with the classical differential calculi
%on $SL(N)$ extends
%to higher order calculi.
%&{\rm (i)}&\SS(\RR)=\ker(\id-\si)=\im(\id +\si).\\
%&{\rm (iii)}& \d\omega_\ig=-\sum_{\jg<\kg}C_{\jg\kg}^\ig\omega_\jg
%\wedge\omega_\kg,\,\ig\in I\quad\mbox{ (Maurer-Cartan-formula) }.


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%




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