# On module categories related to $Sp(N-1)\subset Sl(N)$

Hans Wenzl  Department of Mathematics
University of California
San Diego, California<EMAIL_ADDRESS>
###### Abstract.

Let $V={\mathbb{C}}^{N}$ with $N$ odd. We construct a $q$-deformation of ${\rm
End}_{Sp(N-1)}(V^{\otimes n})$ which contains ${\rm
End}_{U_{q}{\mathfrak{s}l}_{N}}(V^{\otimes n})$. It is a quotient of an
abstract two-variable algebra which is defined by adding one more generator to
the generators of the Hecke algebras $H_{n}$. These results suggest the
existence of module categories of $Rep(U_{q}{\mathfrak{s}l}_{N})$ which may
not come from already known coideal subalgebras of $U_{q}{\mathfrak{s}l}_{N}$.
We moreover indicate how this can be used to construct module categories of
the associated fusion tensor categories as well as subfactors, along the lines
of previous work for inclusions $Sp(N)\subset SL(N)$.

IN MEMORIAM OF VAUGHAN JONES WITH GRATITUDE

The problem of classifying module categories of a given tensor category arises
in different contexts such as conformal field theory and the study of
subfactors. A lot of progress has been made for module categories of fusion
categories coming from quantum groups (or Wess-Zumino-Witten models) in recent
work by Edie-Michell (see [4]), building on work of Jones, Ocneanu, Gannon,
Schopieray, Evans and Pugh and others. But in many cases, a detailed
description of those module categories such as fusion rules, algebras etc is
still not available.

It is well-known that $Rep(H)$ is a module category of $Rep(G)$ for an
inclusion of groups $H\subset G$. The basic idea here is to find a subgroup
$H$ of a Lie group $G$ for which one can find analogs of the module $Rep(H)$
in fusion categories related to $G$. This was successfully carried out in [24]
and [25] for inclusions $Sp(N)\subset SL(N)$, $N$ even, and $O(N)\subset
SL(N)\times{\mathbb{Z}}_{2}$ for arbitrary $N>1$. It allowed a detailed
description of these module categories in terms of the well-known
combinatorics of these groups (see Section 5). The current paper roughly
contains the analogous results of the paper [24] for the inclusions
$Sp(N-1)\subset SL(N)$ for $N$ odd.

Here is our approach in more detail, formulated for the group $G=SL(N)$ for
simplicity. We denote the quotient tensor category of tilting modules of the
quantum group $U_{q}{\mathfrak{s}l}_{N}$ modulo negligible modules for $q$ a
root of unity by $\overline{Rep}(U_{q}{\mathfrak{s}l}_{N})$. It is often also
denoted by $SU(N)_{k}$ for $q^{2}$ a primitive $(N+k)^{th}$ root of unity. We
study the following questions: Find a subgroup $H\subset SL(N)$ for which we
can find

(a) a $q$-deformation of $Rep(H)$ which is a module category of
$Rep(U_{q}{\mathfrak{s}l}_{N})$,

(b) a quotient of said $q$-deformation for $q$ a root of unity which is a
module category of

$\overline{Rep}(U_{q}{\mathfrak{s}l}_{N})$,

(c) a subfactor corresponding to the module category in (b) if it is
unitarizable.

Before describing the results in this paper in more detail, we would like to
make a few general remarks about this approach. It is known that Question (a)
can be solved if $H$ is the group of fixed points under a period 2
automorphism via coideal subalgebras, see work of Letzter [13] and of Noumi
and Sugitani [15]. It follows from work in [14] that the examples in [25]
indeed correspond to special cases of the work in [13] and [15]. However, it
is also clear that not all module categories constructed via coideal algebras
allow solutions of Questions (b) and (c). This can be seen e.g. for $SL(3)$,
where all module categories of the corresponding fusion categories are known
due to work of Gannon and Evans and Pugh, see [6].

As stated in the title, we consider the inclusion of $Sp(N-1)\subset SL(N)$
for $N$ odd. One of the main results of this paper is the construction of a
sequence of two-parameter algebras $C_{n}=C_{n}^{(N)}(q)$ which contain the
Hecke algebras $H_{n}(q)$ of type $A_{n-1}$ as subalgebras. For $q\neq\pm 1$
it suffices to add one more generator $e$, corresponding to the projection
onto the trivial $Sp(N-1)$ submodule ${\bf 1}\subset V$, where
$V={\mathbb{C}}^{N}$ is the vector representation of $SL(N)$. In order to get
the correct algebra in the classical limit $q\to\pm 1$, we also need
additional generators to obtain nontrivial morphisms between the two copies of
the trivial representation in $V^{\otimes 2}$. We give a presentation of these
algebras via generators and relations and an explicit basis. Another important
result is the proof of the existence of an extension of the Markov trace for
$H_{n}$ to the algebras $C_{n}$ which satisfies a generalized Markov
condition.

We now give a more detailed description of the results of this paper which
will also explain how these algebras can be used to construct module
categories. We fix notations in the first section and prove a number of
combinatorial and algebraic results concerning ${\rm End}_{Sp(N-1)}(V^{\otimes
n})$. We derive relations for a $q$-deformation of ${\rm
End}_{Sp(N-1)}(V^{\otimes n})$ which contains the Hecke algebra $H_{n}$ as a
subalgebra in the second section. These relations are essentially forced by
the fusion respectively restriction rules for $Sp(N-1)\subset SL(N)$ and a
generalized Markov condition (see Condition 2.9 or the discussion here for
Section 4). In particular, we show that only two solutions are possible for
fixed $q\neq\pm 1$ and $N>1$ odd (see Remark 2.6). Hence we have two possible
choices $C_{n,\pm}$ of extensions of $H_{n}$ subject to our conditions. As
they are closely related, see Remark 2.9, we will often just use the notation
$C_{n}$ for either of these cases. We also define a version of $C_{n}$
depending on two parameters $q$ and $p$, which specializes to the original
version for $p=q^{N}$. It is then shown that for $N>2n$ the dimension of these
algebras is at most $\dim{\rm End}_{Sp(N-1)}(V^{\otimes n})$ and explicit
spanning sets are determined. We define representations of $C_{n}^{(N)}$ into
${\rm End}(V^{\otimes n})$ in the third section. It is shown that their images
in the classical limit $q=1$ concide with ${\rm End}_{Sp(N-1)}(V^{\otimes
n})$. We conclude from this that the given spanning sets in the previous
section are actually bases. Section 4 contains a proof that we can extend the
Markov trace $tr$ of the Hecke algebra $H_{n}$ to a trace on $C_{n}$
satisfying the generalized Markov condition $tr(cg_{n})=tr(c)tr(g_{n})$ for
all $c\in C_{n}$. In the last section, we first briefly describe how this
paper has been influenced by the work of Vaughan Jones, even though this will
be pretty obvious to experts anyways. We then indicate how our algebras can be
used to construct module categories and subfactors. To do so, we consider the
quotients $\overline{H}_{n}$ and $\overline{C}_{n}$ of the algebras $H_{n}$
and $C_{n}$ modulo the annihilator ideals of the Markov trace $tr$. The
objects of the category and of the module category are given by idempotents in
$\overline{H}_{n}$ and $\overline{C}_{n}$ respectively, and the module action
comes from the natural inclusion map $C_{n}\otimes H_{m}\to C_{n+m}$. To
illustrate this, we restate results from [24] and [25] in the language of
module categories which was not used there. This shows, in particular, that
Problems (a)-(c) have been solved for the inclusions considered in those
papers. We finally discuss how the approach in this paper can be used to give
detailed descriptions for a large class of module categories.

$Acknowledgements:$ I would like to thank Gail Letzter and Cain Edie-Michell
for useful conversations.

##  1\. Fusion rules for the embedding of $Sp(N-1)\subset SL(N)$, $N$ odd

###  1.1. Fusion rules

Let $V={\mathbb{C}}^{N}$ with $N$ odd. We fix a symplectic bilinear form $(\
,\ )^{\prime}$ on $V$ with 1-dimensional kernel spanned by the nonzero vector
$v_{o}$ and a complement $V^{\prime}$ of $v_{o}$ on which the form $(\ ,\
)^{\prime}$ is nondegenerate. This defines an embedding $Sp(N-1)\subset SL(N)$
such that $V$ decomposes into the direct sum
$V^{\prime}\oplus{\mathbb{C}}v_{o}$ as an $Sp(N-1)$ module. Recall that the
finite-dimensional simple representations of $Sp(N-1)$ are labelled by Young
diagrams with $\leq(N-1)/2$ rows, with the $(N-1)$-dimensional simple
representation $V^{\prime}$ labeled by the Young diagram with one box. If
$V_{\lambda}$ is a simple representation of $Sp(N-1)$ labeled by the Young
diagram $\lambda$, we have

(1.1) $V_{\lambda}\otimes
V^{\prime}\cong\bigoplus_{\mu\leftrightarrow\lambda}V_{\mu},$

where $\mu$ ranges over all Young diagrams which can be obtained from
$\lambda$ by either adding or removing a box. One deduces from this the tensor
product rule

(1.2) $V_{\lambda}\otimes V\cong
V_{\lambda}\oplus\bigoplus_{\mu\leftrightarrow\lambda}V_{\mu},$

with $\mu$ as in 1.1.

###  1.2. Bratteli diagrams and path bases

The inclusions of the algebras

$...\subset{\rm End}_{Sp(N-1)}(V^{\otimes n})\subset{\rm
End}_{Sp(N-1)}(V^{\otimes n+1})\subset\ ...$

are conveniently described by a Bratteli diagram. It follows from the tensor
product rules 1.2 that its vertices at level $n$ are labeled by Young diagrams
$\lambda$ with $|\lambda|\leq n$, where $|\lambda|$ denotes the number of
boxes in the Young diagram. A diagram $\lambda$ at level $n$ is connected with
a diagram $\mu$ at level $n+1$ if $\lambda$ differs from $\mu$ by at most one
box. The multiplicity of $V_{\lambda}$ in $V^{\otimes n}$ is then given by the
number of paths of length $n$ from level 0 to level $n$. As we only have
multiplicities 0 or 1, we obtain a basis of ${\rm Hom}(V_{\lambda},V^{\otimes
n})$ labeled by the paths of length $n$ which end in $\lambda$ for any
irreducible $Sp(N-1)$ module $V_{\lambda}$ labeled by $\lambda$. This basis is
unique up to rescaling by nonzero scalars. Below is the part of the Bratteli
diagram containing level 1 and level 2.

$\emptyset$[1][2][1,1]$\emptyset$[1]Inclusion diagram${\rm
End}_{Sp(N-1)}(V^{\otimes 2})$${\rm End}_{Sp(N-1)}(V)$

###  1.3. Multiplicities for large $N$

The tensor product rule 1.2 allows us to calculate the multiplicity
$m_{n,\lambda}$ of the simple module $V_{\lambda}$ in $V^{\otimes n}$. If
$N>2n$, there are no restrictions on Young diagrams, and we can give closed
formulas for the multiplicities $m_{n,\lambda}$. To do so, we define integers
$h_{r}$ inductively by $h_{0}=1$, $h_{1}=1$ and

(1.3) $h_{r+1}=h_{r}+rh_{r-1}.$

We denote by $d_{\lambda}$ the dimension of the simple $S_{n}$ module labeled
by the Young diagram $\lambda$, where the number of boxes $|\lambda|$ of
$\lambda$ is equal to $n$. There exists a well-known explicit formula for it
in terms of the hook lengths of $\lambda$. We will need the following well-
known identities

(1.4) $\sum_{\mu<\lambda}d_{\mu}\ =\ d_{\lambda}\quad{\rm
and}\quad\sum_{\nu>\lambda}d_{\nu}=(n+1)d_{\lambda},$

where $|\lambda|=n$, and $\mu$ and $\nu$ range over all Young diagrams which
can be obtained by removing a box from $\lambda$ (for $\mu$) or adding a box
to $\lambda$ (for $\nu$).

######  Proposition 1.1.

(a) The multiplicity $m_{n,\lambda}$ of the simple module $V_{\lambda}$ in
$V^{\otimes n}$ for $N>2n$ is equal to
$h_{n-|\lambda|}\binom{n}{|\lambda|}d_{\lambda}$.

(b) We have the identity

$\sum_{|\lambda|=n-r}m_{n,\lambda}d_{\lambda}\ =\ h_{r}\frac{n!}{r!}$

(c) If $N>2n$, we have $\dim{\rm End}_{Sp(N-1)}(V^{\otimes
n})=\sum_{r=0}^{n}h_{r}^{2}\frac{n!}{r!}\binom{n}{r}$.

$Proof.$ We will prove (a) by induction on $n$, with the claim easily checked
for $n=1$. It follows from 1.2 that $V_{\lambda}\subset V_{\mu}\otimes V$ if
and only if $\mu=\lambda$ or $\mu$ is obtained by removing or adding a box
from/to $\lambda$. In each of theses cases $V_{\lambda}$ appears with
multiplicity 1. It follows that

$m_{n,\lambda}\ =\ m_{n-1,\lambda}\ +\ \sum_{\mu<\lambda}m_{n-1,\mu}\ +\
\sum_{\nu>\lambda}m_{n-1,\nu}.$

Using the induction assumption, the identities 1.3, 1.4 and the identity
$\binom{n-1}{k+1}(k+1)=\binom{n-1}{k}(n-1-k)$, we obtain

$\displaystyle m_{n,\lambda}\ $
$\displaystyle=h_{n-|\lambda|-1}\binom{n-1}{|\lambda|}d_{\lambda}\ +\
\sum_{\mu<\lambda}h_{n-|\lambda|}\binom{n-1}{|\lambda|-1}d_{\mu}\ +\
\sum_{\nu>\lambda}h_{n-|\lambda|-2}\binom{n-1}{|\lambda|+1}d_{\nu}$
$\displaystyle=h_{n-|\lambda|-1}\binom{n-1}{|\lambda|}d_{\lambda}\ +\
h_{n-|\lambda|}\binom{n-1}{|\lambda|-1}d_{\lambda}\ +\
h_{n-|\lambda|-2}\binom{n-1}{|\lambda|+1}d_{\lambda}(|\lambda|+1)$
$\displaystyle=\ d_{\lambda}[h_{n-|\lambda|-1}\binom{n-1}{|\lambda|}\ +\
h_{n-|\lambda|}\binom{n-1}{|\lambda|-1}\ +\
h_{n-|\lambda|-2}\binom{n-1}{|\lambda|}(n-1-|\lambda|)$ $\displaystyle=\
d_{\lambda}[\binom{n-1}{|\lambda|}(h_{n-|\lambda|-1}+h_{n-|\lambda|-2}(n-1-|\lambda|))\
+\ h_{n-|\lambda|}\binom{n-1}{|\lambda|-1}$ $\displaystyle=\
d_{\lambda}h_{n-|\lambda|}(\binom{n-1}{|\lambda|}+\binom{n-1}{|\lambda|-1}).$

This proves part (a). Part (b) follows from this and the identity
$\sum_{|\lambda|=n-r}d_{\lambda}^{2}=(n-r)!$. Part(c) similarly follows from
this, part (a) and $\dim{\rm End}_{Sp(N-1)}(V^{\otimes n})=\sum_{|\lambda|\leq
n}m_{n,\lambda}^{2}$.

###  1.4.  Description of ${\rm End}_{Sp(N-1)}(V^{\otimes n})$

We denote by $E\in{\rm End}(V)$ the projection onto the trivial representation
${\mathbb{C}}v_{o}$ of $Sp(N-1)$ with kernel $V^{\prime}$, and by $U$ the
antisymmetrization map

$U:\ v\otimes w\ \mapsto\ v\wedge w=v\otimes w-w\otimes v,\quad v,w\in V.$

The elements $U_{i}\in{\rm End}(V^{\otimes n})$ are defined for $1\leq i<n$ by

(1.5) $U_{i}\ =\ 1\otimes 1\otimes\ ...\ \otimes U\ \otimes\ ...\ \otimes 1,$

where $U$ acts on the $i$-th and $(i+1)$-st factors in $V^{\otimes n}$.
Observe that the flip $G:v\otimes w\in V^{\otimes 2}\mapsto w\otimes v$ is
related to $U$ by the simple formula $G=1-U$. We now extend the bilinear form
$(\ ,\ )^{\prime}$ to a non-degenerate bilinear form $(\ ,\ )$ on $V$ by
defining

(1.6) $(v_{o},v_{o})=1\quad{\rm
and}\quad(v_{o},v^{\prime})=0=(v^{\prime},v_{o})\quad{\rm for\ all}\
v^{\prime}\in V^{\prime}.$

If $(v_{i}^{\prime})$ and $(w_{i}^{\prime})$ are dual bases of $V^{\prime}$
with respect to $(\ ,\ )^{\prime}$, we obtain the canonical vector
$v_{o}\otimes v_{o}+\sum_{i}v_{i}^{\prime}\otimes w_{i}^{\prime}\in V^{\otimes
2}$ for the form $(\ ,\ )$. We define the element $F\in{\rm End}(V^{\otimes
2})$ by

$F(v\otimes w)\ =\ (v,w)\ (v_{o}\otimes v_{o}+\sum_{i}v_{i}^{\prime}\otimes
w_{i}^{\prime}).$

The elements $F_{i}$ and $G_{i}$ in ${\rm End}(V^{\otimes n})$ are defined in
the same way as $U_{i}$ in 1.5. It is well-known that the element $F$ can be
used to calculate the trace of an element $A\in{\rm End}(V)$ by

$Tr(A)F=F(A\otimes 1)F.$

Let us now decompose $V^{\otimes n}$ into the direct sum of three $Sp(N-1)$
submodules as follows. Write $V^{\otimes
n}=\bigoplus_{\lambda}m_{\lambda}V_{\lambda}$, where $m_{\lambda}$ is the
multiplicity of the simple module $V_{\lambda}$ in $V^{\otimes n}$. For given
$n$, we call the diagram $\lambda$ an old / recent / new diagram in
$V^{\otimes n}$ if $V_{\lambda}$ has appeared for the first time in
$V^{\otimes m}$ with $m\leq n-2$ / $m=n-1$ / $m=n$. We define

$V^{\otimes n}_{old}\ =\ \bigoplus_{\lambda\ old}m_{\lambda}V_{\lambda},$

and $V^{\otimes n}_{rec}$ and $V^{\otimes n}_{new}$ accordingly. Then it
follows from the tensor product rule 1.2 that $\lambda$ is an old / recent/
new diagram if $|\lambda|\leq n-2$/$|\lambda|=n-1$ / $|\lambda|=n$.

######  Theorem 1.2.

(see e.g. [23], Proposition 4.10) Let $V$ be a finite-dimensional self-dual
$G$-module. Then ${\rm End}_{G}(V^{\otimes n}_{old})$ is given by a Jones
basic construction for ${\rm End}_{G}(V^{\otimes n-2})\subset{\rm
End}_{G}(V^{\otimes n-1})$. Moreover, it coincides with the two-sided ideal in
${\rm End}_{G}(V^{\otimes n})$ generated by $F_{n-1}$, which is spanned by
elements of the form $aF_{n-1}b$, with $a,b\in{\rm End}_{G}(V^{\otimes
n-1}\otimes 1)$.

######  Theorem 1.3.

The algebra ${\rm End}_{Sp(N-1)}(V^{\otimes n})$ is generated by the symmetric
group $S_{n}$, acting via permutations of the factors of $V^{\otimes n}$, the
element $E\otimes 1_{n-1}$ and the element $F_{1}$.

$Proof.$ The proof goes by induction on $n$, with $n=1$ obviously true. The
claim follows for ${\rm End}(V^{\otimes n})_{old}$ from Theorem 1.2 and the
induction assumption. If $V_{\lambda}\subset V^{\otimes n}_{new}$, i.e.
$|\lambda|=n$, it follows from the tensor product rules that
$V_{\lambda}\subset(V^{\prime})^{\otimes n}$. In this case, the claim follows
from Brauer’s classical result. It also implies that $W_{\lambda}={\rm
Hom}(V_{\lambda},V^{\otimes n})$ is an irreducible $S_{n}$ module. To deal
with the remaining cases, observe that the elements $G_{i}$, $1\leq i<n$ and
$T=(1-2E)\otimes 1_{n-1}$ satisfy the relations of the Weyl group $W(B_{n})$
of type $B_{n}$. Hence the quotient modulo the ideal generated by $F_{n-1}$ is
also a quotient of the group algebra of $W(B_{n})$. We will finish the proof
in the next section after a brief review of the representation theory of
$W(B_{n})$.

###  1.5. Weyl group of type $B_{n}$

The Weyl group $W(B_{n})$ of type $B_{n}$ can be defined via generators $t$
and $s_{i}$, $1\leq i<n$ and relations such that the $s_{i}$ are simple
reflections of the symmetric group (e.g. we can take $s_{i}=(i,i+1)$) and such
that $t$ commutes with $s_{i}$ for $i>1$ and satisfies
$s_{1}ts_{1}t=ts_{1}ts_{1}$. It is well-known that it is isomorphic to the
semidirect product $({\mathbb{Z}}/2)^{n}\rtimes S_{n}$, with $t$ corresponding
to $(1,0,...,0)\in({\mathbb{Z}}/2)^{n}$, and with $S_{n}$ permuting the
coordinates of elements of $({\mathbb{Z}}/2)^{n}$. The irreducible
representations of $W$ are labeled by pairs of Young diagrams $(\lambda,\mu)$
with $|\lambda|+|\mu|=n$. They can be constructed as follows (see e.g. [18],
Chapter 8 for details):

Let $\phi$ be a character of $({\mathbb{Z}}/2)^{n}$ such that
$\phi(\varepsilon_{i})=-1$ for $i\leq r$ and $\phi(\varepsilon_{i})=1$ for
$i>r$; here $\varepsilon_{i}$ is the $i$-th unit vector in
$({\mathbb{Z}}/2)^{n}$. The centralizer of $\phi$ consists of all $g\in
W(B_{n})$ such $\phi(gxg^{-1})=\phi(x)$ for all $x\in({\mathbb{Z}}/2)^{n}$. It
is easy to see that for our choice of $\phi$ the centralizer is equal to
$({\mathbb{Z}}/2)^{n}\rtimes(S_{r}\times S_{n-r})$. Let $W_{\lambda}$ and
$W_{\mu}$ be irreducible representations of $S_{r}$ and $S_{n-r}$. Then
$W_{\lambda}\otimes W_{\mu}$ becomes an irreducible representation of
$({\mathbb{Z}}/2)^{n}\rtimes(S_{r}\times S_{n-r})$, where the action of
$x\in({\mathbb{Z}}/2)^{n}$ is given by the scalar $\phi(x)$. It can then be
shown that inducing this representation up to $W(B_{n})$ yields an irreducible
representation of $W(B_{n})$ of dimension $\binom{n}{r}d_{\lambda}d_{\mu}$.

Conclusion of proof of Theorem 1.3 Let $|\lambda|=n-1$. Then $E\otimes
E\otimes 1_{n-2}$, and hence also $F_{1}$ acts as 0 on ${\rm
Hom}(V_{\lambda},V^{\otimes n})$, with $V_{\lambda}$ an irreducible
$Sp(N-1)$-module. Hence we can view ${\rm Hom}(V_{\lambda},V^{\otimes n})$ as
a $W(B_{n})$-module on which $E\otimes 1_{n-1}$ acts nontrivially; indeed, the
module ${\mathbb{C}}v_{o}\otimes V_{\lambda}\cong V_{\lambda}$ is in the image
of $E\otimes 1_{n-1}$. As ${\mathbb{C}}v_{0}\otimes
V_{\lambda}\subset{\mathbb{C}}v_{o}\otimes(V^{\prime})^{\otimes n-1}$, it also
follows that $E_{i}$ acts as 0 on ${\mathbb{C}}v_{o}\otimes V_{\lambda}$ for
$i>1$. Hence the action of $({\mathbb{Z}}/2)^{n}$ on ${\mathbb{C}}v_{o}\otimes
V_{\lambda}$ is given by the functional
$\phi:x\in({\mathbb{Z}}/2)^{n}\mapsto(-1)^{x_{1}}$. We obtain that ${\rm
Hom}(V_{\lambda},V^{\otimes n})$ contains an irreducible $W(B_{n})$-module
labeled by $([1],\lambda)$. By the previous discussion it has dimension $n\
d_{\lambda}=w_{n,\lambda}=\dim{\rm Hom}(V_{\lambda},V^{\otimes n})$. Hence
$W(B_{n})$ and therefore also $C_{n}$ acts irreducibly on ${\rm
Hom}(V_{\lambda},V^{\otimes n})$.

##  2\. A $q$-deformation of ${\rm End}_{Sp(N-1)}(V^{\otimes n})$

The goal of this paper is to study $q$ deformations of $Rep(Sp(N-1))$ at the
categorical level which are compatible with the deformation of $Rep(Sl(N))$ to
$U_{q}{\mathfrak{s}l}_{N}$ and with the embedding $Sp(N-1)\subset Sl(N)$. As
we shall see, this leads to a structure different from
$Rep(U_{q}{\mathfrak{s}p}_{N-1})$.

###  2.1. Hecke algebras

The Hecke algebra $H_{n}=H_{n}(q)$ of type $A_{n-1}$ is defined via generators
$g_{i}$, $1\leq i<n$ and relations

$g_{i}g_{i+1}g_{i}\ =\ g_{i+1}g_{i}g_{i+1},\quad 1\leq i<n-1,$

together with $g_{i}^{2}=(q-q^{-1})g_{i}+1$ and $g_{i}g_{j}=g_{j}g_{i}$ for
$|i-j|\neq 1$. Defining

$u_{i}=q1-g_{i},$

the relations above translate to $u_{i}^{2}=(q+q^{-1})u$,
$u_{i}u_{j}=u_{j}u_{i}$ for $|i-j|\neq 1$ and

(2.1) $u_{i}u_{i+1}u_{i}-u_{i}\ =\ u_{i+1}u_{i}u_{i+1}-u_{i+1}.$

We replace $Rep(Sl(N))$ by the representation category of the Drinfeld-Jimbo
quantum group $U_{q}{\mathfrak{s}l}_{N}$. It was shown in [9] that the
generalization of the map $U$ of the previous subsection can then be defined
with respect to a basis $\\{v_{i}\\}$ for $V$ by $U(v_{i}\otimes v_{i})=0$ and
by defining its restriction to the ordered basis vectors $v_{i}\otimes v_{j}$
and $v_{j}\otimes v_{i}$, $i<j$ by the matrix

(2.2) $\ U\ =\ \left[\begin{matrix}q^{-1}&-1\\\ -1&q\end{matrix}\right].$

The elements $U_{i}\in{\rm End}(V^{\otimes n})$ are defined for $1\leq i<n$ as
in 1.5.

###  2.2. Dimensions and traces

We review some basics about dimension functions for representations of quantum
groups, see [19] and [10], XIV.4 for more details. Let $W$ be an
$U_{q}{\mathfrak{s}l}_{N}$-module with dual module $W^{*}$. Then there exist
canonical morphisms

$b_{W}:{\bf 1}\to W\otimes W^{*}\quad{\rm and}\quad d^{\prime}_{W}:W\otimes
W^{*}\to{\bf 1}$

such that for $a\in{\rm End}(W)$ we define

(2.3) $Tr_{q}(a)=d_{W}^{\prime}(a\otimes 1)b_{W}.$

We remark that while $Tr_{q}$ does not satisfy the trace property
$Tr_{q}(ab)=Tr_{q}(ba)$ in general, its restriction to ${\rm
End}_{U_{q}{\mathfrak{s}l}_{N}}(W)$ is indeed a trace. There exists an element
$q^{2\rho}\in U_{q}{\mathfrak{s}l}_{N}$ such that $Tr_{q}(a)=Tr(aq^{2\rho})$.
We will only need to know its action on $W=V^{\otimes n}$, where we have

(2.4) $Tr_{q}(a)\ =\ Tr(aD^{\otimes n}),\quad a\in{\rm End}(V^{\otimes
n}),\quad D=diag(q^{2i-N-1}).$

More generally, we can define a partial trace (also referred to as a
contraction, or a conditional expectation) $\mathcal{E}_{X}:{\rm End}(X\otimes
W)\to{\rm End}(X)$ by

(2.5) $\mathcal{E}_{X}(a)\ =\ (1_{X}\otimes d_{W}^{\prime})(a\otimes
1_{W^{*}})(1_{X}\otimes b_{W}),$

which satisfies $Tr_{q}(a)=Tr_{q}(\mathcal{E}_{X}(a))$. If we take
$X=W=V={\mathbb{C}}^{N}$, the vector representation of
$U_{q}{\mathfrak{s}l}_{N}$, it follows from the discussion above that
$\mathcal{E}_{X}(U)=Tr_{q}(U)1_{V}$, as $V$ is irreducible. One deduces from
this more generally that

(2.6) $\mathcal{E}_{n}(U_{n})\ =\ Tr(U_{n})1,$

where $\mathcal{E}_{n}$ is the partial trace from ${\rm End}(V^{\otimes n+1})$
to ${\rm End}(V^{\otimes n})$.

As usual, we define the $q$-number $[k]=(q^{k}-q^{-k})/(q-q^{-1})$. Then the
$q$-dimension of the irreducible $U_{q}{\mathfrak{s}l}_{N}$ module $V_{\mu}$
with highest weight $\mu$ is given by

(2.7) $\dim_{q}V_{\mu}\ =\ \prod_{1\leq i<j\leq
N}\frac{[\mu_{i}-\mu_{j}+j-i]}{[j-i]}.$

The dimension of a simple $U_{q}{\mathfrak{s}p}_{2k}$-module $V_{\lambda}$
labeled by the Young diagram $\lambda$ is given by

(2.8) $\dim_{q}V_{\lambda}\ =\
\prod_{i<j}\frac{[\lambda_{i}-\lambda_{j}+j-i][\lambda_{i}+\lambda_{j}+2k+2-i-j]}{[j-i][2k+2-i-j]}\
\prod_{i=1}^{k}\frac{[2\lambda_{i}+2k+2-2i]}{[2k+2-2i]}.$

######  Remark 2.1.

Let $H_{n}^{(N)}$ be the image of the Hecke algebra $H_{n}$ in the
representation in $V^{\otimes n}$. Then we refer to the trace $tr$ on $H_{n}$
such that $tr(p_{\mu})=\dim_{q}V_{\mu}/[N]^{n}$ (as in 2.7) for a minimal
idempotent $p_{\mu}$ in the direct summand of $H_{n}$ labeled by $\mu$ as the
Markov trace on $H_{n}^{(N)}$. It was shown in [21] that the quotient
$\overline{H}_{n}$ modulo the annihilator ideal of $tr$ is indeed isomorphic
to $H_{n}^{(N)}$ for $q$ not a root of unity.

###  2.3. Posing the question

We can now make the above mentioned problem of finding a $q$-deformation for
the embedding $Sp(N-1)\subset SL(N)$ more precise as follows: We fix $N$ and
denote by $H_{n}^{(N)}$ the image of the Hecke algebra $H_{n}$ in the
representation in $V^{\otimes n}$. Then we want to find extensions
$C_{n}=C_{N}^{(N)}$ of $H_{n}^{(N)}$ such that

(a) $C_{n}\subset C_{n+1}$,

(b) $H_{n}^{(N)}\subset C_{n}\cong{\rm End}_{SL(N)}(V^{\otimes})\subset{\rm
End}_{Sp(N-1)}(V^{\otimes})$ at least for $q$ not a root of unity,

(c) (Markov property) There exists a normalized trace $tr$ on $C_{n}$
extending the Markov trace on the Hecke algebra $H_{n}^{(N)}$ compatible with
the embedding $C_{n}\subset C_{n+1}$ for all $n$ such that $tr(e)=1/[N]$ and

(2.9) $tr(cg_{n})=tr(c)tr(g_{n})\quad{\rm for\ all\ }c\in C_{n}.$

The following lemma will only be needed for $n\leq 2$ in this paper (but more
later).

######  Lemma 2.2.

If a trace $tr$ on $C_{n}$ as in (c) exists, its value on a minimal idempotent
$p_{\lambda}$ in the direct summand of $C_{n}$ labeled by $\lambda$ is equal
to $\dim_{q}V_{\lambda}/[N]^{n}$, with $\dim_{q}V_{\lambda}$ given in 2.8.

$Proof$. We view $D$ as the representation of an element of $Sp(N-1)$ with
eigenvalues $q^{\pm 2j}$, $1\leq j\leq(N-1)/2$. These are exactly the
eigenvalues of the element $q^{2\rho^{\prime}}$ for $\rho^{\prime}$ half the
sum of the roots of $Sp(N-1)$ in its vector representation $V^{\prime}$. Hence
the value of $tr$ for $\tilde{p}_{\lambda}$ is given by the symplectic
character $\chi^{\lambda}(q^{2\rho^{\prime}})/[N]^{n}$, where
$\tilde{p}_{\lambda}$ is a minimal idempotent in ${\rm
End}_{Sp(N-1)}(V^{\otimes n})$. It is well-known that an irreducible
$SL(N)$-module labeled by $\mu$ decomposes as a direct sum of
$Sp(N-1)$-modules labeled by Young diagrams with fewer boxes than $\mu$ except
possibly for $\mu$ itself (see [3]). By assumption $H_{n}\subset C_{n}$ has
the same inclusion pattern as ${\rm End}_{SL(N)}(V^{\otimes n})\subset{\rm
End}_{Sp(N-1)}(V^{\otimes n})$. Hence $tr(p_{\lambda})$ is already determined
by the values $tr(p_{\nu})$ for minimal idempotents $p_{\nu}\in C_{n}$ labeled
by diagrams $\nu$ with $|\nu|<|\lambda|$ and by the value of $tr$ on minimal
idempotents of $H_{n}$. Hence we can show the claim by induction on
$|\lambda|$.

###  2.4. First structure coefficients

As a simple but important special case of the tensor product rule 1.2, we
obtain that $V^{\otimes 2}$ decomposes as an $Sp(N-1)$-module into the direct
sum

(2.10) $V^{\otimes 2}\ \cong\ 2\cdot{\bf 1}\oplus 2V^{\prime}\oplus
V_{2\Lambda_{1}}\oplus V_{\Lambda_{2}};$

here ${\bf 1}$ is the trivial representation and $2\Lambda_{1}$ and
$\Lambda_{2}$ denote the Young diagrams with two boxes in the same row
respectively in the same column; see also the Bratteli diagram in Section 1.2.
The labeling is chosen such that $V_{\Lambda_{2}}$ is the nontrivial
irreducible representation appearing in the antisymmetrization of $V^{\otimes
2}$. It follows from the tensor product rules 1.2 that exactly one copy of
${\bf 1}$ and of $V$ appears in the symmetrization of $V$, and hence also
exactly one copy of each of these representations appears in the
antisymmetrization of $V^{\otimes 2}$. It follows from 2.10 that $C_{2}$ has
two nonisomorphic 2-dimensional irreducible representations, and two one-
dimensional representations. If $e$ denotes the projection onto the trivial
representation, we can choose bases for the two 2-dimensonal representations
for which the elements $e$ and $u$ are given by

(2.11) $e\ \mapsto\ \left[\begin{matrix}1&0\\\ 0&0\end{matrix}\right]\ \oplus\
\left[\begin{matrix}1&0\\\ 0&0\end{matrix}\right],$ (2.12) $u_{1}\ \mapsto\
\left[\begin{matrix}a&\sqrt{ab}\\\ \sqrt{ab}&b\end{matrix}\right]\ \oplus\
\left[\begin{matrix}c&\sqrt{cd}\\\ \sqrt{cd}&d\end{matrix}\right].$

Using 1.2 again, we obtain (for $N\geq 5$) that

(2.13) $V^{\otimes 3}\ \cong\ 4\cdot{\bf 1}\oplus 6V\oplus....$

For $N=3$, we have to make the following adjustments: In Eq 2.10, the
representation $V_{\Lambda_{2}}$ does not appear, and in Eq 2.13 the
representation $V$ only appears with multiplicity 5.

######  Lemma 2.3.

The matrix entries above satisfy $a+b=[2]=c+d$ and, if $ab\neq 0$,
$(a-c)^{2}=1$. Moreover, there exists a 4-dimensional representation of the
Hecke algebra $H_{3}$ with $u_{1}$ given by the matrices in 2.12 and $u_{2}$
by the same matrix blocks, with the second and third basis vectors permuted.

$Proof.$ The first statement follows from the fact that each of the matrix
blocks in 2.12 has rank 1 and its only possible nonzero eigenvalue is $[2]$.
We now consider the representation of the Hecke algebra $H_{3}$ on the four-
dimensional space ${\rm Hom}({\bf 1},V^{\otimes 3})$. It follows from the
definition of path bases that $u_{1}$ can be represented by the 4 by 4 matrix
$\rho(u_{1})$ with the two diagonal blocks as in Eq 2.12. As
${\mathbb{C}}v_{0}\otimes V^{\otimes 2}\cong V^{\otimes 2}$ as an $Sp(N-1)$
module, we can also assume that $u_{2}$ is represented by the first matrix
block in 2.12 on the span of the first and third path, while it is given by a
2 by 2 block with diagonal entries $c^{\prime}$ and $d^{\prime}$ on the
remaining two paths. Checking the Hecke algebra relation 2.1 for the (1,1)
entry, we deduce that $c^{\prime}=c$, and hence also $d^{\prime}=d$. Checking
relation 2.1 for entries (1,2) and (2,1), one deduces that the off-diagonal
entries in the $c^{\prime}$,$d^{\prime}$ block of $u_{2}$ are equal to
$\sqrt{cd}$. This determines the matrix for $u_{2}$ as claimed. But then the
(21) entry of both the left and the right hand side of relation 2.1 reads as

$\sqrt{ab}(a^{2}+bc-1)\ =\ \sqrt{ab}(ac+cd).$

Substituting $b=[2]-a$ and $d=[2]-c$, and dividing by $\sqrt{ab}$, we obtain
$(a-c)^{2}=1$, as claimed.

######  Corollary 2.4.

Let $e_{(3)}$ be the projection onto the path $0\to 0\to 0\to 0$.

(a) If $a=c+1$, then $(u_{1}u_{2}-u_{1})e_{(3)}=(u_{2}u_{1}-u_{2})e_{(3)}$,
or, equivalently, $(g_{1}g_{2}+g_{1})e_{(3)}=(g_{2}g_{1}+g_{2})e_{(3)}$,

(b) If $a=c+1$, then $(u_{1}u_{2}+u_{1})e_{(3)}=(u_{2}u_{1}+u_{2})e_{(3)}$,
or, equivalently, $(g_{1}g_{2}-g_{1})e_{(3)}=(g_{2}g_{1}-g_{2})e_{(3)}$.

$Proof.$ This follows from the explicit matrices as described in the proof of
Lemma 2.3.

###  2.5. Relations from Markov property

Recall that the Markov property 2.9 requires $tr(cu_{n})=tr(c)tr(u_{n})$ for
any $c\in C_{n}$.

######  Lemma 2.5.

The Markov property only holds if the matrix entries in 2.12 are as follows
(where $N=2k+1$):

$a\ =\ \frac{[k](q^{1/2}+q^{-1/2})}{[k+1/2]},\hskip 60.00009ptc\ =\
\frac{[k-1/2]}{[k+1/2]},\hskip 55.00008pt{\rm if\ }a=c+1,$

$a\ =\ \frac{-(q^{1/2}-q^{-1/2})(q^{k}-q^{-k})}{q^{k+1/2}+q^{-k-1/2}},\hskip
20.00003ptc\ =\ \frac{q^{k-1/2}+q^{-k+1/2}}{q^{k+1/2}+q^{-k-1/2}},\hskip
20.00003pt{\rm if\ }a=c-1.$

$Proof.$ It follows from the definitions and Lemma LABEL:tracerestricction
that $tr(e)=1/[N]$ and

$tr(u)\ =[2]\ \frac{[N][N-1]}{[2]}\frac{1}{[N]^{2}}\ =\ \frac{[N-1]}{[N]}.$

Using the explicit matrix representations 2.11 and 2.12 and the weights of the
traces in Lemma 2.2, we obtain

$tr(ue)\ =\ \frac{1}{[N]^{2}}(a+c([N]-1)).$

It follows from the Markov property $tr(ue)=tr(u)tr(e)$, with $N=2k+1$ that

$a+c([2k+1]-1)=[2k].$

By Lemma 2.3, we have $a=c\pm 1$. Let us consider the case $a=c+1$.
Substituting this into the last equation we obtain

$c=\frac{[2k]-1}{[2k+1]}\ =\ \frac{[k-1/2]}{[k+1/2]}\ =\
\frac{q^{k-1/2}-q^{-k+1/2}}{q^{k+1/2}-q^{-k-1/2}}.$

The formula for $a$ follows from $a=c+1$. The case $a=c-1$ goes similarly.

######  Remark 2.6.

The previous lemma implies that there are at most two module categories
$\mathcal{M}$ of $Rep(U_{q}{\mathfrak{s}l}_{N})$ for our given fusion rules if
they allow categorical traces which also satisfy the compatibility condition
$Tr_{M}(\mathcal{E}_{M}(a))=Tr_{M\otimes W}(a)$ for $a\in{\rm
End}_{\mathcal{M}}(M\otimes W)$, where $\mathcal{E}_{M}$ is defined for an
object $M$ in $\mathcal{M}$ and an object $W\in Rep(U_{q}{\mathfrak{s}l}_{N})$
as in 2.5.

###  2.6. Correction for $q\to 1$

We assume in this section that $a=c-1$. It follows from Lemma 2.5 that $ab=0$
for $q=1$ in this case. This would make the representation in the first matrix
block in 2.12 reducible. This can be avoided by introducing the elements
$u_{12}$ and $u_{21}$ defined below; they correspond to intertwiners between
the two copies of the trivial representation in $V^{\otimes 2}$. We shall show
later that one can make sense of them also at $q=1$. It will be convenient to
define

(2.14) $[N]_{+}=\frac{q^{N/2}-q^{-N/2}}{q^{1/2}-q^{-1/2}}\quad{\rm
and}\quad[N]_{-}=\frac{q^{N/2}+q^{-N/2}}{q^{1/2}+q^{-1/2}}.$

Observe that

$b\ =\ [2]-a\ =\ \frac{q^{k+1}+q^{-k-1}}{[N]_{-}},$
$\sqrt{ab}=\frac{-i(q^{1/2}-q^{-1/2})}{[N]_{-}}\sqrt{[N]-1},$

where we used the identity $[k](q^{k+1}+q^{-k-1})=[N]-1$, and where the choice
of sign for $\sqrt{ab}$ will turn out to be immaterial. Let $e_{(2)}$ be the
subprojection of $e$ which is nonzero only in the first matrix block in 2.11.
We define the element

(2.15) $\displaystyle u_{21}\ $ $\displaystyle=\
\frac{[N]_{-}}{-i(q^{1/2}-q^{-1/2})}(1-e_{(2)})u_{1}e_{(2)}$ $\displaystyle=\
\frac{i[N]_{-}}{-q^{1/2}-q^{-1/2}}u_{1}e_{(2)}+i(q^{1/2}-q^{-1/2})[k]e_{(2)},$

where we used $e_{(2)}u_{1}e_{(2)}=ae_{(2)}$. We similarly define
$u_{12}=u_{21}^{t}$ by the same expression as in 2.15 with $u_{1}$ and
$e_{(2)}$ interchanged. By construction, it follows that the elements $u_{21}$
and $u_{12}$ are represented by the matrices $u_{21}=\sqrt{[N]-1}E_{21}$ and
$u_{12}=\sqrt{[N]-1}E_{12}$, where $E_{ij}$ are matrix units in the first
$2\times 2$ matrix block in 2.12. The following results follow immediately
from this.

######  Lemma 2.7.

We have $u_{12}u_{21}=([N]-1)e_{(2)}$,
$u_{21}u_{12}e_{2)}=0=e_{(2)}u_{21}u_{12}$ and the element
$P=e_{(2)}+u_{12}+u_{21}+u_{21}u_{12}$ satisfies $P^{2}=[N]P$.

###  2.7. Summary of relations

The following is a preliminary definition of the algebras $C_{n}$. The
precise, but less intuitive definition of the algebras $C_{n}$ in various
versions will be given in the following section.

######  Definition 2.8.

Fix $N=2k+1$. Then we define the algebra $C_{n,\pm}=C^{(N)}_{n,\pm}$ via
generators $e$, $u_{i}$, $1\leq i<n$ with the following relations:

(a) The elements $u_{i}$ satisfy the Hecke algebra relations,

(b) We have a sequence of idempotents $e_{(r)}$ defined inductively by
$e_{(0)}=1$, $e_{(1)}=e$ and

$e_{(r+1)}\ =\
e_{(r)}u_{r}e_{(r)}-\frac{q^{(N-2)/2}-q^{-(N-2)/2}}{q^{N/2}-q^{-N/2}}e_{(r)},\quad{\rm
for\ }C_{n,+},$ $e_{(r+1)}\ =\
\frac{q^{(N-2)/2}+q^{-(N-2)/2}}{q^{N/2}+q^{-N/2}}e_{(r)}-e_{(r)}u_{r}e_{(r)},\quad{\rm
for\ }C_{n,-}.$

(c) For $j<r$ we have
$(u_{j-1}u_{j}-u_{j-1})e_{(r)}=(u_{j}u_{j-1}-u_{j})e_{(r)}$ for $C_{n,+}$ and
$(u_{j-1}u_{j}+u_{j-1})e_{(r)}=(u_{2}u_{1}+u_{2})e_{(r)}$ for $C_{n,-}$.

######  Remark 2.9.

1\. In spite of fractions in the exponents, it is not hard to check that the
relations only depend on $q$ and not on the choice of a square root $q^{1/2}$.
E.g. we have

$\frac{q^{(N-2)/2}-q^{-(N-2)/2}}{q^{N/2}-q^{-N/2}}\ =\ \frac{q^{N-1}+q^{N-2}+\
...\ +q^{1-N}}{q^{N}+q^{N-1}+\ ...\ +q^{-N}}.$

2\. Let $e_{\pm}(q)$ and $u_{i,\pm}(q)$ be the generators of $C_{n,\pm}(q)$
for a given choice of $q$ respectively. Then we can check that the maps

$e_{-}(q)\mapsto e_{+}(-q),\hskip 30.00005ptu_{i,-}(q)\mapsto-
u_{i,+}(-q),\quad 1\leq i<n,$

define an isomorphism between $C_{n,-}(q)$ and $C_{n,+}(-q)$.

###  2.8. Alternative definitions of the algebras $C_{n}$

We make the following adjustments for the precise definition of the algebra
$C_{n}=C_{n}(p,q)$ as an algebra depending on two variables $p$ and $q$.
First, we substitute $p=q^{N}$ in the previous relations. Secondly, we
introduce additional generators which will only be relevant for the important
classical limit $q\to 1$ for $C_{n,-}$ (respectively for $q\to-1$ for
$C_{n,+}$).

######  Definition 2.10.

(Two variable definition) We define the algebra
$C_{n}=C_{n,\pm}=C_{n,\pm}(p,q)$ over the field ${\mathbb{C}}(p,q)$ of
rational functions in the variables $p$ and $q$ as follows: We have generators
$u_{i}$, $e_{(r)}$ for $1\leq i,r<n$ with the following relations:

(a) The elements $u_{i}$ satisfy the Hecke algebra relations,

(b) The elements $e_{(r)}$ are idempotents which satisfy the relations

$e_{(r+1)}\ =\
e_{(r)}u_{r}e_{(r)}-\frac{p^{1/2}q^{-1}-qp^{-1/2}}{p^{1/2}-p^{-1/2}}e_{(r)},\quad{\rm
for\ }C_{n,+},$ $e_{(r+1)}\ =\
\frac{p^{1/2}q^{-1}+qp^{-1/2}}{p^{1/2}+p^{-1/2}}e_{(r)}-e_{(r)}u_{r}e_{(r)},\quad{\rm
for\ }C_{n,-}.$

(c) For $j<r$ we have
$(u_{j-1}u_{j}-u_{j-1})e_{(r)}=(u_{j}u_{j-1}-u_{j})e_{(r)}$ for $C_{n,+}$ and
$(u_{j-1}u_{j}+u_{j-1})e_{(r)}=(u_{2}u_{1}+u_{2})e_{(r)}$ for $C_{n,-}$.

######  Remark 2.11.

Similarly as e.g. for the algebras defined in [1], this definition is not
convenient if we are interested in obtaining the classical limits for E$q\to
1$ and $p=q^{N}\to 1$. This can be addressed by introducing additional
generators. It will be shown in Section 3 that one can make sense of these
additional elements if $q\to 1$.

######  Definition 2.12.

(Extended definition for $C_{n}(q^{N},q)$) We now let $p=q^{N}$ as before. We
add to the usual generators $u_{i}$, $e_{(r)}$ for $1\leq i,r<n$ also the
elements

$\frac{1}{q^{1/2}+q^{-1/2}}u_{i}e_{(r)},\
\frac{1}{q^{1/2}+q^{-1/2}}e_{(r)}u_{i},\quad 1\leq i<r<n,\quad{\rm for\
}C_{n,+},$ $\frac{1}{q^{1/2}-q^{-1/2}}u_{i}e_{(r)},\
\frac{1}{q^{1/2}-q^{-1/2}}e_{(r)}u_{i},\quad 1\leq i<r<n,\quad{\rm for\
}C_{n,-}.$

The relations are the same as in Definition 2.8.

###  2.9. Basic structure results

We will prove existence of nontrivial representations of the algebras
$C_{n,\pm}$ in the next section.

######  Proposition 2.13.

(a) The map $\Phi$ given by $e\mapsto e_{(r+1)}$, $u_{i}\mapsto
e_{(r)}u_{r+i}$ defines a homomorphism of $C_{n-r}$ onto
$e_{(r)}C_{n}e_{(r)}$.

(b) The algebras $C_{n,\pm}$ are spanned by the $H_{n}-H_{n}$ bimodules
$H_{n}e_{(r)}H_{n}$, $0\leq r\leq n$. In particular, they are finite
dimensional.

(c) The span $I_{r}$ of $\cup_{s\geq r}H_{n}e_{(s)}H_{n}$ is a two-sided ideal
of $C_{n}$ for $1\leq s\leq n$.

$Proof.$ The homomorphism property in (a) follows directly from the relations.
For (b), we observe that the claimed spanning set contains the generators of
$C_{n}$. It hence suffices to show that multiplying it by a generator from the
right or left will still produce an element in the span. This is obviously
true for the generators $u_{i}$. We prove the claim for multiplication by $e$
by induction on $n$, with the statement obviously true for $n=1$. Observe that
the Hecke algebra $H_{n}$ is spanned by elements of the form $au_{1}b$ or $a$,
with $a,b\in H_{2,n}$, where $H_{2,n}$ is the subalgebra generated by
$u_{2},u_{3},\ ...\ u_{n-1}$ and 1. We then have, using $e_{(r)}=ee_{(r)}$

$e(au_{1}b)e_{(r)}=a(eu_{1}e)be_{(r)}=c\ abe_{(r)}+ae_{(2)}be_{(r)}.$

But now $e_{(2)}be_{(r)}\in e_{(2)}H_{2,n}e_{(2)}\subset\Phi(eH_{n-1}e)$ by
(a). Hence, by induction assumption, we have

$e_{(2)}be_{(r)}\in span(\cup_{s}\Phi^{-1}(H_{n-1}e_{(s)}H_{n-1})\subset
span\cup_{s}H_{n-1}e_{(s)}H_{n-1}.$

The proof for multiplication by $e$ from the right goes completely
analogously. This finishes the proof of statement (b). The surjectivity
statement in (a) now follows from (b), as $\Phi$ maps $e_{(s)}$ to
$e_{(s+1)}$. Our proof of (b) also implies statement (c).

###  2.10. Dimension estimates

We will explicitly construct spanning sets for the algebras $C_{n}$ which will
later be shown to be bases. To do so, we shall use two well-known facts about
Hecke algebras, here only formulated for Hecke algebras for type $A$ (see [8]
for details). Let $s_{i}$, $1\leq i<n$ be a set of simple reflections for the
symmetric group $S_{n}$ (say $s_{i}=(i,i+1)$). Then any element $w\in S_{n}$
can be written as a product of simple reflections. Any such expression for $w$
with the minimal number of factors is called a reduced word, and the number of
factors is called the length $\ell(w)$ of $w$. In the case of the symmetric
group, the length $\ell(w)$ can also be defined as the number of pairs $i<j$
such that $w(i)>w(j)$. Replacing the elements $s_{i}$ by generators $g_{i}$ in
such an expression defines an element $h_{w}\in H_{n}$ which does not depend
on the choice of reduced expression for $w$.

It is easy to see that the shortest elements in the left cosets of
$S_{r}\subset S_{n}$ are given by permutations $w$ which satisfy $w(i)<w(j)$
for any $1\leq i<j\leq r$, and that the shortest elements in the left cosets
of $S_{r}\times S_{n-r}\subset S_{n}$ are given by permutations $w$ which also
satisfy the additional conditions $w(i)<w(j)$ for all $r<i<j\leq n$. It is
well-known (and easy to check for these cases) that each such coset contains a
unique element of lowest length.

######  Definition 2.14.

If $w\in S_{n}$ and $h_{w}$ the corresponding element in $H_{n}$, we define
$h_{w}^{T}=h_{w^{-1}}$. If $\mathcal{S}\subset H_{n}$ we define
$\mathcal{S}^{T}=\\{h^{T},h\in\mathcal{S}\\}$. Using these conventions, we
define $\mathcal{C}_{n,r}=\\{h_{w}\\}$, where $w$ ranges over the shortest
elements in the left cosets of $S_{r}\subset S_{n}$, and we define
$\mathcal{D}_{n,r}=\\{h_{w}\\}$, where now $w$ ranges over the shortest
elements of the left cosets of $S_{r}\times S_{n-r}\subset S_{n}$. Finally, we
define ${\mathcal{B}}_{r}$ inductively by ${\mathcal{B}}_{0}=\emptyset$,
${\mathcal{B}}_{1}=\\{1\\}$ and

${\mathcal{B}}_{r+1}\ =\ {\mathcal{B}}_{r}\ \cup\
\bigcup_{j=1}^{r}g_{j}g_{j+1}\ ...\ g_{r}{\mathcal{B}}_{r-1}.$

We remark that $|\mathcal{C}_{n,r}|=[S_{n}:S_{r}]=n!/r!$ and that
$|\mathcal{D}_{n,r}|=\binom{n}{r}$, and that $\mathcal{D}_{n,r}^{T}$ contains
the elements $h_{w}$ with $w$ running through the shortest elements of the
right cosets of $S_{r}\times S_{n-r}\subset S_{n}$.

######  Lemma 2.15.

(a) We have $|{\mathcal{B}}_{r}|=h_{r}$, with $h_{r}$ as in 1.3.

(b) The set $\mathcal{C}_{n,r}{\mathcal{B}}_{r}e_{(r)}$ spans $H_{n}e_{(r)}$.

$Proof.$ Statement (a) follows from the definitions of $h_{r}$ in 1.3 and
${\mathcal{B}}_{r}$. To establish statement (b), let us first prove it in the
special case $n=r$ by induction on $r$. This is obviously true for $r=1$. We
now prove by downward induction from $s=r-1$ to $s=1$ that

$None$ $H_{r}g_{r}g_{r-1}\ ...\ g_{s}e_{(r+1)}\ \subset
H_{r}e_{(r+1)}+H_{r}g_{r}e_{(r+1)}.$

For $s=r-1$ it follows from relation (c) that
$g_{r}g_{r-1}e_{(r+1)}=(g_{r-1}g_{r}-g_{r-1}+g_{r})e_{(r+1)}$. This implies
$H_{r}g_{r}g_{r-1}e_{(r+1)}$ is contained in the right hand side of the claim.
The induction step for $s<r-1$ is shown in the same way. As
$H_{r+1}=H_{r}+\sum_{s=1}^{r}H_{r}g_{r}g_{r-1}\ ...\ g_{s}$, it follows that

(2.16) $\displaystyle H_{r+1}e_{(r+1)}\ $ $\displaystyle=\ H_{r}e_{(r+1)}\ +\
H_{r}g_{r}e_{(r+1)}$ $\displaystyle=\ H_{r}e_{(r+1)}\ +\
\sum_{j=1}^{r}H_{r-1}g_{r-1}g_{r-2}\ ...g_{j}e_{(r-1)}g_{r}e_{(r+1)},$

where the summand for $j=r$ is defined to be equal to
$H_{r-1}e_{(r-1)}g_{r}e_{(r+1)}$. The claim for $n=r$ follows from $(*)$ and
the definition of ${\mathcal{B}}_{r+1}$. The general claim for $n>r$ follows
from this and the fact that
$H_{n}=\mathcal{C}_{n,r}H_{r}=\bigcup_{h_{w}\in\mathcal{C}_{n,r}}h_{w}H_{r}$.

######  Lemma 2.16.

We have the following inequalities:

(a) $\dim H_{n}e_{(r)}\leq\frac{n!}{r!}h_{r}$,

(b) $\dim H_{n}e_{(r)}H_{n}\leq\frac{n!}{r!}\binom{n}{r}h_{r}^{2}$,

(c) We have $\dim C_{n}\leq\sum_{r=0}^{n}\frac{n!}{r!}\binom{n}{r}h_{r}^{2}$.

$Proof.$ It follows from the inductive definition of $\mathcal{C}_{r,n}$ that
$|\mathcal{C}_{r,n}|=n!/r!$ This and Lemma 2.15 imply claim (a). The
inequality in (b) follows from (a) and the surjective map
$H_{n}\otimes_{H_{r+1,n}}H_{n}\to H_{n}e_{(r)}H_{n}$. Finally, claim (c)
follows from the fact that $C_{n}$ is spanned by the subspaces
$H_{n}e_{(r)}H_{n}$, $0\leq r\leq n$.

###  2.11. An explicit spanning set

We use the notations from Definition 2.14.

######  Proposition 2.17.

The set
${\mathcal{B}}(r)=\mathcal{C}_{r,n}{\mathcal{B}}_{r,n}e_{(r)}{\mathcal{B}}_{n,r}^{T}\mathcal{D}_{r,n}^{T}$
spans $H_{n}e_{(r)}H_{n}$ for $1\leq r\leq n$, and hence
${\mathcal{B}}=\bigcup{\mathcal{B}}(r)$ spans $C_{n}$.

$Proof.$ We hace already proved in Lemma 2.15 that set
$\mathcal{C}_{n,r}{\mathcal{B}}_{r}e_{(r)}$ spans $H_{n}e_{(r)}$. One can show
the same way that the set $e_{(r)}{\mathcal{B}}^{T}(r)$ spans $e_{(r)}H_{r}$.
The claim now follows from this and the fact that $\mathcal{D}_{n,r}^{T}$
contains the elments of minimal lengths for all right cosets of $H_{r}\times
H_{n-r}\subset H_{n}$.

##  3\. Tensor product representations

We will give explicit representations of the algebras $C_{n}=C_{n,+}$ in this
section. In view of Remark 2.9, these representations can be easily modified
to representations of the algebras $C_{n,-}$. As before, let
$V={\mathbb{C}}^{N}$ with $N=2k+1$ and let $\\{v_{i}\\}$ denote the standard
basis of ${\mathbb{C}}^{N}$. We define $v_{o}=\sum_{i=1}^{N}\alpha_{i}v_{i}$,
with $\alpha_{i}=q^{(k+1-i)/2}/\|v_{o}\|$, where
$\|v_{o}\|^{2}=\sum\alpha_{i}^{2}$.

###  3.1. A matrix for $e$

We define the $N$ by $N$ matrix $E=(e_{ij})$ by $e_{ij}=\alpha_{i}\alpha_{j}$.
Moreover, we modify the Hecke algebra representation in 2.2 to $u\ \mapsto\
U$, where the matrix $U$ is defined by

(3.1) $U_{|span\\{v_{i}\otimes v_{j},v_{j}\otimes v_{i}\\}}\ =\
\left[\begin{matrix}q^{-1}&1\\\ 1&q\end{matrix}\right],\quad i<j.$

with the matrices $U_{i}$ defined as in 1.5. We then consider the map $\Phi$
which maps the generators of $C_{n}$ into ${\rm End}(V^{\otimes n})$ given by

(3.2) $\Phi:\quad e\mapsto E\otimes 1_{n-1},\hskip 30.00005ptu_{i}\mapsto
U_{i},\quad 1\leq i<n.$

######  Lemma 3.1.

The map $\Phi$ is compatible with relation (b), mapping the element $e_{(n)}$
to $E^{\otimes n}$. In particular, we have

$E\otimes E\ =\ (E\otimes 1)U(E\otimes 1)\ -\
\frac{q^{k-1/2}-q^{-k+1/2}}{q^{k+1/2}-q^{-k-1/2}}\ E\otimes 1.$

$Proof.$ The vector $v_{o}$ spans the image of $E$ by definition. We then
calculate

$U(v\otimes v_{j})\ =\ \sum_{i=1}^{j-1}\alpha_{i}(q^{-1}v_{i}\otimes
v_{j}+v_{j}\otimes v_{i})\ +\ \sum_{i=j+1}^{N}\alpha_{i}(qv_{i}\otimes
v_{j}+v_{j}\otimes v_{i}),$ $(E\otimes 1)U(v\otimes v_{j})\ =\
\sum_{i=1}^{j-1}\alpha_{i}^{2}q^{-1}v_{i}\otimes
v_{j}+\alpha_{i}\alpha_{j}v_{j}\otimes v_{i}\ +\
\sum_{i=j+1}^{N}\alpha_{i}^{2}qv_{i}\otimes
v_{j}+\alpha_{i}\alpha_{j}v_{j}\otimes v_{i}\ =$ $=\
\alpha_{j}\sum_{i=1}^{N}v\otimes\alpha_{i}v_{i}\ +\ \beta v\otimes e_{j},$

where

$\beta\ =\ -\alpha_{j}^{2}\ +\ \sum_{i=1}^{j-1}\alpha_{i}^{2}q^{-1}\ +\
\sum_{i=j+1}^{N}\alpha_{i}^{2}q.$

It is now straightforward to check that
$\alpha_{j}\sum_{i=1}^{N}v\otimes\alpha_{i}v_{i}=(E\otimes E)(v\otimes v_{j})$
and

$\beta=\frac{q^{k-1}+q^{k-2}+\ ...\ +q^{1-k}}{\|v_{o}\|^{2}}\ =\
\frac{q^{k-1/2}-q^{-k+1/2}}{q^{k+1/2}-q^{-k-1/2}},$

where the last equality is obtained by multiplying both numerator and
denominator by $q^{1/2}-q^{-1/2}$. This proves the second statement in the
claim. We can now show by induction on $n$ that $\Phi$ can be extended to a
homomorphism which maps $e_{(n)}$ to $E^{\otimes n}$.

###  3.2. Checking relation (c)

Let $\\{v_{i}\\}$ and $v_{o}$ be as above. Then we have

######  Lemma 3.2.

$(U_{1}U_{2}-U_{1})v^{\otimes 3}\ =\ (U_{2}U_{1}-U_{2})v^{\otimes 3}.$

$Proof$. This is a straightforward calculation. It is manageably tedious if
one checks it separately on the span of all possible permutations of
$v_{i}\otimes v_{j}\otimes v_{m}$ for any choice of indices $i,j$ and $m$. If
all indices are mutually distinct, one obtains two $6\times 6$ matrices for
$U_{1}$ and $U_{2}$, both with three $2\times 2$ blocks. Moreover, the
coefficient for each of these vectors in the basis expansion of $v^{\otimes
3}$ is equal to $\alpha_{i}\alpha_{j}\alpha_{m}$. Hence it suffices to check
the claim for these two $6\times 6$ matrices, applied to the vector
$(1,1,1,1,1,1)^{T}$, which is not very hard. The case where two indices
coincide is done similarly and easier, only involving $3\times 3$ matrices.

######  Remark 3.3.

If we define the algebra $\tilde{C}_{3}$ like the algebra $C_{3}$ without
relation (c), it can be shown that $\tilde{C}_{3}$ modulo its radical is
isomorphic to $C_{3}$.

###  3.3. Classical limits

It will be more convenient to consider the representations for $C_{n,-}$, i.e.
we basically replace $q$ by $-q$ in the matrix $\Phi(e)$ and in the
coefficients of the vector $v_{o}$, see Remark 2.9. We are going to show that
the elements $\frac{1}{q^{1/2}-q^{-1/2}}u_{i}e_{(r)}$ still make sense in our
representation even at $q=1$.

######  Lemma 3.4.

(a) The matrix coefficients of $\frac{1}{q^{1/2}-q^{-1/2}}U_{i}E_{(r)}$ in the
tensor product representation of $C_{n,-}$ are well-defined also at $q=1$, up
to the choice of the square root $q^{1/2}$.

(b) The elements $E^{\otimes 2}$, $\frac{1}{q^{1/2}-q^{-1/2}}U_{1}E^{\otimes
2}$ and $\frac{1}{q^{1/2}-q^{-1/2}}E^{\otimes 2}U_{1}$ generate an algebra
which is isomorphic to the $2\times 2$ matrices if $[N]\neq 1$.

$Proof.$ We calculate

$Uv_{o}^{\otimes 2}\ =\ \sum_{i<j}\alpha_{i}\alpha_{j}(q^{-1}-1)v_{i}\otimes
v_{j}+(q-1)v_{j}\otimes v_{i}).$

It follows that also the coefficients in the expression for
$\frac{1}{q^{1/2}-q^{-1/2}}Uv_{o}^{\otimes 2}$ are in ${\mathbb{Z}}[q^{\pm
1/2}]$. As all columns of $E^{\otimes 2}$ are proportional to $v_{o}^{\otimes
2}$, claim (a) follows.

Recall the definition of $u_{21}$ in Definition 2.15. It follows from part (a)
and the equations below 2.14 that $\Phi(u_{21})$ and $\Phi(u_{12})$ are well-
defined and that

$\Phi(u_{21}u_{12})\ =\
\frac{-[N]_{-}^{2}}{(q^{1/2}-q^{-1/2})^{2}}(1-E^{\otimes 2})UE^{\otimes
2}U(1-E^{\otimes 2})$

has the nonzero eigenvalue

$\frac{-[N]_{-}^{2}a_{-}([2]-a_{-})}{(q^{1/2}-q^{-1/2})^{2}}\ =\
[k](q^{k+1}+q^{-k-1})\ =\ [N]-1.$

###  3.4. An embedding of $Sp(N-1)$ into $Sl(N)$

Let $A$ be the $N\times N$ matrix (with $N=2k+1$ odd) defined by

$a_{ij}=\begin{cases}(-q^{-1})^{k+1-(i+j)/2}&if\
i<j,\cr-(-q^{-1})^{k+1-(i+j)/2}&if\ i>j.\end{cases}$

As usual, we assume fixed choices of $(-q)^{1/2}$ and of $(-q^{-1})^{1/2}$ in
all these formulas such that their product is equal to $-1$.

######  Lemma 3.5.

The matrix $A$ has rank $N-1$ in a neighborhood of $q=1$, with kernel
$v_{o}=\sum_{j=1}^{2k+1}(-q)^{(k+1-j)/2}v_{j}$. Hence we obtain a symplectic
form $(v,w)=v^{T}Aw$ for $q=1$ whose restriction to any complement
$V^{\prime}$ of $v_{o}$ is nondegenerate.

$Proof.$ We check that

$\displaystyle(Av_{o})_{i}\ $ $\displaystyle=\
\sum_{j=1}^{i-1}-(-q^{-1})^{k+1-(i+j)/2}(-q)^{(k+1-j)/2}\ +\
\sum_{j=i+1}^{2k+1}(-q^{-1})^{k+1-(i+j)/2}(-q)^{(k+1-j)/2}$ $\displaystyle=\
\sum_{j=2}^{2k+1}(-q^{-1})^{(k+1-i)/2}(-1)^{k+1-j}\ =\ 0.$

To determine the rank at $q=1$, we observe that after conjugation by the
diagonal matrix $D=diag((-1)^{(k+1-i)/2})$ the matrix entries become equal to
$a_{ij}=(-1)^{k+1-i}$ for $i<j$ and $a_{ij}=-a_{ji}$ for $i>j$. It is now easy
to see that the transformed matrix has eigenvectors $(1,0,0,...,\pm 1)$,
$(0,1,0,...,\pm 1,0)$ etc. This proves the claim about the rank.

######  Proposition 3.6.

Fix $N=2k+1$ and let $V={\mathbb{C}}^{N}$. If $q=1$, the representation of
$C_{n,-}$ into ${\rm End}(V^{\otimes n})$ surjects onto ${\rm
End}_{Sp(N-1)}(V^{\otimes n})$, where the embedding of $Sp(N-1)\subset Sl(N)$
is defined via the symplectic form given by the matrix $A$ in Lemma 3.5 at
$q=1$. In particular, $\dim_{{\mathbb{C}}(q)}C_{n,-}(q^{N},q)\geq\dim{\rm
End}_{Sp(N-1)}(V^{\otimes n})$.

$Proof.$ The image of $C_{n,-}$ in ${\rm End}(V^{\otimes n})$ at $q=1$
contains the usual action of the symmetric group $S_{n}$ on $V^{\otimes n}$
and the projection $E\in{\rm End}(V)$ onto ${\mathbb{C}}v_{o}$. By Lemma 3.4,
it also acts as a full $2\times 2$ matrix algebra on ${\rm Hom}({\bf
1},V^{\otimes 2})$. So, in particular, it also must contain the projection
$F$. It follows from Theorem 1.3 that $C_{n,-}(q)$ maps surjectively onto
${\rm End}_{Sp(N-1)}(V^{\otimes n})$ for $q=1$. This implies the estimate
about the dimensions.

###  3.5. A basis for $C_{n}(p,q)$

It will be convenient to consider the 2-variable version of $C_{n}$, as
defined in Definition 2.10.

######  Theorem 3.7.

The spanning set in Proposition 2.17 is a basis for the two-variable version
$C_{n}(p,q)$, viewed as an algebra over the field of rational functions in $p$
and $q$. In particular, we have $\dim
C_{n}(p,q)=\sum_{r=0}^{n}h_{r}^{2}\frac{n!}{r!}\binom{n}{r}$.

$Proof.$ It follows from Proposition 3.6 and Lemma 2.16,(c) that $\dim
C_{n,-}(q^{N},q)$ is equal to $\dim{\rm End}_{Sp(N-1)}(V^{\otimes n})$ if
$N>2n$. Hence the spanning set in Proposition 2.17 is a basis for these
values. In particular, this is true for the two-variable version of
$C_{n,-}(p,q)$ if $p=q^{N}$ with $N>2n$ odd. Hence it is true in general by
Zariski density. The claim can be similarly shown for $C_{n,+}(p,q)$ using
Remark 2.9.

##  4\. Markov traces

Recall that we defined $Tr_{q}$ on ${\rm End}(V^{\otimes n})$ in Section 2.2
by $Tr_{q}(a)=Tr(aD^{\otimes n})$, where $a\in{\rm End}(V^{\otimes n})$ and
$D=diag(q^{2i-N-1})$, see 2.3 and also 2.4. We define a functional $\phi$ on
$C_{n}=C_{n}^{(N)}$ as a normalized pull-back of $Tr_{q}$ by

(4.1) $\phi(c)\ =\ \frac{1}{[N]^{n}}Tr_{q}(\Phi(c)),\quad c\in C_{n}.$

Observe that $\phi(1)=1$. It is the goal of this section to show that the
functional $\phi$ defines a trace $tr$ on $C_{n}$ which satisfies the Markov
condition 2.9.

######  Lemma 4.1.

The functional $\phi$ has the following properties.

(a) The restriction of $\phi$ to $H_{n}$ defines a trace.

(b) $\phi(cg_{n-1})=\phi(c)\phi(g_{n-1})$ for all $c\in C_{n-1}$.

(c) $\phi(e_{(r)})=\phi(e)^{r}=\frac{1}{[N]^{r}}$.

(d) $\phi(ch)=\phi(hc)$ for all $h\in H_{n}$, $c\in C_{n}$

$Proof.$ Part (a) follows from the discussion in Section 2.2, and part (b)
follows from 2.6. We also have for $n=r=1$ and for $C_{n,+}$

$Tr_{q}(E)=Tr(ED)\ =\ \sum_{i=1}^{N}q^{2i-N-1}q^{(N+1)/2-i}/{[N]_{+}}\ =\
\sum_{i=1}^{N}q^{i-(N+1)/2}/{[N]_{+}}\ =\ 1.$

One deduces from this that

$Tr_{q}(E^{\otimes r})\ =\ Tr((ED)^{\otimes r})Tr(D^{\otimes n-r})\ =\
[N]^{n-r},$

from which follows claim (c). As $\Phi(h)$ commutes with $D^{\otimes n}$ for
all $h\in H_{n}$, we have

$Tr_{q}(\Phi(ch))=Tr(\Phi(c)\Phi(h)D^{\otimes n})=Tr_{q}(\Phi(c)D^{\otimes
n}\Phi(h))=Tr_{q}(\Phi(h)\Phi(c)D^{\otimes n})=Tr_{q}(\Phi(hc)).$

###  4.1. Technical lemmas

It follows directly from the relations that the map

(4.2) $\Theta_{n}:g_{i}\mapsto g_{n-i},\quad 1\leq i<n$

induces an automorphism of the Hecke algebra $H_{n}$ which will be denoted by
the same letter. Also observe that if $w_{o}\in S_{n}$ is defined by
$w_{o}(i)=n-i$, and we denote the corresponding map on $V^{\otimes n}$ given
via permutation of the factors by the same letter, we have

(4.3) $w_{o}U_{i}w_{o}=U_{n-i}(q^{-1}),$

where $U_{i}(q^{-1})$ is given by the same matrix as $U_{i}$, with every
occurrence of $q$ replaced by $q^{-1}$.

######  Lemma 4.2.

We have

$Tr_{q}(\Phi(h_{1})E^{\otimes n}\Phi(h_{2}))=Tr(\Phi(h_{1})E^{\otimes
n}(q^{-1})\Phi(h_{2}))=Tr(\Phi(\Theta_{n}(h_{1}))E^{\otimes
n}\Phi(\Theta_{n}(h_{2}))).$

$Proof.$ To avoid cumbersome notation, we denote $H_{i}=\Phi(h_{i})$ in this
proof. Then we have

$Tr_{q}(H_{1}E^{\otimes n}H_{2})\ =\ Tr(H_{1}(D^{1/2}ED^{1/2})^{\otimes
n}H_{2})\ =\ Tr(H_{1}E^{\otimes n}(q^{-1})H_{2}),$

from which follows the first equality in the statement. Now observe that the
structure coefficients in the defining relations of $C_{n}$ are invariant
under $q\leftrightarrow q^{-1}$. Hence we also obtain a representation of
$C_{n}$ via the assignment

$e\mapsto E(q^{-1}),\hskip 30.00005ptu_{i}\mapsto U_{i}(q^{-1}),\quad 1\leq
i<n.$

As $E^{\otimes n}HE^{\otimes n}=\alpha E^{\otimes n}$ for some scalar
$\alpha$, it also follows $E^{\otimes n}(q^{-1})H(q^{-1})E^{\otimes
n}(q^{-1})=\alpha E^{\otimes n}(q^{-1})$ for the same scalar. Applying this to
$H=H_{2}H_{1}$, we obtain

$Tr(H_{1}E^{\otimes n}H_{2})=Tr(E^{\otimes}H_{2}H_{1}E^{\otimes n})=\alpha=$
$=Tr(E^{\otimes}(q^{-1})H_{2}H_{1}(q^{-1})E^{\otimes n}(q^{-1}))\ =\
Tr(\Theta_{n}(H_{1})E^{\otimes}(q^{-1})\Theta_{n}(H_{2})).$

Replacing $H_{i}$ by $\Theta(H_{i})$ for $i=1,2$ in the equation above now
proves the second equality in the statement.

######  Lemma 4.3.

We have $e_{(n)}he_{(n)}=e_{(n)}\Theta_{n}(h)e_{(n)}$ for all $h\in H_{n}$.

$Proof.$ The claim is proved by induction on $n$, with $n=1,2$ being trivially
true. For the induction step from $n-1$ to $n$ first observe that for any
$A\in{\rm End}(V^{\otimes n-1})$ we have

$E^{\otimes n}(A\otimes 1)E^{\otimes n}=E^{\otimes n-1}AE^{\otimes n-1}\otimes
E=E\otimes E^{\otimes n-1}AE^{\otimes n-1}=E^{\otimes n}(1\otimes A)E^{\otimes
n},$

as $E^{\otimes n-1}AE^{\otimes n-1}$ is a scalar multiple of $E^{\otimes
n-1}$. We define the homomorphism $sh:H_{n-1}\to H_{n}$ via
$sh(u_{i})=u_{i+1}$. If $H\in\Phi(H_{n-1})$, it follows that $E^{\otimes
n}sh(H)E^{\otimes n}=E^{\otimes n}HE^{\otimes n}$. Moreover, by induction
assumption, we have

$E^{\otimes n}(\Theta_{n-1}(H)\otimes 1)E^{\otimes n}=E^{\otimes n}(H\otimes
1)E^{\otimes n}.$

Hence we have

(4.4)
$e_{(n)}he_{(n)}=e_{(n)}sh(\Theta_{n-1}(h))e_{(n)}=e_{(n)}\Theta_{n}(h)e_{(n)},$

which proves the claim for $h\in H_{n-1}$. Let now $h=h^{\prime}g_{n-1}$ with
$h^{\prime}\in H_{n-1}$. We first observe that for $h\in H_{n}$,
$e_{(n-1)}he_{(n-1)}$ is a linear combination of $\alpha e_{(n-1)}+\beta
e_{(n)}$. Using $e_{(n)}e_{(n-1)}=e_{(n)}$, one deduces easily that

$e_{(n)}he_{(n-1)}=e_{(n)}he_{(n)}.$

If $e_{(n-1)}h^{\prime}e_{(n-1)}=\gamma e_{(n-1)}$, we calculate

$e_{(n)}he_{(n)}=e_{(n)}h^{\prime}e_{(n-2)}g_{n-1}e_{(n)}=\gamma
e_{(n)}g_{n-1}e_{(n)}=\gamma(c^{\prime}+1)e_{(n)}.$

On the other hand,

$e_{(n)}\Theta_{n}(h)e_{(n)}=e_{(n)}\Theta_{n}(h^{\prime})g_{1}e_{(n)}=e_{(n)}\Theta_{n}(h^{\prime})eg_{1}e_{(n)},$

where we used that $\Theta_{n}(h^{\prime})\in H_{2,n}$ commutes with $e$. But
then

$e_{(n)}\Theta_{n}(h)e_{(n)}=e_{(n)}\Theta_{n}(h^{\prime})(c^{\prime}+1)e_{(n)}=\gamma(c^{\prime}+1)e_{(n)},$

by 4.4. We now prove the claim for $h=h^{\prime}g_{n-1}g_{n-2}...g_{n-s}$ by
induction on $s$, with the case for $s=1$ just shown. Using the relation
$(g_{n-s+1}g_{n-s}-g_{n-s+1})e_{(n)}=(g_{n-s}g_{n-s+1}-g_{n-s})e_{(n)}$, we
obtain

$e_{(n)}h^{\prime}g_{n-1}g_{n-2}...g_{n-s}e_{(n)}=$
$=e_{(n)}(h^{\prime}g_{n-s}-h^{\prime})g_{n-1}g_{n-2}...g_{n-s+1}e_{(n)}+e_{(n)}(h^{\prime}g_{n-s})g_{n-1}g_{n-2}...g_{n-s+2}e_{(n)}.$

The claim now holds for each summand on the right hand side by induction
assumption. After applying $\Theta_{n}$ to it, it can be easily shown that it
is equal to $e_{(n)}\Theta(h)e_{(n)}$.

###  4.2. Trace property of $\phi$

We will use the following simple observation. Let $A$ be a semisimple algebra,
and let $I\subset A$ be a two-sided ideal. If $\psi:A\to{\mathbb{C}}$ is a
functional satisfying $\psi(cd)=\psi(dc)$ for all $c,d\in I$, then we also
have $\psi(ca)=\psi(ac)$ for all $a\in A$ and $c\in I$. Indeed, we can write
$a=a_{I}+a_{J}$ with $a_{I}\in I$ and $a_{J}\in J$ where $J$ is a two-sided
ideal of $A$ such that $IJ=0$. The claim follows from $a_{J}c=0=ca_{J}$.

######  Theorem 4.4.

The functional $\phi$ satisfies the Markov property
$\phi(cg_{n-1})=\phi(c)\phi(g_{n-1})$ for all $c\in C_{n-1}$ and the trace
property $\phi(cd)=\phi(dc)$ for all $c,d\in C_{n}$. Hence there exists a
trace on $C_{n}$ satisfying Condition 2.9.

$Proof.$ The first claim follows from Lemma 4.1,(b). We will prove the second
claim by induction on $n$, which is certainly true for the abelian algebra
$C_{1}$. Let $I_{r}=\bigoplus_{s\geq r}H_{n}e_{(s)}H_{n}$. We will prove that
the restriction of $\phi$ to $I_{r}$ satisfies the trace property by downwards
induction. We define the functional $\alpha:H_{n}\to{\mathbb{C}}$ by

$e_{(n)}he_{(n)}=\alpha(h)e_{(n)}.$

It follows from Lemma 4.2 and Lemma 4.3 that

${[N]^{n}}\phi(h_{1}e_{(n)}h_{2})=Tr(\Phi(\Theta_{n}(h_{1})E^{\otimes
n}\Phi(\Theta_{n}(h_{2}))=$ $=Tr(E^{\otimes
n}\Phi(\Theta_{n}(h_{2}h_{1}))(E^{\otimes n})=Tr(E^{\otimes
n}\Phi(h_{2}h_{1})E^{\otimes n})=\alpha(h_{2}h_{1}).$

Then we calculate

$[N]^{n}\phi(a_{1}e_{(n)}a_{2}b_{1}e_{(n)}b_{2})=[N]^{n}\alpha(a_{2}b_{1})\phi(a_{1}e_{(n)}b_{2})=\alpha(a_{2}b_{1})\alpha(b_{2}a_{1}).$

One calculates in the same way that also
$[N]^{n}\phi(b_{1}e_{(n)}b_{2}a_{1}e_{(n)}a_{2})=\alpha(a_{2}b_{1})\alpha(b_{2}a_{1})$.
This proves the claim for $r=n$. For the induction step, we first prove

(4.5) $\phi(he_{(s)})=\phi(e_{(s)}he_{(s)})\quad{\rm for\ all\ }h\in
H_{n},s\geq r.$

For $s>r$, this is clear as the restriction of $\phi$ to $H_{n}e_{(s)}H_{n}$
is a trace by induction assumption. For $s=r$, we will prove the claim by
induction on $m\geq r$, which has already been proved for $m=r$. For the
induction step, it suffices to prove the claim for elements of the form
$h=h_{1}g_{m}h_{2}$, $h_{1},h_{2}\in H_{m}$. But by Markov property, we have

$\phi(h_{1}g_{m}h_{2}e_{(r)})=\phi(g_{m})\phi(h_{1}h_{2}e_{(r)})=\phi(g_{m})\phi(e_{(r)}h_{1}h_{2}e_{(r)})=\phi(e_{(r)}h_{1}g_{m}h_{2}e_{(r)}),$

where we used the induction assumption for $h_{1}h_{2}\in H_{m}$.

Let now $h\in H_{n}$ and $k\in I_{r+1}$ such that $ke_{(r)}=e_{(r)}k=k$. Then
it follows from 4.5 that

(4.6)
$\phi(hk)=\phi(hke_{(r)})=\phi(e_{(r)}hke_{(r)})=\phi(e_{(r)}he_{(r)}k),$

where we used that $\phi$ is a trace on $I_{r+1}$. Let now $a_{i},b_{i}\in
H_{n}$, $i=1,2$. Then we can write
$e_{(r)}a_{2}b_{1}e_{(r)}=(h^{\prime}+k)e_{(r)}=e_{(r)}(h^{\prime}+k)$ with
$h^{\prime}\in H_{r+1,n}$ and $k\in I_{r+1}$. We then calculate, using 4.6

$\phi(a_{1}e_{(r)}a_{2}b_{1}e_{(r)}b_{2})=\phi(a_{1}e_{(r)}(h^{\prime}+k)e_{(r)}b_{2})=$
$=\phi(e_{(r)}b_{2}a_{1}e_{(r)}(h^{\prime}+k))=\phi(e_{(r)}b_{2}a_{1}e_{(r)}a_{2}b_{1}e_{(r)}).$

We similarly calculate

$\phi(b_{1}e_{(r)}b_{2}a_{1}e_{(r)}a_{2})=\phi(e_{(r)}a_{2}b_{1}e_{(r)}b_{2}a_{1}e_{(r)}).$

The trace property now follows from the induction assumption, using the fact
that $e_{(r)}C_{n}e_{(r)}\cong C_{n-r}$, see Proposition 2.13

##  5\. Conclusions and future research

###  5.1. Historical context

When the author of this paper visited Columbia University as a postdoc in the
first half of 1986, Vaughan suggested as a project with my host, J. Birman,
that we try to find an algebraic interpretation of the Kauffman polynomial.
This resulted in the definition of a new algebra (independently discovered by
J. Murakami) which turned out to be a $q$-deformation of Brauer’s centralizer
algebra, see [3], [1]. This algebra was subsequently used to construct
subfactors of type $BCD$, among other applications. A different
$q$-deformation of Brauer’s centralizer algebra was found in [24], see also
[14]. It contained the Hecke algebra $H_{n}$ as a subalgebra. It was shown in
[25] that it could also be used to construct subfactors, as well as module
categories of $Rep(U_{q}{\mathfrak{s}l}_{N})$ and of the related fusion tensor
categories $\overline{Rep}(U_{q}{\mathfrak{s}l}_{N})$ for $q$ a root of unity
(they are often also referred to as $SU(N)_{k}$). This will be sketched below.
In particular, we could explicitly calculate the indices and first principal
graphs of these subfactors. This, in turn, also allows us to give an explicit
description of the algebras corresponding to these module categories. So while
the current paper is purely algebraic, it is closely related to research in
which Vaughan was interested. In particular, the idea of a Markov trace which
plays a crucial role in finding the relations for the algebras $C_{n}$ goes
back to him. It should also be noted that the first module categories for
fusion categories related to $SU(2)$ were already constructed by Vaughan and
his collaborators in [7].

###  5.2. Markov traces, module categories and subfactors

It was shown in [21] that the quotient $\overline{H}_{n}(q)$ of $H_{n}(q)$
modulo the annihilator of the Markov trace $tr$ is semisimple for all $n$,
with $tr$ as in Remark 2.1. We expect the same to be true for the quotient
$\overline{C}_{n}(q)$ modulo the annihilator ideal of its extension, which was
shown to exist in Theorem 4.4. E.g. it is not hard to see that for $q$ not a
root of unity the quotient is isomorphic to the image of $C_{n}(q)$ in its
representation in ${\rm End}(V^{\otimes n})$, see Section 3. Assuming this,
the construction of the module category goes as follows:

It was shown in [11] (see also [20] for a variation of this construction) that
$Rep(U_{q}{\mathfrak{s}l}_{N})$ for $q$ not a root of unity and
$\overline{Rep}(U_{q}{\mathfrak{s}l}_{N})$ for $q$ a root of unity can be
reconstructed from the quotients $\overline{H}_{n}(q)$ modulo the annihilator
ideal of a suitable version of the Markov trace. Here the objects are given by
idempotents of $\overline{H}_{n}(q)$. We similarly define the module category
$\mathcal{M}$ whose objects are idempotents in $\overline{C}_{n}(q)$. If
$p_{M}\in\overline{C}_{m}(q)$ and $p_{H}\in\overline{H}_{n}(q)$ are
idempotents, we define the module action by

$p_{M}\otimes p_{H}\ :=\ p_{M}sh_{m}(p_{H}),$

where the algebra homomorphism $sh_{m}:H_{n}(q)\to C_{n+m}(q)$ is defined by
$sh_{m}(g_{i})=g_{i+m}\in C_{n+m}(q)$. It follows from the relations that
$p_{M}\otimes p_{H}$ is an idempotent in $\overline{C}_{n+m}(q)$. Finally, if
the quotients $\overline{H}_{n}(q)$ and $\overline{C}_{n}(q)$ allow compatible
$C^{*}$ structures, we can construct subfactors
$\mathcal{N}\subset\mathcal{M}$ from the inclusions
$\lim_{n\to\infty}\overline{H}_{n}(q)\subset\overline{C}_{n}(q)$ following the
procedure in [21], Section 1.

###  5.3. Results in [24] and [25]

We give an outline of the results in these papers which give a good idea of
the results to be expected in the approach outlined in the previous
subsection. A $q$-version $Br_{n}(q)$ of Brauer’s centralizer algebra (see
[3]) was defined in [24] by again adding one more generator $e$ to the
generators of the Hecke algebras $H_{n}(q)$. As in this paper, the relations
were forced by the condition that the extension of the Markov trace on
$H_{n}(q)$ to the algebras $Br_{n}(q)$ satisfy an analog of the Markov
condition 2.9. Subfactors were constructed from these algebras as outlined in
the previous section. Their indices and first principal graphs are given in
[25] Sections 3F and 3G. Instead of copying the results there, we just state
an easy consequence of these results which only appears implicitly in [25]:

Let $q=e^{\pi i/(N+k)}$. Then the category constructed from the quotients
$\overline{H}_{n}(q)$ is equivalent to the fusion category $SU(N)_{k}$ (or
$\overline{Rep}(U_{q}{\mathfrak{s}l}_{N})$ in the notation of this paper). For
simplicity, we assume a trivial twist (see [11] or [20] for details). It is
well-known that the simple objects of $SU(N)_{k}$ are labeled by the Young
diagrams $\lambda$ with $\leq N-1$ rows such that $\lambda_{1}\leq k$. Recall
that a module category over a tensor category can be defined via an algebra
object in the given tensor category (see [17]). We will reformulate the
following theorem in a somewhat more conceptual way in Remark 5.2.

######  Theorem 5.1.

Let $N$ be even. Then $SU(N)_{k}$ has an algebra object $A=Ind_{Ad}({\bf 1})$,
where $Ind_{Ad}({\bf 1})$ is the direct sum of simple objects $V_{\lambda}$
such that $N|\ |\lambda|$ and the number of boxes in each column of $\lambda$
is even.

$Proof.$ We consider the inclusion of von Neumann factors
$\mathcal{N}\subset\mathcal{M}$ constructed in [25], Theorem 3.4 for case (c)
listed before that theorem. It follows from the explicit description of its
principal graph in [25], Section 3G that the von Neumann algebra
$\mathcal{M}$, viewed as an $\mathcal{N}-\mathcal{N}$ bimodule, decomposes
into a direct sum of simple $\mathcal{N}-\mathcal{N}$ bimodules labeled by
exactly the Young diagrams which appear in $Ind_{Ad}({\bf 1})$. As
$\mathcal{M}$ has a multiplication, it follows that $A$ in the statement is an
algebra object in the category of $\mathcal{N}-\mathcal{N}$ bimodules. It is
known that this category is equivalent to $Ad(SU(N)_{k}$, the subcategory of
$SU(N)_{k}$ whose simple objects are labeled by Young diagrams $\lambda$ such
that $N|\ |\lambda|$ (see e.g. [22], Theorem 4.4).

######  Remark 5.2.

1\. It is well-known that the restriction to $Sp(N)$ of a simple
$SU(N)$-module labeled by the Young diagram $\lambda$ contains the trivial
representation of $Sp(N)$ if and only if the number of boxes in each column of
$\lambda$ is even. Hence the algebra in Theorem 5.1 can be viewed as a natural
analog in $SU(N)_{k}$ of the induction of the trivial representation of
$PSp(N)$ to $PSU(N)$. It would seem plausible that similar algebras exist
which correspond to inducing the trivial representation of $PSp(N)$ or $Sp(N)$
to quotients of $SU(N)$ modulo a subgroup of its center. This seems to be
compatible with Edie-Michell’s classification results of module categories of
$SU(N)_{k}$, see [4] and [5]. This possible generalization of our result
became evident after conversations with Edie-Michell.

2\. Similarly, one can construct module categories and algebras from cases (a)
and (b) before [25], Theorem 3.4. As they are related to embeddings of the
full orthogonal group $O(N)$, we would need as larger group the group
$SU(N)\times{\mathbb{Z}}/2$ of unitary matrices $u$ with $|\det(u)|=1$. It
should be possible to obtain module categories of $SU(N)_{k}$ from this via a
${\mathbb{Z}}/2$ orbifold construction. One obtains algebra objects for these
cases in the same way as it was done in Theorem 5.1. Again, one would expect
algebra objects and module categories corresponding to each quotient group
$SU(N)/Z$, where $Z$ is a subgroup of the center of $SU(N)$.

3\. A complete realization of all module categories for all fusion tensor
categories of type $SU(3)_{k}$ has been given in [6]. Using their results, one
can show that the general approach outlined here also works in the setting of
this paper for the special case $N=3$, i.e. for the embedding of $Sp(2)\subset
SL(3)$. Indeed, the explicit calculations in [6] were useful in the initial
phase of finding relations for our algebras. Here the algebra object coming
from the subfactor constructed there would be the direct sum of all simple
objects in $SU(3)_{k}$ labeled by Young diagrams $\lambda$ with $3|\
|\lambda|$. These subfactors seem to be closely connected to subfactors
constructed by F. Xu in [27].

###  5.4. Classification of module categories of WZW-fusion categories

A lot of progress in classifying module categories has recently been made by
Edie-Michell [4], building on the works of Ocneanu, Gannon, Schopieray, Evans
and Pugh, and others. Very roughly speaking these module categories can be
divided into exceptional and non-exceptional module categories. It appears
that we can find realizations for all non-exceptional module categories of
fusion tensor categories of type $SU(N)_{k}$ using the construction sketched
in this paper and its generalizations in Remark 5.2 together with the orbifold
construction. This was done in collaboration with Edie-Michell. It would also
be interesting to find out whether non-exceptional module categories of fusion
categories of other Lie types could similarly be realized via the
constructions in this paper for certain subgroups in connection with
orbifolds.

###  5.5. Co-ideal subalgebras

As mentioned in the introduction, module categories of a Drin- feld-Jimbo
quantum group $U_{q}\mathfrak{g}$ can be defined for the sub-Lie algebra
$\mathfrak{h}$ consisting of the fixed points of an order 2 Lie algebra
automorphism, see [13] and [15]. It would be interesting to see whether our
(proposed) module category of $U_{q}{\mathfrak{s}l}_{N}$ could be realized by
a suitable co-ideal deformation of the universal enveloping algebras
$U{\mathfrak{s}p}_{N-1}\subset U{\mathfrak{s}l}_{N}$.

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