FRT Construction and Equipped Quantum Linear Spaces
Sergio D. Grillo
%EndAName Centro At\'{omico Bariloche and Instituto Balseiro}
\ 8400-S. C. de Bariloche
\ Argentina
We show there exists a rigid monoidal category formed out by quantum spaces with an additional structure, such that FRT bialgebras and corresponding rectangular generalizations are its internal coEnd and coHom objects, respectively. This enable us to think of them as the coordinate rings of `quantum spaces of homomorphisms' that preserve the mentioned structure. The well known algebra epimorphisms between FRT bialgebras and Manin quantum semigroups translate into `inclusions' of the corresponding quantum spaces, as the space of endomorphisms of a metric linear space $\mathbf{V}$ is included in $gl\left( \mathbf{V}\right) $. Our study is mainly developed for quadratic quantum spaces, and later generalized to the conic case.
math0205334.tex

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\begin{document}

\author{Sergio D. Grillo \\
%EndAName
\textit{Centro At\'{o}mico Bariloche and Instituto Balseiro}\\
\textit{\ 8400-S. C. de Bariloche}\\
\textit{\ Argentina}}
\title{\emph{FRT} Construction and Equipped Quantum Linear Spaces}
\date{February 2003 }
\maketitle

\begin{abstract}
We show there exists a rigid monoidal category formed out by quantum spaces
with an additional structure, such that \emph{FRT }bialgebras and
corresponding rectangular generalizations are its internal coEnd and coHom
objects, respectively. This enable us to think of them as the coordinate
rings of `quantum spaces of homomorphisms' that preserve the mentioned
structure. The well known algebra epimorphisms between \emph{FRT} bialgebras
and Manin quantum semigroups translate into `inclusions' of the
corresponding quantum spaces, as the space of endomorphisms of a metric
linear space $\mathbf{V}$ is included in $gl\left( \mathbf{V}\right) $. Our
study is mainly developed for quadratic quantum spaces, and later
generalized to the conic case.
\end{abstract}

\section{Introduction}

Given a finite dimensional $\Bbbk $-vector space $\mathbf{V}$ and a linear
endomorphism $\Bbb{R}$ of $\mathbf{V\otimes V}$, a universal bialgebra $%
A\left( \Bbb{R}\right) $ can be constructed \cite{kas}. The assignment of $%
A\left( \Bbb{R}\right) $ to each pair $\left( \mathbf{V},\Bbb{R}\right) $ is
known as \emph{FRT} \emph{construction} \cite{frt}. Every $A\left( \Bbb{R}%
\right) $ is a generically non commutative quadratic algebra generated by a
finite dimensional coalgebra, more precisely, by linearly independent
coefficients of a multiplicative matrix $\mathbf{t}$ \cite{A}. This is why
they are called \emph{quantum matrix bialgebras}. Given a basis $\left\{
v_{i}\right\} $ of $\mathbf{V}$, coefficients of $\mathbf{t}$ are identified
with the elements $t_{i}^{j}=v^{j}\otimes v_{i}$ of $\mathbf{V}^{\ast
}\otimes \mathbf{V}$, and the quadratic relations they must satisfy are
sometimes written 
\[
\Bbb{R}_{ij}^{kl}\;t_{k}^{n}\,t_{l}^{m}-t_{i}^{k}\,t_{j}^{l}\;\Bbb{R}%
_{kl}^{nm};\;\;\;i,j,n,m=1,...,\dim \mathbf{V,} 
\]
being $\Bbb{R}_{ij}^{kl}\in \Bbbk $ the coefficients of $\Bbb{R}$ in the
given basis.

Quantum matrix bialgebras are the `dual' version of quantum universal
enveloping algebras, such as Drinfeld-Jimbo \cite{dj} quantized Lie
bialgebras $U_{q}\left( \frak{g}\right) $.\footnote{%
It is worth mentioning we are not asking for $\Bbb{R}$ to be a Yang-Baxter
operator. If this were the case, $\Bbb{R}$ would indicate the so called $R$%
-matrix.} On the other hand, they are quotient of Manin \emph{quantum }(%
\emph{semi})\emph{groups} $\underline{end}\left[ \mathcal{V}\right] $ \cite
{man0}, i.e. the internal coEnd objects of the monoidal category $\mathrm{QA}
$ of quadratic algebras. In other words, there exists a bialgebra
epimorphism $\underline{end}\left[ \mathcal{V}\right] \twoheadrightarrow
A\left( \Bbb{R}\right) $ in $\mathrm{QA}$. The relationship between $\Bbb{R}$
and $\mathcal{V}$ will be discussed later. By now, let us write $\mathcal{V}%
\Vdash \Bbb{R}$ when they are related. Using geometric language, each object 
$\underline{end}\left[ \mathcal{V}\right] $ is interpreted as the coordinate
ring of a non commutative algebraic variety, or \emph{quantum linear space},
living in the opposite category $\mathrm{QA}^{op}$. It represents the
quantum semigroup of endomorphisms corresponding to the quantum space $%
\mathcal{V}^{op}$. Thus, epimorphism above gives rise to a monic $A\left( 
\Bbb{R}\right) ^{op}\hookrightarrow \underline{end}\left[ \mathcal{V}\right]
^{op}$ enabling us to regard $A\left( \Bbb{R}\right) $ as the coordinate
ring of a quantum subspace of $\underline{end}\left[ \mathcal{V}\right]
^{op} $. Of course, Manin construction also includes \emph{quantum spaces of
homomorphisms} $\underline{hom}\left[ \mathcal{W},\mathcal{V}\right] $,
indicating by $\mathcal{W}$ a quadratic algebra generated by a vector
subspace $\mathbf{W}$. They have as \emph{FRT} analogue the \emph{%
rectangular quantum matrix algebras} $A\left( \Bbb{R}:\Bbb{S}\right) $ \cite
{mm}\cite{maj}, in the sense that there exist algebra epimorphisms $%
\underline{hom}\left[ \mathcal{W},\mathcal{V}\right] \twoheadrightarrow
A\left( \Bbb{R}:\Bbb{S}\right) $ leading us to a geometric interpretation as
described before. $\Bbb{S}$ denotes a linear map $\mathbf{W}\otimes \mathbf{W%
}\rightarrow \mathbf{W}\otimes \mathbf{W}$ such that $\mathcal{W}\Vdash \Bbb{%
S}$. Given a basis $\left\{ w_{i}\right\} $ of $\mathbf{W}$, the algebras $%
A\left( \Bbb{R}:\Bbb{S}\right) $ are generated by symbols $%
t_{i}^{j}=w^{j}\otimes v_{i}\in \mathbf{W}^{\ast }\otimes \mathbf{V}$
satisfying\footnote{%
We are using the convention of ref. \cite{phh} to evaluate the maps $\Bbb{R}$
and $\Bbb{S}$ appearing in the rectangular quantum matrix algebras, instead
of the one used in \cite{mm}.} 
\begin{equation}
\Bbb{R}_{ij}^{kl}\;t_{k}^{n}\,t_{l}^{m}-t_{i}^{k}\,t_{j}^{l}\;\Bbb{S}%
_{kl}^{nm};\;\;\;i,j=1,...,\dim \mathbf{V},\;\;\;n,m=1,...,\dim \mathbf{W}.
\label{rqa}
\end{equation}
These algebras were studied in detail in \cite{phh}, where $\Bbb{R}$ and $%
\Bbb{S}$ are Yang-Baxter operators of Hecke type.

This paper was mainly motived by the following question induced by
`inclusions' $A\left( \Bbb{R}:\Bbb{S}\right) ^{op}\hookrightarrow \underline{%
hom}\left[ \mathcal{W},\mathcal{V}\right] ^{op}$: Do the quadratic algebras $%
A\left( \Bbb{R}:\Bbb{S}\right) $ represent homomorphisms between quantum
spaces supplied with some additional structure, i.e. spaces that are not
characterized just by its respective coordinate rings ?

In order to answer this question we encode Manin and \emph{FRT}
constructions, reformulating and generalizing the latter, in the unifying
language of rigid monoidal categories \cite{dm}. We show that bialgebras $%
A\left( \Bbb{R}\right) $ can be seen as internal coEnd objects contained in
certain rigid monoidal category $\left( \mathrm{EQA},\boxtimes \right) $,
the \emph{equipped quantum spaces}, formed out by pairs $\frak{V}=\left( 
\mathcal{V};\Bbb{R}\right) $ with $\mathcal{V}\in \mathrm{QA}$ and $\mathcal{%
V}\Vdash \Bbb{R}$. More precisely, there exists a surjective embedding $%
\mathsf{U}:\mathrm{EQA}\hookrightarrow \mathrm{QA}:\left( \mathcal{V};\Bbb{R}%
\right) \mapsto \mathcal{V}$, and a related opposite $\mathsf{U}^{op}:%
\mathrm{EQA}^{op}\hookrightarrow \mathrm{QA}^{op}$, such that for each pair $%
\frak{V}$, the object $\underline{hom}\left[ \frak{V},\frak{V}\right] =%
\underline{end}\left[ \frak{V}\right] $ is \emph{functored} to $A\left( \Bbb{%
R}\right) $. In general, coHom objects $\underline{hom}\left[ \frak{W},\frak{%
V}\right] $, with $\frak{W}=\left( \mathcal{W};\Bbb{S}\right) $, are sent to 
$A\left( \Bbb{R}:\Bbb{S}\right) $. Moreover, from the general formalism of
rigid monoidal categories follows existence and associativity properties of 
\emph{rectangular comultiplication} maps defined in \cite{mm}, and also
existence of algebra epimorphisms 
\begin{equation}
\underline{hom}_{\mathrm{QA}}\left[ \mathsf{U}\frak{W},\mathsf{U}\frak{V}%
\right] \twoheadrightarrow \mathsf{U}\underline{hom}_{\mathrm{EQA}}\left[ 
\frak{W},\frak{V}\right] ,\;\;\;\forall \frak{W},\frak{V}\in \mathrm{EQA},
\label{inc}
\end{equation}
from which previously mentioned `inclusions' are deduced. We conclude, each
algebra $A\left( \Bbb{R}:\Bbb{S}\right) $ is the coordinate ring of the
space $\underline{hom}\left[ \frak{W},\frak{V}\right] ^{op}\in \mathrm{EQA}%
^{op}$ of homomorphisms between spaces $\frak{W}^{op}$ and $\frak{V}^{op}$.
Thus, such spaces are described by their respective coordinate rings $%
\mathsf{U}\underline{hom}\left[ \frak{W},\frak{V}\right] =A\left( \Bbb{R}:%
\Bbb{S}\right) $, $\mathsf{U}\frak{W}$ and $\mathsf{U}\frak{V}$ (given by
quadratic algebras), and an additional data. We also show $\mathrm{EQA}$ is
equivalent to a category whose objects are pairs $\left( \mathbf{V},\Bbb{R}%
\right) $ as described above, in such a way that we can write $\left( 
\mathcal{V};\Bbb{R}\right) \equiv \left( \mathbf{V},\Bbb{R}\right) $ if $%
\mathcal{V}$ is generated by $\mathbf{V}$. Hence, we are assigning a
bialgebra $A\left( \Bbb{R}\right) =\mathsf{U}\underline{end}\left[ \frak{V}%
\right] $ to each pair $\left( \mathbf{V},\Bbb{R}\right) $ in a universal
way.

\section{Quantum linear spaces}

In what follows $\Bbbk $ indicates some of the numerics fields, $\Bbb{R}$ or 
$\Bbb{C}$. The usual tensor product on $\Bbbk \mathrm{-Alg}=\mathrm{Alg}$%
\textrm{\ }and $\mathrm{Vct}_{\Bbbk }=\mathrm{Vct}$ (the categories of
unital associative $\Bbbk $-algebras and of $\Bbbk $-vector spaces,
respectively) is denoted by $\otimes $. \textrm{Vct}$_{f}$ indicates the
full subcategory of $\mathrm{Vct}$ formed out by finite dimensional vector
spaces.

Originally \cite{man0}, Manin defined quantum spaces as opposite objects to
quadratics algebras. The latter are pairs $\left( \mathbf{A}_{1},\mathbf{A}%
\right) $, with $\mathbf{A}\in \mathrm{Alg}$ generated by $\mathbf{A}_{1}$
in $\mathrm{Vct}_{f}$, such that the canonical epimorphism $\mathbf{A}%
_{1}^{\otimes }\twoheadrightarrow \mathbf{A}$ has as kernel a bilateral
ideal algebraically generated by a subspace of $\mathbf{A}_{1}^{\otimes 2}$.
As usual, $\mathbf{A}_{1}^{\otimes }=\tbigoplus_{n\in \Bbb{N}_{0}}\mathbf{A}%
_{1}^{\otimes n}$ denotes the tensor algebra of\textbf{\ }$\mathbf{A}_{1}$
(being $\Bbb{N}_{0}\doteq \Bbb{N}\cup \left\{ 0\right\} $). To be more
explicit, for every quadratic algebra $\left( \mathbf{A}_{1},\mathbf{A}%
\right) $ there exists a subspace $\mathbf{R}\subset \mathbf{A}_{1}^{\otimes
2}$ such that 
\[
\ker \left[ \mathbf{A}_{1}^{\otimes }\twoheadrightarrow \mathbf{A}\right] =I%
\left[ \mathbf{R}\right] =\mathbf{A}_{1}^{\otimes }\cdot \mathbf{R}\cdot 
\mathbf{A}_{1}^{\otimes }. 
\]
In general, we note by $I\left[ \mathbf{X}\right] \subset \mathbf{A}%
_{1}^{\otimes }$ the bilateral ideal generated by a set $\mathbf{X}\subset 
\mathbf{A}_{1}^{\otimes }$. For instance, each algebra $A\left( \Bbb{R}:\Bbb{%
S}\right) $ defines a quantum space 
\[
A\left( \Bbb{R}:\Bbb{S}\right) \equiv \left( \mathbf{W}^{\ast }\otimes 
\mathbf{V},A\left( \Bbb{R}:\Bbb{S}\right) \right) . 
\]
The kernel of its related canonical epimorphism is generated by the elements
given in Eq. $\left( \ref{rqa}\right) $. The category $\mathrm{QA}$, as
mentioned before, has above pairs as objects and as arrows $\left( \mathbf{A}%
_{1},\mathbf{A}\right) \rightarrow \left( \mathbf{B}_{1},\mathbf{B}\right) $
algebra homomorphisms $\mathbf{A}\rightarrow \mathbf{B}$ that preserve
generating spaces, that is to say, $\mathbf{A}\rightarrow \mathbf{B}$
restricted to $\mathbf{A}_{1}$ defines a linear map $\mathbf{A}%
_{1}\rightarrow \mathbf{B}_{1}$.

In \cite{man1}, Manin extended the concept to arbitrary finitely generated
algebras, i.e. pairs $\left( \mathbf{A}_{1},\mathbf{A}\right) $ as above,
but without restrictions on their respective canonical epimorphisms $\mathbf{%
A}_{1}^{\otimes }\twoheadrightarrow \mathbf{A}$. We shall indicate \textrm{%
FGA }the category formed out by these pairs. Its arrows are again algebra
homomorphisms preserving the generating linear spaces. Thus, $\mathrm{QA}$
is a full subcategory of \textrm{FGA}. Note that arrows $\alpha :\left( 
\mathbf{A}_{1},\mathbf{A}\right) \rightarrow \left( \mathbf{B}_{1},\mathbf{B}%
\right) $ in \textrm{FGA }are characterized by linear maps $\alpha _{1}:%
\mathbf{A}_{1}\rightarrow \mathbf{B}_{1}$ such that 
\[
\alpha _{1}^{\otimes }\left( \ker \left[ \mathbf{A}_{1}^{\otimes
}\twoheadrightarrow \mathbf{A}\right] \right) \subset \ker \left[ \mathbf{B}%
_{1}^{\otimes }\twoheadrightarrow \mathbf{B}\right] .
\]
In $\mathrm{QA}$, if $\mathbf{R}$ and $\mathbf{S}$ are the subspaces
generating the respective kernels, last condition reads $\alpha
_{1}^{\otimes 2}\left( \mathbf{R}\right) \subset \mathbf{S}$. In \cite{gm}
we study another full subcategory of \textrm{FGA}, namely $\mathrm{CA}$, the
conic algebras or conic quantum spaces. Its objects $\left( \mathbf{A}_{1},%
\mathbf{A}\right) $ are such that $\mathbf{A}$ is a graded algebra and $%
\mathbf{A}_{1}$ is its subspace of homogeneous elements of degree one, or
equivalently, its related ideal (i.e. the kernel of its canonical
epimorphism) is a graded subalgebra of $\mathbf{A}_{1}^{\otimes }$. Examples
of them, beside quadratics, are the so called $m$-th quantum spaces, whose
associated ideals are generated by a subspace of $\mathbf{A}_{1}^{\otimes m}$%
, for some $m\geq 2$. The latter, in turn, form a full subcategory $\mathrm{%
CA}^{m}$ of $\mathrm{CA}$, leading us to the full inclusions $\mathrm{CA}%
^{m}\subset \mathrm{CA}\subset \mathrm{FGA}$. Of course, $\mathrm{QA}=%
\mathrm{CA}^{2}$.

The monoid we consider on these categories is the bifunctor $\circ $, given
on objects by 
\begin{equation}
\left( \mathbf{A}_{1},\mathbf{A}\right) \circ \left( \mathbf{B}_{1},\mathbf{B%
}\right) =\left( \mathbf{A}_{1}\otimes \mathbf{B}_{1},\mathbf{A}\circ 
\mathbf{B}\right) ,  \label{mon}
\end{equation}
with $\mathbf{A}\circ \mathbf{B}$ the subalgebra of $\mathbf{A}\otimes 
\mathbf{B}$ generated by $\mathbf{A}_{1}\otimes \mathbf{B}_{1}$. On arrows,
it assigns to $\alpha $ and $\beta $, with domains $\left( \mathbf{A}_{1},%
\mathbf{A}\right) $ and $\left( \mathbf{B}_{1},\mathbf{B}\right) $,
respectively, the algebra morphism $\alpha \circ \beta =\left. \alpha
\otimes \beta \right| _{\mathbf{A}\circ \mathbf{B}}$. The unit object is $%
\mathcal{I}=\left( \Bbbk ,\Bbbk \right) $ in $\mathrm{FGA}$ and $\mathcal{K}%
=\left( \Bbbk ,\Bbbk ^{\otimes }\right) $ in $\mathrm{CA}$ and every $%
\mathrm{CA}^{m}$. Let us mention the forgetful functor $\mathsf{F}:\mathrm{%
FGA}\rightarrow \mathrm{Alg}:\left( \mathbf{A}_{1},\mathbf{A}\right) \mapsto 
\mathbf{A}$ preserves the units, since $\mathsf{F}\mathcal{I}=\Bbbk $, but
is not monoidal. Nevertheless, it is easy to check that algebra inclusions $%
i_{\mathcal{A},\mathcal{B}}:\mathbf{A}\circ \mathbf{B}\hookrightarrow 
\mathbf{A}\otimes \mathbf{B}$, related to quantum spaces $\mathcal{A}=\left( 
\mathbf{A}_{1},\mathbf{A}\right) $ and $\mathcal{B}=\left( \mathbf{B}_{1},%
\mathbf{B}\right) $, define a natural transformation $\mathsf{F}\,\circ
\,\rightarrow \otimes \,\left( \mathsf{F}\times \mathsf{F}\right) $. (Note
that $\mathsf{F}\left( \mathcal{A}\circ \mathcal{B}\right) =\mathbf{A}\circ 
\mathbf{B}$ and $\mathsf{F}\mathcal{A}\otimes \mathsf{F}\mathcal{B}=\mathbf{A%
}\otimes \mathbf{B}$.) That is to say, for any couple of arrows $\alpha
,\beta \in \mathrm{FGA}$, with $\alpha :\mathcal{A}\rightarrow \mathcal{C}$
and $\beta :\mathcal{B}\rightarrow \mathcal{D}$, the diagram 
\begin{equation}
\begin{diagram}[s=2.5em] \QTR{bf}{A}\circ \QTR{bf}{B} &
\rInto^{i_{\QTR{cal}{A},\QTR{cal}{B}}} & \QTR{bf}{A}\otimes \QTR{bf}{B} \\
\dTo^{\QTR{sf}{F}\left( \alpha \circ \beta \right) }& &
\dTo_{\QTR{sf}{F}\alpha \otimes \QTR{sf}{F}\beta } \\ \QTR{bf}{C}\circ
\QTR{bf}{D} & \rInto^{i_{\QTR{cal}{C},\QTR{cal}{D}}} & \QTR{bf}{C}\otimes
\QTR{bf}{D} \\ \end{diagram}  \label{fa}
\end{equation}
is commutative. When restricted to $\mathrm{CA}$ and every $\mathrm{CA}^{m}$%
, the above natural transformation holds, but $\mathsf{F}$ does not respects
units, because $\mathsf{F}\mathcal{K}=\Bbbk ^{\otimes }$. However, the
canonical projection $\Bbbk ^{\otimes }\twoheadrightarrow \Bbbk $ defines
epimorphisms $p_{\mathcal{A}}:\Bbbk ^{\otimes }\otimes \mathbf{A}%
\twoheadrightarrow \Bbbk \otimes \mathbf{A}$, with $\mathsf{F}\mathcal{A}=%
\mathbf{A}$, that make commutative the diagrams 
\begin{equation}
\begin{diagram}[w=3.5em] \Bbbk ^{\otimes }\circ \QTR{bf}{A} &
\rInto^{i_{\QTR{cal}{K},\QTR{cal}{A}}} & \Bbbk ^{\otimes }\otimes
\QTR{bf}{A} & \rOnto^{p_{\QTR{cal}{A}}} & \Bbbk \otimes \QTR{bf}{A} \\ &
\rdTo~{\backsimeq } & & \ruTo~{\backsimeq} & \\ & & \QTR{bf}{A} & & \\
\end{diagram}  \label{fu}
\end{equation}
Indicating by $e$ the generator of $\Bbbk $, then 
\[
\Bbbk ^{\otimes }=\Bbbk \left[ e\right] \;\;\;\;and\;\;\;p_{\mathcal{A}%
}\left( e^{n}\otimes a\right) =e\otimes a.
\]
The isomorphisms $\mathbf{A}\backsimeq \Bbbk \otimes \mathbf{A}$ and $%
\mathbf{A}\backsimeq \Bbbk ^{\otimes }\circ \mathbf{A}$ are the functorial
isomorphisms related to the left unital constraint in $\mathrm{Alg}$ and $%
\mathrm{CA}$, respectively. Of course, a diagram analogous to $\left( \ref
{fu}\right) $ but with $\Bbbk $ on the right is also fulfilled.

\bigskip

There exist internal coHom objects on each one of this monoidal categories.
For instance, for $\mathcal{A}=\left( \mathbf{A}_{1},\mathbf{A}\right) $ and 
$\mathcal{B}=\left( \mathbf{B}_{1},\mathbf{B}\right) $ in $\mathrm{CA}$
(resp. $\mathrm{CA}^{m}$), they are given by graded algebras $\underline{hom}%
\left[ \mathcal{B},\mathcal{A}\right] $ generated by $\mathbf{B}_{1}^{\ast
}\otimes \mathbf{A}_{1}$ and constrained by homogeneous relations (resp. of $%
m$-th order). For more details, see \cite{gm}.

\section{The equipped quantum spaces}

Consider $\mathcal{A}=\left( \mathbf{A}_{1},\mathbf{A}\right) \in \mathrm{QA}
$ and a linear map $\Bbb{R}:\mathbf{A}_{1}^{\otimes 2}\rightarrow \mathbf{A}%
_{1}^{\otimes 2}$.

\begin{definition}
We say $\Bbb{R}$ is \textbf{compatible} with $\mathcal{A}$, and use the
shorthand notation $\mathcal{A}\Vdash $ $\Bbb{R}$, if 
\[
\ker \left[ \mathbf{A}_{1}^{\otimes }\twoheadrightarrow \mathbf{A}\right] =I%
\left[ \func{Im}\Bbb{R}\right] .
\]
A pair $\left( \mathcal{A};\Bbb{R}\right) $ such that $\mathcal{A}\Vdash $ $%
\Bbb{R}$ will be called \textbf{equipped quantum space} \emph{(}or equipped
quadratic algebra\emph{)} with \textbf{structure }$\Bbb{R}$. If a morphism
of quantum spaces $\alpha :\mathcal{A}\rightarrow \mathcal{B}$, with $%
\mathcal{A}\Vdash $ $\Bbb{R}$ and $\mathcal{B}\Vdash \Bbb{S}$, satisfies $%
\alpha _{1}^{\otimes 2}\,\Bbb{R}=\Bbb{S}\,\alpha _{1}^{\otimes 2}$, we say
that $\alpha $ preserves structures $\Bbb{R}$ and $\Bbb{S}$. The category
formed out by equipped quantum spaces and structure preserving arrows will
be denoted $\mathrm{EQA}$.\ \ \ $\blacksquare $
\end{definition}

From now on, we reserve the name \emph{pair }only for equipped quantum
spaces (in contrast to last section where we used it for ordinary ones).

A simple characterization of equipped quantum spaces is given by the
following result.

\begin{proposition}
The category $\mathrm{EQA}$ is equivalent to one whose objects are pairs $%
\left( \mathbf{V},\Bbb{R}\right) $, with $\mathbf{V}\in \mathrm{Vct}_{f}$
and $\Bbb{R}:\mathbf{V}^{\otimes 2}\rightarrow \mathbf{V}^{\otimes 2}$ a
linear map, and whose arrows $\left( \mathbf{V},\Bbb{R}\right) \rightarrow
\left( \mathbf{W},\Bbb{S}\right) $ are linear homomorphisms $l:\mathbf{V}%
\rightarrow \mathbf{W}$ such that $l^{\otimes 2}\,\Bbb{R}=\Bbb{S}%
\,l^{\otimes 2}$.
\end{proposition}

\textbf{Proof:} The equivalence is defined by functors 
\begin{equation}
\mathsf{f}:\left( \mathcal{A};\Bbb{R}\right) \mapsto \left( \mathbf{A}_{1},%
\Bbb{R}\right) \;\;\;and\;\;\;\mathsf{g}:\left( \mathbf{V},\Bbb{R}\right)
\mapsto \left( \left( \mathbf{V},\left. \mathbf{V}^{\otimes }\right/ I\left[ 
\func{Im}\Bbb{R}\right] \right) ;\Bbb{R}\right) .  \label{fg}
\end{equation}
On arrows, $\mathsf{f}\alpha =\alpha _{1}$ and $\mathsf{g}l$ is the
extension of $l$ to an algebra homomorphism. If $l$ goes from $\left( 
\mathbf{V},\Bbb{R}\right) $ to $\left( \mathbf{W},\Bbb{S}\right) $, since 
\begin{eqnarray*}
l^{\otimes }\left( I\left[ \func{Im}\Bbb{R}\right] \right) &=&\mathbf{W}%
^{\otimes }\cdot l^{\otimes }\left( \func{Im}\Bbb{R}\right) \cdot \mathbf{W}%
^{\otimes }=\mathbf{W}^{\otimes }\cdot \left( \func{Im}l^{\otimes 2}\,\Bbb{R}%
\right) \cdot \mathbf{W}^{\otimes } \\
&=&\mathbf{W}^{\otimes }\cdot \left( \func{Im}\Bbb{S}\,l^{\otimes 2}\right)
\cdot \mathbf{W}^{\otimes }\subset \mathbf{W}^{\otimes }\cdot \func{Im}\Bbb{S%
}\,\cdot \mathbf{W}^{\otimes }=I\left[ \func{Im}\Bbb{S}\right] ,
\end{eqnarray*}
then such extension is well defined.

Natural equivalence $\mathsf{f}\circ \mathsf{g}\backsimeq id$ is immediate.
The functorial isomorphisms for equivalence $\mathsf{g}\circ \mathsf{f}%
\backsimeq id$ are given by the algebra isomorphisms $\mathbf{A}\backsimeq
\left. \mathbf{A}_{1}^{\otimes }\right/ I\left[ \func{Im}\Bbb{R}\right] $,
which are well defined provided $\ker \left[ \mathbf{A}_{1}^{\otimes
}\twoheadrightarrow \mathbf{A}\right] =I\left[ \func{Im}\Bbb{R}\right]
.\;\;\;\blacksquare $

\bigskip

Because of this equivalence, we identify the objects of both categories.
That is to say, we understand pairs $\left( \mathbf{A}_{1},\Bbb{R}\right) $
and $\left( \mathcal{A};\Bbb{R}\right) $ as the same thing, indicating both
categories by \textrm{EQA}.\footnote{%
Note that the category $\mathcal{YB}$ defined in \cite{mm}, formed out by
pairs $\left( \mathbf{V},\Bbb{R}\right) $ such that $\Bbb{R}$ is a
Yang-Baxter solution of $q$-Hecke type, is a full subcategory of $\mathrm{EQA%
}$.} Since to deal with pairs $\left( \mathbf{V},\Bbb{R}\right) $ is often
easier than to deal with $\left( \mathcal{A};\Bbb{R}\right) $, we shall work
out our constructions mainly in terms of the former. Naturally, they also
provide a more direct contact to \emph{FRT }construction.

\subsection{Products and duals}

A monoidal structure and an involution can be attached to \textrm{EQA }in
the following way. Let us consider the canonical algebra isomorphism $%
\varphi _{\mathbf{V},\mathbf{W}}$ between $\left[ \mathbf{V}\otimes \mathbf{W%
}\right] ^{\otimes }$ and 
\[
\mathbf{V}\circ \mathbf{W}\doteq \bigoplus\nolimits_{n\in \Bbb{N}_{0}}%
\mathbf{V}^{\otimes n}\otimes \mathbf{W}^{\otimes n}, 
\]
the subalgebra of $\mathbf{V}^{\otimes }\otimes \mathbf{W}^{\otimes
}=\bigoplus\nolimits_{n,m}\mathbf{V}^{\otimes n}\otimes \mathbf{W}^{\otimes
m}$ generated by $\mathbf{V}\otimes \mathbf{W}$. The restriction of $\varphi
_{\mathbf{V},\mathbf{W}}$ to $\left( \mathbf{V}\otimes \mathbf{W}\right)
^{\otimes 2}$, which we also denote $\varphi _{\mathbf{V},\mathbf{W}}$, is
given by 
\[
v\otimes w\otimes v^{\prime }\otimes w^{\prime }\mapsto v\otimes v^{\prime
}\otimes w\otimes w^{\prime };\;\;\forall v,v^{\prime }\in \mathbf{V}%
,\;w,w^{\prime }\in \mathbf{W}. 
\]
We define the bifunctor $\boxtimes :\mathrm{EQA}\times \mathrm{EQA}%
\rightarrow \mathrm{EQA}$ as 
\begin{equation}
\left( \mathbf{V},\Bbb{R}\right) \times \left( \mathbf{W},\Bbb{S}\right)
\mapsto \left( \mathbf{V}\otimes \mathbf{W},\Bbb{R}\boxtimes \Bbb{S}\right)
,\;\;\;k\times l\mapsto k\boxtimes l\doteq k\otimes l,  \label{A}
\end{equation}
being 
\begin{equation}
\Bbb{R}\boxtimes \Bbb{S}\doteq \varphi _{\mathbf{V},\mathbf{W}}^{-1}\,\left( 
\Bbb{R}\otimes \Bbb{I}+\Bbb{I}\otimes \Bbb{S}\right) \,\varphi _{\mathbf{V},%
\mathbf{W}}  \label{B}
\end{equation}
(taking into account the restriction of $\varphi _{\mathbf{V},\mathbf{W}}$). 
$\Bbb{I}$ denotes the identity endomorphism of the corresponding vector
spaces. Identifying $\mathbf{V}\circ \mathbf{W}$ and $\left[ \mathbf{V}%
\otimes \mathbf{W}\right] ^{\otimes }$ we shall write $\Bbb{R}\boxtimes \Bbb{%
S}\thickapprox \Bbb{R}\otimes \Bbb{I}+\Bbb{I}\otimes \Bbb{S}$.
Straightforwardly, the bifunctor $\boxtimes $ defines a symmetric monoidal
structure with unit object $\frak{K}=\left( \Bbbk ,\Bbb{O}\right) $, being $%
\Bbb{O}$ the null endomorphism of $\Bbbk ^{\otimes 2}$, i.e. $\func{Im}\Bbb{O%
}=\left\{ 0\right\} $. The functorial isomorphisms $\tau _{\frak{V},\frak{W}%
}:\frak{V}\boxtimes \frak{W}\backsimeq \frak{W}\boxtimes \frak{V}$ related
to symmetry, with $\frak{V}=\left( \mathbf{V},\Bbb{R}\right) $ and $\frak{W}%
=\left( \mathbf{W},\Bbb{S}\right) $, are given by canonical flipping maps $%
\mathbf{V}\otimes \mathbf{W}\backsimeq \mathbf{W}\otimes \mathbf{V}$, $%
v\otimes w\mapsto w\otimes v$. The ones related to unit are $\ell _{\frak{V}%
}:v\in \mathbf{V}\mapsto e\otimes v$ and $r_{\frak{V}}:v\in \mathbf{V}%
\mapsto v\otimes e$, indicating by $e$ the generator of $\Bbbk $.

Let us define the contravariant functor $\dagger :\mathrm{EQA}\rightarrow 
\mathrm{EQA}$, 
\begin{equation}
\dagger \,:\left( \mathbf{V},\Bbb{R}\right) \mapsto \frak{V}^{\dagger
}\doteq \left( \mathbf{V}^{\ast },\Bbb{R}^{\dagger }\right) \doteq \left( 
\mathbf{V}^{\ast },-\Bbb{R}^{\ast }\right) ,\;\;\;\;\dagger \,:l\mapsto
l^{\dagger }\doteq l^{\ast },  \label{C}
\end{equation}
being $\mathbf{V}^{\ast }$ the dual of $\mathbf{V}$, and $\Bbb{R}^{\ast }$
the transpose map w.r.t. the usual extension to $\mathbf{V}^{\otimes 2}$ of
the pairing between $\mathbf{V}$ and $\mathbf{V}^{\ast }$. It is clear that $%
\dagger ^{2}=\dagger \dagger $ is naturally equivalent to $id_{\mathrm{EQA}}$%
. In particular, $\frak{V}^{\dagger \dagger }\backsimeq \frak{V}$, $\forall 
\frak{V}\in \mathrm{EQA}$. The relation between $\boxtimes $ and $\dagger $
can be summarized by equations 
\[
\left( \frak{V}\boxtimes \frak{W}\right) ^{\dagger }\backsimeq \frak{V}%
^{\dagger }\boxtimes \frak{W}^{\dagger },\;\frak{K}^{\dagger }\backsimeq 
\frak{K}. 
\]

In terms of pairs $\left( \mathcal{A};\Bbb{R}\right) $, $\boxtimes $ and $%
\dagger $ are given by 
\[
\left( \mathcal{A};\Bbb{R}\right) \times \left( \mathcal{B};\Bbb{S}\right)
\mapsto \left( \mathcal{A}\boxtimes \mathcal{B};\Bbb{R}\boxtimes \Bbb{S}%
\right) ,\;\;\;\alpha \times \beta \mapsto \alpha \boxtimes \beta ,
\]
and 
\[
\left( \mathcal{A};\Bbb{R}\right) \mapsto \left( \mathcal{A}^{\dagger };\Bbb{%
R}^{\dagger }\right) \doteq \left( \left( \mathbf{A}_{1}^{\ast },\mathbf{A}%
_{1}^{\ast \otimes }/I\left[ \func{Im}\Bbb{R}^{\ast }\right] \right) ;-\Bbb{R%
}^{\ast }\right) ,\;\;\;\;\alpha \mapsto \alpha ^{\dagger },
\]
respectively, being $\mathcal{A}\boxtimes \mathcal{B}\doteq \left( \mathbf{A}%
_{1}\otimes \mathbf{B}_{1},\left. \left[ \mathbf{A}_{1}\otimes \mathbf{B}_{1}%
\right] ^{\otimes }\right/ \func{Im}\left[ \Bbb{R}\boxtimes \Bbb{S}\right]
\right) $. The arrows $\alpha \boxtimes \beta $ and $\alpha ^{\dagger }$ are
the extension of $\alpha _{1}\otimes \beta _{1}$ and $\alpha _{1}^{\ast }$
to an algebra map. The unit object for $\boxtimes $ is $\left( \mathcal{K};%
\Bbb{O}\right) $.

\subsection{The embedding $\mathrm{EQA}\hookrightarrow \mathrm{QA}$}

Now, we study the relationship between $\mathrm{EQA}$ and $\mathrm{QA}$ as
monoidal categories. There exists an obvious forgetful functor between these
categories.

\begin{proposition}
The function $\left( \mathcal{A};\Bbb{R}\right) \mapsto \mathcal{A}$ defines
a surjective embedding $\mathsf{U}:\mathrm{EQA}\hookrightarrow \mathrm{QA}$.
\end{proposition}

\textbf{Proof}: We just need to show the function is surjective, i.e. given $%
\mathcal{A}\in \mathrm{QA}$, there exists a compatible map $\Bbb{R}$ such
that $\mathsf{U}\left( \mathcal{A};\Bbb{R}\right) =\mathcal{A}$. Let $I\left[
\mathbf{R}\right] $ be the ideal related to $\mathcal{A}$. Consider a
decomposition $\mathbf{A}_{1}^{\otimes 2}=\mathbf{R}\oplus \mathbf{R}^{c}$,
with associated projections $\Bbb{P}$ such that $\func{Im}\Bbb{P}=\mathbf{R}$%
. Since $I\left[ \mathbf{R}\right] =I\left[ \func{Im}\Bbb{P}\right] $, then $%
\mathcal{A}\Vdash \Bbb{P}$ and the proposition follows.\ \ \ \ $\blacksquare 
$

\bigskip

On pairs $\left( \mathbf{V},\Bbb{R}\right) $ the embedding is given by $%
\left( \mathbf{V},\Bbb{R}\right) \mapsto \left( \mathbf{V},\left. \mathbf{V}%
^{\otimes }\right/ I\left[ \func{Im}\Bbb{R}\right] \right) $ (see second
part of Eq. $\left( \ref{fg}\right) $). The surjectivity is up to
isomorphisms in $\mathrm{QA}$. The functor $\mathsf{U}$ obviously preserves
the unit objects, in fact 
\[
\mathsf{U}\frak{K}=\mathsf{U}\left( \Bbbk ,\Bbb{O}\right) =\left( \Bbbk
,\left. \Bbbk ^{\otimes }\right/ I\left[ \func{Im}\Bbb{O}\right] \right)
=\left( \Bbbk ,\Bbbk ^{\otimes }\right) =\mathcal{K}, 
\]
but is not monoidal. Nevertheless,

\begin{proposition}
There exist functorial epimorphisms $\mathsf{U}\left( \frak{V}\boxtimes 
\frak{W}\right) \twoheadrightarrow \mathsf{U}\frak{V}\circ \mathsf{U}\frak{W}
$, $\frak{V},\frak{W}\in \mathrm{EQA}$, defining a natural transformation $%
\mathsf{U\,}\boxtimes \mathsf{\,}\rightarrow \circ \mathsf{\,}\left( \mathsf{%
U}\times \mathsf{U}\right) $.
\end{proposition}

\textbf{Proof:} It is clear that, given pairs $\frak{V}$ and $\frak{W}$, we
have 
\[
\begin{array}{l}
\func{Im}\Bbb{R}\boxtimes \Bbb{S}\thickapprox \varphi _{\mathbf{V},\mathbf{W}%
}\left( \func{Im}\Bbb{R}\boxtimes \Bbb{S}\right) = \\ 
\\ 
=\func{Im}\left[ \Bbb{R}\otimes \Bbb{I}+\Bbb{I}\otimes \Bbb{S}\right]
\subset \func{Im}\Bbb{R}\otimes \mathbf{W}^{\otimes 2}+\mathbf{V}^{\otimes
2}\otimes \func{Im}\Bbb{S},
\end{array}
\]
and accordingly, 
\[
\varphi _{\mathbf{V},\mathbf{W}}\left( I\left[ \func{Im}\Bbb{R}\boxtimes 
\Bbb{S}\right] \right) \subset \left( I\left[ \func{Im}\Bbb{R}\right]
\otimes \mathbf{W}^{\otimes }+\mathbf{V}^{\otimes }\otimes I\left[ \func{Im}%
\Bbb{S}\right] \right) \cap \mathbf{V}\circ \mathbf{W}. 
\]
The first ideal in above inclusion is related to the quantum space $\mathsf{U%
}\left( \frak{V}\boxtimes \frak{W}\right) $, and the latter to $\mathsf{U}%
\frak{V}\circ \mathsf{U}\frak{W}$, since it defines the algebra 
\[
\left( \left. \mathbf{V}^{\otimes }\right/ I\left[ \func{Im}\Bbb{R}\right]
\right) \circ \left( \left. \mathbf{W}^{\otimes }\right/ I\left[ \func{Im}%
\Bbb{S}\right] \right) . 
\]
Note the corresponding algebras are quotient of $\left[ \mathbf{V}\otimes 
\mathbf{W}\right] ^{\otimes }$. Hence, for every couple $\frak{V},\frak{W}%
\in \mathrm{EQA}$, we have an epimorphism $p_{\frak{V},\frak{W}}:\mathsf{U}%
\left( \frak{V}\boxtimes \frak{W}\right) \twoheadrightarrow \mathsf{U}\frak{V%
}\circ \mathsf{U}\frak{W}$.

By straightforward calculations, it can be checked commutativity of diagrams 
\begin{equation}
\begin{diagram} \QTR{sf}{U}\left( \QTR{frak}{V}\boxtimes
\QTR{frak}{W}\right) & \rOnto^{p_{\QTR{frak}{V},\QTR{frak}{W}}} &
\QTR{sf}{U}\QTR{frak}{V}\circ \QTR{sf}{U}\QTR{frak}{W} \\
\dTo^{\QTR{sf}{U}\left( \alpha \boxtimes \beta \right) }& &
\dTo_{\QTR{sf}{U}\alpha \circ \QTR{sf}{U}\beta } \\ \QTR{sf}{U}\left(
\QTR{frak}{X}\boxtimes \QTR{frak}{Y}\right) &
\rOnto^{p_{\QTR{frak}{X},\QTR{frak}{Y}}} & \QTR{sf}{U}\QTR{frak}{X}\circ
\QTR{sf}{U}\QTR{frak}{Y} \\ \end{diagram}  \label{ua}
\end{equation}
for every couple of arrows $\alpha :\frak{V}\rightarrow \frak{X}$ and $\beta
:\frak{W}\rightarrow \frak{Y}$ in $\mathrm{EQA}$.\ \ \ $\blacksquare $

\bigskip

This result, together with the ones relating monoids $\circ $ and $\otimes $%
, will be useful in order to construct the rectangular comultiplications
maps.

\section{Rectangular quantum matrix algebras}

Now, the central result. We shall show the following theorem later, in a
more general context.

\begin{theorem}
The monoidal category $\left( \mathrm{EQA},\boxtimes ,\frak{K}\right) $ is 
\textbf{rigid}, and has $\dagger $ as duality functor. For every $\frak{V}%
=\left( \mathbf{V},\Bbb{R}\right) $ in $\mathrm{EQA}$, the evaluation and
coevaluation arrows, $ev_{\frak{V}}:\frak{V}^{\dagger }\boxtimes \frak{V}%
\rightarrow \frak{K}$ and $coev_{\frak{V}}:\frak{K}\rightarrow \frak{V}%
\boxtimes \frak{V}^{\dagger }$, respectively, are given by the corresponding
maps for $\mathbf{V}$ in the rigid monoidal category $\left( \mathrm{Vct}%
_{f},\otimes ,\Bbbk \right) $. \ \ \ $\blacksquare $
\end{theorem}

We can define the internal coHom object related to a couple $\frak{W},\frak{V%
}\in \mathrm{EQA}$ as $\underline{hom}\left[ \frak{W},\frak{V}\right] \doteq 
\frak{W}^{\dagger }\boxtimes \frak{V}$, and take 
\[
\delta _{\frak{V},\frak{W}}\doteq \tau _{\frak{W},\underline{hom}\left[ 
\frak{W},\frak{V}\right] }\,\left( coev_{\frak{W}}\boxtimes I\right) \,\ell
_{\frak{V}}:\frak{V}\rightarrow \underline{hom}\left[ \frak{W},\frak{V}%
\right] \boxtimes \frak{W} 
\]
as the (left) coevaluation arrow. Its well known universality property says:
given $\frak{H}\in \mathrm{EQA}$ and $\varphi :\frak{V}\rightarrow \frak{H}%
\boxtimes \frak{W}$, there exists a unique morphism $\alpha :\underline{hom}%
\left[ \frak{W},\frak{V}\right] \rightarrow \frak{H}$ making commutative the
diagram 
\begin{equation}
\begin{diagram} & & \QTR{frak}{V} & & \\ & \ldTo^{\delta
_{\QTR{frak}{V},\QTR{frak}{W}}} & & \rdTo^{\varphi } & \\
\underline{hom}\left[ \QTR{frak}{W},\QTR{frak}{V}\right] \boxtimes
\QTR{frak}{W}& & \rTo^{\alpha \boxtimes I}&
&\QTR{frak}{H}\boxtimes\QTR{frak}{W} \\ \end{diagram}  \label{diaun}
\end{equation}
From $\left( \ref{diaun}\right) $ and general properties of monoidal
categories follow the existence of comultiplication 
\begin{equation}
\underline{hom}\left[ \frak{W},\frak{V}\right] \rightarrow \underline{hom}%
\left[ \frak{U},\frak{V}\right] \boxtimes \underline{hom}\left[ \frak{W},%
\frak{U}\right] ,\;\forall \frak{U},\frak{V},\frak{W}\in \mathrm{EQA},
\label{com}
\end{equation}
given by $\Delta _{\frak{U},\frak{V},\frak{W}}=\tau _{\underline{hom}\left[ 
\frak{W},\frak{U}\right] ,\underline{hom}\left[ \frak{U},\frak{V}\right]
}\,\left( I\boxtimes coev_{\frak{U}}\boxtimes I\right) \,\left( I\boxtimes
\ell _{\frak{V}}\right) $, and counit arrows 
\begin{equation}
\varepsilon _{\frak{V}}=ev_{\frak{V}}:\underline{end}\left[ \frak{V}\right]
\rightarrow \frak{K},\;\forall \frak{V}\in \mathrm{EQA}.  \label{coun}
\end{equation}
Coevaluations are particular comultiplications. Indeed, since $\underline{hom%
}\left[ \frak{K},\frak{V}\right] =\frak{V}$, $\forall \frak{V}\in \mathrm{EQA%
}$, it can be seen that $\delta _{\frak{V},\frak{W}}=\Delta _{\frak{W},\frak{%
V},\frak{K}}$. On the other hand, if $\frak{U}=\frak{V}=\frak{W}$, $\Delta _{%
\frak{V}}=\Delta _{\frak{V},\frak{V},\frak{V}}$ and $\varepsilon _{\frak{V}}$
gives $\underline{end}\left[ \frak{V}\right] $ a coalgebra structure in $%
\mathrm{EQA}$, and $\delta _{\frak{V}}=\delta _{\frak{V},\frak{V}}$ makes $%
\frak{V}$ an $\underline{end}\left[ \frak{V}\right] $-corepresentation in
the same category.

We add that arrows $\left( \ref{com}\right) $ and $\left( \ref{coun}\right) $
satisfy usual associativity and unit constraints, expressed by commutativity
of the following diagrams\footnote{%
Of course, we have a similar commutative diagram where $\frak{K}$ is on the
right.} 
\begin{equation}
\begin{diagram} \underline{hom}\left[ \QTR{frak}{X},\QTR{frak}{V}\right]
\boxtimes \underline{hom}\left[ \QTR{frak}{W},\QTR{frak}{X}\right] &\rTo&
\underline{hom}\left[ \QTR{frak}{Y},\QTR{frak}{V}\right] \boxtimes
\underline{hom}\left[ \QTR{frak}{X},\QTR{frak}{Y}\right] \boxtimes
\underline{hom}\left[ \QTR{frak}{W},\QTR{frak}{X}\right]\\ \uTo& &\uTo \\
\underline{hom}\left[ \QTR{frak}{W},\QTR{frak}{V}\right] &\rTo&
\underline{hom}\left[ \QTR{frak}{Y},\QTR{frak}{V}\right] \boxtimes
\underline{hom}\left[ \QTR{frak}{W},\QTR{frak}{Y}\right]\\ \end{diagram}
\label{evco}
\end{equation}
\medskip 
\begin{equation}
\begin{diagram} \QTR{frak}{K}\boxtimes \underline{hom}\left[
\QTR{frak}{W},\QTR{frak}{V}\right] & &\lTo & & \underline{end}\left[
\QTR{frak}{V}\right] \boxtimes \underline{hom}\left[
\QTR{frak}{W},\QTR{frak}{V}\right] \\ & \luTo~{\backsimeq } & &\ruTo & \\ &
& \underline{hom}\left[ \QTR{frak}{W},\QTR{frak}{V}\right] & &\\
\end{diagram}  \label{idco}
\end{equation}

\bigskip

Consider pairs $\frak{V}=\left( \mathbf{V},\Bbb{R}\right) $ and $\frak{W}%
=\left( \mathbf{W},\Bbb{S}\right) $ in $\mathrm{EQA}$, and take basis $%
\left\{ v_{i}\right\} $ and $\left\{ w_{i}\right\} $ of $\mathbf{V}$ and $%
\mathbf{W}$, respectively. The image under $\mathsf{U}$ of the internal
coHom object 
\[
\underline{hom}\left[ \frak{W},\frak{V}\right] =\frak{W}^{\dagger }\boxtimes 
\frak{V}=\left( \mathbf{W}^{\ast }\otimes \mathbf{V},\Bbb{S}^{\dagger
}\boxtimes \Bbb{R}\right) 
\]
is a quadratic algebra generated by $\mathbf{W}^{\ast }\otimes \mathbf{V}$
and obeying relations $I\left[ \func{Im}\Bbb{S}^{\dagger }\boxtimes \Bbb{R}%
\right] $. Writing $t_{i}^{j}=w^{j}\otimes v_{i}$, straightforwardly 
\begin{equation}
\func{Im}\Bbb{S}^{\dagger }\boxtimes \Bbb{R}=span\left[ \Bbb{R}%
_{ij}^{kl}\;t_{k}^{n}\otimes t_{l}^{m}-t_{i}^{k}\otimes t_{j}^{l}\;\Bbb{S}%
_{kl}^{nm}\right] _{i,j,n,m}.  \label{raa}
\end{equation}
Comparing $\left( \ref{rqa}\right) $ with $\left( \ref{raa}\right) $, we
have $\mathsf{U}\underline{hom}\left[ \frak{W},\frak{V}\right] =A\left( \Bbb{%
R}:\Bbb{S}\right) $. In particular, $\mathsf{U}\underline{end}\left[ \frak{V}%
\right] =A\left( \Bbb{R}\right) $. Thus, the algebras $A\left( \Bbb{R}:\Bbb{S%
}\right) $ (resp. bialgebras $A\left( \Bbb{R}\right) $) are the coordinate
ring of an equipped quantum space with structure $\Bbb{S}^{\dagger
}\boxtimes \Bbb{R}$ (resp. $\Bbb{R}^{\dagger }\boxtimes \Bbb{R}$),
representing the space of homomorphisms from $\frak{W}^{op}$ to $\frak{V}%
^{op}$.

\subsection{Rectangular comultiplication and counit maps}

Now, we are going to construct rectangular comultiplication and counit maps
defined in \cite{mm} for algebras $A\left( \Bbb{R}:\Bbb{S}\right) $. This
can be done in steps below:

\begin{enumerate}
\item  Apply the functor $\mathsf{U}$ to comultiplications given in $\left( 
\ref{com}\right) $ to obtain the maps 
\[
\mathsf{U}\underline{hom}\left[ \frak{W},\frak{V}\right] \rightarrow \mathsf{%
U}\left( \underline{hom}\left[ \frak{U},\frak{V}\right] \boxtimes \underline{%
hom}\left[ \frak{W},\frak{U}\right] \right) .
\]

\item  Compose it with the functorial epimorphism 
\[
\mathsf{U}\left( \underline{hom}\left[ \frak{U},\frak{V}\right] \boxtimes 
\underline{hom}\left[ \frak{W},\frak{U}\right] \right) \twoheadrightarrow 
\mathsf{U}\underline{hom}\left[ \frak{U},\frak{V}\right] \circ \mathsf{U}%
\underline{hom}\left[ \frak{W},\frak{U}\right] .
\]

\item  Apply the forgetful functor $\mathsf{F}:\left( \mathbf{A}_{1},\mathbf{%
A}\right) \mapsto \mathbf{A}$ to this composition. This gives us an algebra
homomorphism 
\[
\mathsf{FU}\underline{hom}\left[ \frak{W},\frak{V}\right] \rightarrow 
\mathsf{F}\left( \mathsf{U}\underline{hom}\left[ \frak{U},\frak{V}\right]
\circ \mathsf{U}\underline{hom}\left[ \frak{W},\frak{U}\right] \right) .
\]

\item  Finally, compose the latter with the functorial inclusion 
\[
\mathsf{F}\left( \mathsf{U}\underline{hom}\left[ \frak{U},\frak{V}\right]
\circ \mathsf{U}\underline{hom}\left[ \frak{W},\frak{U}\right] \right)
\hookrightarrow \mathsf{FU}\underline{hom}\left[ \frak{U},\frak{V}\right]
\otimes \mathsf{FU}\underline{hom}\left[ \frak{W},\frak{U}\right] .
\]
\end{enumerate}

The resulting maps are precisely the arrows $A\left( \Bbb{R}:\Bbb{S}\right)
\rightarrow A\left( \Bbb{R}:\Bbb{T}\right) \otimes A\left( \Bbb{T}:\Bbb{S}%
\right) $ defined in \cite{mm}. For counits:

\begin{enumerate}
\item  Apply $\mathsf{FU}$ to the counit arrow $\varepsilon _{\frak{V}}:%
\underline{end}\left[ \frak{V}\right] \rightarrow \frak{K}$ given in $\left( 
\ref{coun}\right) $, to obtain the arrow $A\left( \Bbb{R}\right) \rightarrow
\Bbbk ^{\otimes }$.

\item  Compose to surjection $\Bbbk ^{\otimes }\twoheadrightarrow \Bbbk $.
\end{enumerate}

The algebra homomorphism $A\left( \Bbb{R}\right) \rightarrow \Bbbk $ we
obtain, together with comultiplication $A\left( \Bbb{R}\right) \rightarrow
A\left( \Bbb{R}\right) \otimes A\left( \Bbb{R}\right) $ gives $A\left( \Bbb{R%
}\right) $ a bialgebra structure. In fact, diagrams $\left( \ref{evco}%
\right) $ and $\left( \ref{idco}\right) $, properly combined with $\left( 
\ref{fa}\right) $, $\left( \ref{fu}\right) $ and $\left( \ref{ua}\right) $,
lead us to associativity and unit constraints for arrows 
\[
A\left( \Bbb{R}:\Bbb{S}\right) \rightarrow A\left( \Bbb{R}:\Bbb{T}\right)
\otimes A\left( \Bbb{T}:\Bbb{S}\right) ,\;\;A\left( \Bbb{R}\right)
\rightarrow \Bbbk . 
\]

Summing up, we have constructed square and rectangular quantum matrices as
internal coHom objects in the rigid category of equipped quantum spaces,
giving a generalization of \emph{FRT} construction in the scenario of Manin
quantum groups.

\subsection{The inclusions $\mathsf{U}\protect\underline{hom}\left[ \frak{W},%
\frak{V}\right] ^{op}\hookrightarrow \protect\underline{hom}\left[ \mathsf{U}%
\frak{W},\mathsf{U}\frak{V}\right] ^{op}$}

Let us show there exist epis $\underline{hom}\left[ \mathsf{U}\frak{W},%
\mathsf{U}\frak{V}\right] \twoheadrightarrow A\left( \Bbb{R}:\Bbb{S}\right) $%
, with $\underline{hom}\left[ \mathsf{U}\frak{W},\mathsf{U}\frak{V}\right] $
the internal coHom object in $\mathrm{QA}$ related to quantum spaces $%
\mathsf{U}\frak{W}$ and $\mathsf{U}\frak{V}$. As we have claimed in the
introduction, this is an indication that quadratic algebras $A\left( \Bbb{R}:%
\Bbb{S}\right) $ represent homomorphisms between structurally richer spaces,
w.r.t. coHom objects in $\mathrm{QA}$.

Consider the evaluation map $\delta _{\frak{V},\frak{W}}:\frak{V}\rightarrow 
\underline{hom}\left[ \frak{W},\frak{V}\right] \boxtimes \frak{W}$ in $%
\mathrm{EQA}$. In the previously given basis $\left\{ v_{i}\right\} $ and $%
\left\{ w_{i}\right\} $ this map is defined by the assignment $v_{i}\mapsto
t_{i}^{j}\otimes w_{j}$, putting again $t_{i}^{j}=w^{j}\otimes v_{i}$. It is
functored by $\mathsf{U}$ to an arrow $\mathsf{U}\frak{V}\rightarrow \mathsf{%
U}\left( \underline{hom}\left[ \frak{W},\frak{V}\right] \boxtimes \frak{W}%
\right) $ which, composed to the functorial epi 
\[
\mathsf{U}\left( \underline{hom}\left[ \frak{W},\frak{V}\right] \boxtimes 
\frak{W}\right) \twoheadrightarrow \mathsf{U}\underline{hom}\left[ \frak{W},%
\frak{V}\right] \circ \mathsf{U}\frak{W}, 
\]
defines in $\mathrm{QA}$ another arrow $\varphi :\mathsf{U}\frak{V}%
\rightarrow \mathsf{U}\underline{hom}\left[ \frak{W},\frak{V}\right] \circ 
\mathsf{U}\frak{W}$, also defined by $v_{i}\mapsto t_{i}^{j}\otimes w_{j}$.
From universality of internal coHom objects in $\mathrm{QA}$ (see Equation $%
\left( \ref{diaun}\right) $), there exists a unique arrow 
\[
\alpha :\underline{hom}\left[ \mathsf{U}\frak{W},\mathsf{U}\frak{V}\right]
\rightarrow \mathsf{U}\underline{hom}\left[ \frak{W},\frak{V}\right] 
\]
such that $\varphi =\left( \alpha \circ I\right) \,\delta _{\mathsf{U}\frak{V%
},\mathsf{U}\frak{W}}$, being $\delta _{\mathsf{U}\frak{V},\mathsf{U}\frak{W}%
}$ the coevaluation associated to $\underline{hom}\left[ \mathsf{U}\frak{W},%
\mathsf{U}\frak{V}\right] \in \mathrm{QA}$. Recalling that (since $\mathsf{U}%
\frak{W}$ and $\mathsf{U}\frak{V}$ are quadratic algebras generated by $%
\mathbf{W}$ and $\mathbf{V}$, resp.) $\underline{hom}\left[ \mathsf{U}\frak{W%
},\mathsf{U}\frak{V}\right] $ is generated by $\mathbf{W}^{\ast }\otimes 
\mathbf{V}$ and $\delta _{\mathsf{U}\frak{V},\mathsf{U}\frak{W}}\left(
v_{i}\right) =t_{i}^{j}\otimes w_{j}$, then $\alpha $ is the identity on
generators. Consequently, due to $\mathsf{U}\underline{hom}\left[ \frak{W},%
\frak{V}\right] $ is also generated by $\mathbf{W}^{\ast }\otimes \mathbf{V}$%
, $\alpha $ is an algebra epimorphism.

Hence, we have shown: For every couple $\frak{W},\frak{V}$ of equipped
quantum space, the quantum space $\underline{hom}\left[ \mathsf{U}\frak{W},%
\mathsf{U}\frak{V}\right] ^{op}$ `contains' $\mathsf{U}\underline{hom}\left[ 
\frak{W},\frak{V}\right] ^{op}$ as a subspace.

\section{The equipped conic quantum spaces}

Given a finite dimensional $\Bbbk $-vector space\textbf{\ }$\mathbf{V}\in $%
\textrm{Vct}$_{f}$, consider the degree cero homogeneous linear
endomorphisms of $\mathbf{V}^{\otimes }$, i.e. 
\[
\Bbb{R}\in End_{\mathrm{Vct}}\left[ \mathbf{V}^{\otimes }\right]
\;\;such\;that\;\;\Bbb{R}\left( \mathbf{V}^{\otimes n}\right) \subset 
\mathbf{V}^{\otimes n}.
\]
Of course, each map $\Bbb{R}$ is defined by a family $\left\{ \Bbb{R}%
_{n}\right\} _{n\in \Bbb{N}_{0}}$ of linear maps $\Bbb{R}_{n}:\mathbf{V}%
^{\otimes n}\rightarrow \mathbf{V}^{\otimes n}$. In terms of these
endomorphisms, all above constructions can be repeated word by word in the
category $\mathrm{CA}$. That is to say, we can define \emph{equipped conic
quantum spaces }as pairs $\left( \mathcal{A},\Bbb{R}\right) $, $\mathcal{A}%
\in \mathrm{CA}$, such that $\mathcal{A}\Vdash \Bbb{R}$, i.e. $\ker \left[ 
\mathbf{A}_{1}^{\otimes }\twoheadrightarrow \mathbf{A}\right] =I\left[ \func{%
Im}\Bbb{R}\right] $. We just must change the defining condition for
morphisms $\left( \mathcal{A},\Bbb{R}\right) \rightarrow \left( \mathcal{B},%
\Bbb{S}\right) $ by $\alpha _{1}^{\otimes }\Bbb{\,R}=\Bbb{S\,}\alpha
_{1}^{\otimes }$, or $\alpha _{1}^{\otimes n}\Bbb{\,R}_{n}=\Bbb{S}_{n}\Bbb{\,%
}\alpha _{1}^{\otimes n}$ for all $n\in \Bbb{N}_{0}$. Let us call $\mathrm{%
ECA}$ the related category. It can be shown, the category of pairs $\frak{V}%
=\left( \mathbf{V},\Bbb{R}\right) $ such that $\func{Im}\Bbb{R}_{1}=\left\{
0\right\} $, with morphisms given by linear maps $l$ such that $l^{\otimes }%
\Bbb{\,R}=\Bbb{S\,}l^{\otimes }$, is equivalent to $\mathrm{ECA}$. The
functors $\boxtimes $ and $\dagger $ are defined by Eqs. $\left( \ref{A}%
\right) $, $\left( \ref{B}\right) $ and $\left( \ref{C}\right) $. $\mathrm{%
EQA}$ can be seen as a full subcategory of $\mathrm{ECA}$ by regarding the
endomorphisms $\Bbb{R}:\mathbf{V}^{\otimes 2}\rightarrow \mathbf{V}^{\otimes
2}$ as the map $\Bbb{R}_{2}$ of an homogeneous endomorphism $\Bbb{R}\in End_{%
\mathrm{Vct}}\left[ \mathbf{V}^{\otimes }\right] $ such that $\Bbb{R}_{n}$
is the null map if $n\neq 2$. This enable us to define the full subcategory
of equipped $m$-th quantum spaces $\mathrm{ECA}^{m}$ in terms of pairs $%
\left( \mathbf{V},\Bbb{R}\right) $ with $\Bbb{R}_{n}$ the null map for all $%
n\neq m$.

Again, function $\left( \mathbf{V},\Bbb{R}\right) \mapsto \left( \mathbf{V}%
,\left. \mathbf{V}^{\otimes }\right/ I\left[ \func{Im}\Bbb{R}\right] \right) 
$ gives rise to a surjective embedding $\mathsf{U}$, now in $\mathrm{CA}$,
such that the functorial epimorphisms $\mathsf{U}\left( \frak{V}\boxtimes 
\frak{W}\right) \twoheadrightarrow \mathsf{U}\frak{V}\circ \mathsf{U}\frak{W}
$ are still valid.

Now, let us prove \textbf{Theor. 4} in this more general setting.

\begin{theorem}
The category of equipped conic quantum spaces is \textbf{rigid }w.r.t. the
monoidal structure $\boxtimes $, and has $\dagger $ as duality functor.
\end{theorem}

\textbf{Proof:}\ Consider an object $\frak{V}=\left( \mathbf{V},\Bbb{R}%
\right) $. We define the evaluation and the coevaluation morphisms, 
\[
ev_{\frak{V}}:\frak{V}^{\dagger }\boxtimes \frak{V}\rightarrow \frak{K}%
\;\;\;\;and\;\;\;\;coev_{\frak{V}}:\frak{K}\rightarrow \frak{V}\boxtimes 
\frak{V}^{\dagger },
\]
as the usual pairing $v\otimes v^{\prime }\mapsto \left\langle v,v^{\prime
}\right\rangle \,e$ and coevaluation of $\mathbf{V}$ and $\mathbf{V}^{\ast }$%
, respectively, being $e$ the generator of $\Bbbk $. We must show that they
are effectively arrows in $\mathrm{E}$\textrm{$CA$}.

Note that the tensor product map $ev_{\frak{V}}^{\otimes }$ defines the
algebra homomorphism $\mathbf{V}^{\ast }\circ \mathbf{V}\rightarrow \Bbbk
^{\otimes }$, 
\[
u\otimes v\in \mathbf{V}^{\ast \otimes n}\otimes \mathbf{V}^{\otimes
n}\mapsto \left\langle u,v\right\rangle \;e^{n}\in \Bbbk ^{\otimes n}, 
\]
where we use $\varphi _{\mathbf{V}^{\ast },\mathbf{V}}$ to identify the
algebras $\mathbf{V}^{\ast }\circ \mathbf{V}$ and $\left[ \mathbf{V}^{\ast
}\otimes \mathbf{V}\right] ^{\otimes }$. By direct calculations (and from
the very definition of $\Bbb{R}^{\dagger }$), it can be seen 
\[
ev_{\frak{V}}^{\otimes }\,\,\Bbb{R}^{\dagger }\boxtimes \Bbb{R}=0=\Bbb{O}%
\,ev_{\frak{V}}^{\otimes }. 
\]
To show the analogous equation for the coevaluation map, let us first
introduce some notation. Let $\left\{ v_{i}\right\} $ be a basis of $\mathbf{%
V}$. Construct for each $n\in \Bbb{N}$ a basis $\left\{ v_{R}\right\} $ of $%
\mathbf{V}^{\otimes n}$, being $R=\left( r_{1},...,r_{n}\right) $ a
multi-index with $1\leq r_{k}\leq \dim \mathbf{V}$, $\forall k=1,...,n$, in
such a way that $v_{R}=v_{r_{1}}\otimes ...\otimes v_{r_{n}}.$ Consider also
the basis $\left\{ v^{R}\right\} $ of $\mathbf{V}^{\ast \otimes n}$ dual to $%
\left\{ v_{R}\right\} $ w.r.t. the usual pairing $\left\langle \cdot ,\cdot
\right\rangle _{n}:\mathbf{V}^{\ast \otimes n}\otimes \mathbf{V}^{\otimes
n}\rightarrow \Bbbk $. In these terms, the algebra homomorphism 
\[
e^{n}\in \Bbbk ^{\otimes }\mapsto v_{R}\otimes v^{R}\in \mathbf{V}\circ 
\mathbf{V}^{\ast } 
\]
(sum over repeated (multi)indices is understood) coincides with the map $%
coev_{\frak{V}}^{\otimes }$. To see that equation 
\[
\Bbb{R}\boxtimes \Bbb{R}^{\dagger }\,coev_{\frak{V}}^{\otimes }=coev_{\frak{V%
}}^{\otimes }\,\Bbb{O} 
\]
holds, we just have to prove $\Bbb{R}\otimes I-I\otimes \Bbb{R}^{\ast }$
evaluated on $v_{R}\otimes v^{R}$ is equal to cero, i.e. 
\[
\left[ \Bbb{R}\left( v_{R}\right) \otimes v^{R}-v_{S}\otimes \Bbb{R}^{\ast
}\left( v^{S}\right) \right] =0. 
\]
Writing $\Bbb{R}\left( v_{R}\right) =\Bbb{R}_{R}^{S}\;v_{S}$, we have for
the transpose $\Bbb{R}^{\ast }\left( v^{S}\right) =v^{R}\;\Bbb{R}_{R}^{S}$,
and consequently the left member of equation above is identically cero. So,
the theorem is proven.\ \ \ $\blacksquare $

\subsubsection*{Acknowledgments}

The author thanks to CNEA and Fundaci\'{o}n Antorchas, Argentina, for
financial support.

\begin{thebibliography}{99}
\bibitem{kas}  C. Kassel, \emph{Quantum Groups}, SpringerVerlag, New York
Inc., 1995.

\bibitem{frt}  N.Yu. Reshetikhin, L.A. Takhtadjian, L.D. Faddeev, \emph{%
Quantization of Lie Groups and Lie Algebras}, Leningrad Math. J. \textbf{1},
193-225 (1990).

\bibitem{A}  E. Abe, \emph{Hopf algebras}, Cambridge Tracks on Math., vol.
74, Cambridge Univ. Press, 1980.

\bibitem{dj}  V. G. Drinfeld, \emph{Quantum groups}, Proc. Int. Congr. Math.
Berkeley, vol. 1, 798-820 (1986); M. Jimbo, \emph{A q-difference analogue of 
}$U\left( \frak{g}\right) $ \emph{and the Yang-Baxter equation}, Lett. Math.
Phys. \textbf{11}, 247-252 (1986).

\bibitem{man0}  Yu. Manin, \emph{Some remarks on Koszul algebras and quantum
groups}, Ann. Inst. Fourier \textbf{33 }(4), 191-205 (1987); Yu. Manin., 
\emph{Quantum groups and non commutative geometry}, CRM, Universit\'{e} de
Montr\'{e}al, 1988.

\bibitem{mm}  S. Majid, M. Markl, \emph{Glueing operation for R-matrices,
quantum groups and link-invariants of Hecke type}, \texttt{hep-th/9308072}.

\bibitem{maj}  S. Majid, \emph{Foundations of quantum group theory},
Cambridge University Press, 1995.

\bibitem{phh}  Pungh Ho Hai, \emph{Realizations of quantum Hom-spaces,
invariant theory and quantum determinantal ideals},\texttt{\ math.QA/0005090.%
}

\bibitem{dm}  P. Deligne, \emph{Cat\'{e}gories tannaki\`{e}nnes}, preprint
(1988).

\bibitem{man1}  Yu. Manin, \emph{Topics in non commutative geometry},
Princeton Univeristy Press, 1991.

\bibitem{gm}  S. Grillo, H. Montani, \emph{Twisted internal coHom objects in
the category of quantum spaces}, \texttt{math.QA/0112233}\textbf{.}
\end{thebibliography}

\end{document}

math0205334.extracted_bib
\begin{thebibliography}{99}
\bibitem{kas}  C. Kassel, \emph{Quantum Groups}, SpringerVerlag, New York
Inc., 1995.

\bibitem{frt}  N.Yu. Reshetikhin, L.A. Takhtadjian, L.D. Faddeev, \emph{%
Quantization of Lie Groups and Lie Algebras}, Leningrad Math. J. \textbf{1},
193-225 (1990).

\bibitem{A}  E. Abe, \emph{Hopf algebras}, Cambridge Tracks on Math., vol.
74, Cambridge Univ. Press, 1980.

\bibitem{dj}  V. G. Drinfeld, \emph{Quantum groups}, Proc. Int. Congr. Math.
Berkeley, vol. 1, 798-820 (1986); M. Jimbo, \emph{A q-difference analogue of 
}$U\left( \frak{g}\right) $ \emph{and the Yang-Baxter equation}, Lett. Math.
Phys. \textbf{11}, 247-252 (1986).

\bibitem{man0}  Yu. Manin, \emph{Some remarks on Koszul algebras and quantum
groups}, Ann. Inst. Fourier \textbf{33 }(4), 191-205 (1987); Yu. Manin., 
\emph{Quantum groups and non commutative geometry}, CRM, Universit\'{e} de
Montr\'{e}al, 1988.

\bibitem{mm}  S. Majid, M. Markl, \emph{Glueing operation for R-matrices,
quantum groups and link-invariants of Hecke type}, \texttt{hep-th/9308072}.

\bibitem{maj}  S. Majid, \emph{Foundations of quantum group theory},
Cambridge University Press, 1995.

\bibitem{phh}  Pungh Ho Hai, \emph{Realizations of quantum Hom-spaces,
invariant theory and quantum determinantal ideals},\texttt{\ math.QA/0005090.%
}

\bibitem{dm}  P. Deligne, \emph{Cat\'{e}gories tannaki\`{e}nnes}, preprint
(1988).

\bibitem{man1}  Yu. Manin, \emph{Topics in non commutative geometry},
Princeton Univeristy Press, 1991.

\bibitem{gm}  S. Grillo, H. Montani, \emph{Twisted internal coHom objects in
the category of quantum spaces}, \texttt{math.QA/0112233}\textbf{.}
\end{thebibliography}