The element $\Phi/(n+1)!$ is the central
idempotent in $B_{n+1}^{n+1}(-2n)$ which corresponds to the trivial representation $\rho_1$
of $B_{n+1}^{n+1}(-2n)$, defined by $\rho_1(s_i)=1$ and $\rho_1(e_i)=0$ for all $i$.
It generates a $1$-dimensional two-sided ideal of $B_{n+1}^{n+1}(-2n)$.

§.§ Presentation of endomorphism algebras

Recall the natural embedding of the Brauer algebra of degree $s$ in that of degree $t$ for any $t>s$.

For each $r>n$, let $\langle \Phi\rangle_r$ be the two-sided ideal in the Brauer algebra
$B_r^r(-2n)$ generated by $\Phi$.

A priori, elements such as $(I_{r-q}\ot A_q\ot I_q)(z\ot X_{q, q})(I_{r-q}\ot U_q\ot I_q)$
are not included in $\langle \Phi\rangle_r$ even if $z\in \langle \Phi\rangle_r$.

We have the following result.

The element $\Sigma(2n+1)$ belongs to $\langle \Phi\rangle_{2n+1}$.

Consider $B_r^r(-2n)$ for $r>n$. Let $E_r^r(k)= \prod_{j=1}^k e_{r-2j+1}$, and define

\[
\begin{aligned}
\Upsilon(r)_k&=\Sigma(r) E_r^r(k) \Sigma(r), \quad k\ge 1,\\
\Upsilon(r)_{\ge k}&=\text{linear span of $\langle \Phi\rangle_r\cup\{\Upsilon(r)_i\mid i\ge k\}$}.
\end{aligned}
\]

We first want to show that

\begin{eqnarray}\label{eq:step}
\Sigma(r)\in \Upsilon(r)_{\ge\left[\frac{r+1-n}{2}\right]}.
\end{eqnarray}

From the formula for $\Phi$, we obtain

\[
\begin{aligned}
r!(n+1)! \Sigma(r)& =\Sigma(r)\left(\left(\Phi - \sum_{k=1}^{\left[\frac{n+1}{2}\right]}a_k
\Xi_k\right)\otimes I_{r-n-1}\right)\Sigma(r).
\end{aligned}
\]

Thus $\Sigma(r)\in \Upsilon(r)_{\ge 1}$.

Note that for any $z\in \Upsilon(r-2k)_{\ge 1}$,
$\Sigma(r)(z\otimes I_{2k}) E_r^r(k) \Sigma(r)$ belongs to $\Upsilon(r)_{\ge k+1}$.
We can always re-write $\Upsilon(r)_k$ as

\[
\Upsilon(r)_k=\frac{1}{(r-2k)!}\Sigma(r)(\Sigma(r-2k)\otimes I_{2k}) E_r^r(k) \Sigma(r).
\]

If $r-2k>n$, then $\Sigma(r-2k)\in \Upsilon(r-2k)_{\ge 1}$. This implies that
$\Upsilon(r)_k\in \Upsilon(r)_{\ge k+1}$ if $r-2k>n$. Hence
\Upsilon(r)_{\ge 1}=\Upsilon(r)_{\ge 2} = \dots = \Upsilon(r)_{\ge\left[\frac{r+1-n}{2}\right]},
and (<ref>) is proved.

Now consider $\Sigma(2n+1)$.
It follows from (<ref>) that $\Sigma(2n+1)^2$ can be expressed as a linear combination of
elements in $\langle \Phi\rangle_{2n+1}$ and also elements of the form

\[
\Sigma(2n+1) E_{2n+1}^{2n+1}(i) \Sigma(2n+1) E_{2n+1}^{2n+1}(j) \Sigma(2n+1),
\quad i, j\ge 1+\left[\frac{n}{2}\right].
\]

Using the symmetrising property of $\Sigma(2n+1)$, we can write this element
as $\Sigma(2n+1)(I_{2n+1-2i}\otimes U_i) \Psi_{i j}(I_{2n+1-2j}\otimes A_j) \Sigma(2n+1)$ with

\[
\Psi_{i j}=
(I_{2n+1-2i}\otimes A_i)\Sigma(2n+1)(I_{2n+1-2j}\otimes U_j).
\]

Now $\Psi_{i j}=0$ for all  $i, j\ge 1+\left[\frac{n}{2}\right]$. Hence
$\Sigma(2n+1)^2$ belongs to $\langle \Phi\rangle_{2n+1}$, and so does also $\Sigma(2n+1)$.

The following results describe the endomorphism algebras $\End_{\Sp(V)}(V^{\otimes r})$ in terms of generators and relations.

The  homomorphism  ${F}_r^r: B_r^r(-2n)\longrightarrow \End_{\Sp(V)}(V^{\otimes r})$
of algebras is injective if $r\le n$. If $r\ge n+1$,
then $\Ker{F}_r^r$ is the two-sided ideal of the Brauer algebra $B_r^r(-2n)$ which is generated by
the element $\Phi$ defined by (<ref>).

Only the second statement requires proof. Thus we assume that $r\ge n+1$.
Consider first the case $r=n+1$. Then there is only one $D(p, q)$ with $p=n$ and $q=0$ (see Figure <ref>).
Using $\Sigma(n+1)=\Phi - \sum_{k=1}^{\left[\frac{n+1}{2}\right]}a_k \Xi_k$, we have

\[
D(n, 0)=\frac{D(n, 0)\Phi}{(n+1)!} - \sum_{k=1}^{\left[\frac{n+1}{2}\right]} a_k\frac{D(n, 0)\Xi_k}{(n+1)!}.
\]

Note that

(100, 80)(0,-30)
(-65, 7)$\frac{D(n, 0)\Xi_k}{(n+1)!}=$
(5, 40)(0, 1)20

(30, 58)$n$
(15, 40)(45, 75)(75, 40)
(35, 40)(45, 54)(55, 40)
(45, 54).
(45, 52).
(45, 50).

(0, 20)(1, 0)80
(0, 20)(0, 1)20
(80, 20)(0, 1)20
(0, 40)(1, 0)80
(30, 28)${2n+1}$

(5, 20)(0, -1)20
(6, 15)...
(16, 20)(0, -1)20

[60](-8, 10)(56, 10)(120, 10)

(19, 20)(24, 3)(29, 20)
(31, 16)...
(42, 20)(47, 3)(52, 20)
(33, 8)$k$
(19, 0)(24, 17)(29, 0)
(31, 4)...
(42, 0)(47, 17)(52, 0)

(60, 20)(90, -100)(108, 60)
(75, 20)(87, -75)(97, 60)
(97, 53)...
(100, 60)$n$

(0, 0)(1, 0)55
(0, 0)(0, -1)20
(55, 0)(0, -1)20
(0, -20)(1, 0)55
(20, -12)${n+1}$
(5, -20)(0, -1)20
(20, -30)...
(50, -20)(0, -1)20

where the dotted-line indicates that the diagram is the composition of
the two diagrams above and below the line. The diagram above the dotted line is the tensor product
of an element in $\langle \Sigma(2n+1)\rangle_{n+1-2k}^1$ with $I_n$.  Since
$\langle \Sigma(2n+1)\rangle_{n+1-2k}^1=0$ for all $k\ge 1$, we have $\frac{D(n, 0)\Xi_k}{(n+1)!}=0$.
This proves $D(n, 0)\in \langle \Phi\rangle_{n+1}$.

Now we use induction on $r$ to show that the theorem holds for $r>n+1$.
If $p=0$, the diagram corresponds to $\Sigma(2n+1)$, which
belongs to $\langle \Phi\rangle_{2n+1}$ by Lemma <ref>.
Assume $n\ge p\ge 1$, and let $r=2n+1-p+q$. Consider $D(p, q)\circ\Sigma(2n+1-p)$
by using the the formula

\[
\Sigma(2n+1-p)=\left(\left(\Phi - \sum_{k=1}^{\left[\frac{n+1}{2}\right]}a_k \Xi_k\right)
\otimes I_{n-p}\right)\frac{\Sigma(2n+1-p)}{(n+1)!}.
\]

We obtain an expression for $D(p, q)$ of the form

\begin{eqnarray}\label{eq:D-Phi}
D(p, q) = \sum_{k\ge 1} c_k D(p, q; k) + D^0,
\end{eqnarray}

where  $c_k$ are scalars, $D^0\in \langle \Phi\rangle_r$, and

(100, 80)(0,0)
(-65, 20)$D(p, q; k) =$
(5, 40)(0, 1)20
(8, 50)...
(20, 40)(0, 1)20

(30, 58)${p-q}$
(22, 40)(36, 70)(52, 40)
(35, 52).
(35, 50).
(35, 48).
(30, 40)(36, 54)(44, 40)

(55, 40)(100, 90)(108, 0)
(70, 40)(95, 70)(97, 0)

(0, 20)(1, 0)80
(0, 20)(0, 1)20
(80, 20)(0, 1)20
(0, 40)(1, 0)80
(30, 28)${2n+1}$

(5, 20)(0, -1)20
(6, 10)...
(16, 20)(0, -1)20

(19, 20)(24, 2)(29, 20)
(31, 16)...
(42, 20)(47, 2)(52, 20)
(33, 8)$k$
(19, 0)(24, 18)(29, 0)
(31, 4)...
(42, 0)(47, 18)(52, 0)

(55, 20)(95, -30)(108, 60)
(70, 20)(95, -10)(97, 60)
(97, 53)...
(98, 60)$p$

(0, 0)(1, 0)55
(0, 0)(0, -1)20
(55, 0)(0, -1)20
(0, -20)(1, 0)55
(15, -12)${2n+1-p}$
(5, -20)(0, -1)20
(20, -30)...
(50, -20)(0, -1)20

(97, 0)(0, -1)40
(108, 0)(0, -1)40

(98, -30)...
(100, -36)$q$

The diagram $D(p, q; k)$ is the composition of

(100, 80)(0,0)
(-50, 30)$D'\otimes |=$
(5, 40)(0, 1)20
(8, 50)...
(20, 40)(0, 1)20

(30, 58)${p-q}$
(22, 40)(36, 70)(52, 40)
(35, 52).
(35, 50).
(35, 48).
(30, 40)(36, 54)(44, 40)

(55, 40)(100, 90)(108, 0)
(70, 40)(95, 70)(97, 0)
(97, 7)...
(98, -5)$q$

(0, 20)(1, 0)80
(0, 20)(0, 1)20
(80, 20)(0, 1)20
(0, 40)(1, 0)80
(30, 28)${2n+1}$

(5, 20)(0, -1)20
(14, 10)...
(33, 20)(0, -1)20

(36, 20)(43, -3)(52, 20)

(55, 20)(95, -30)(108, 60)
(70, 20)(95, -10)(97, 60)
(97, 53)...
(97, 62)${p-1}$
(115, 60)(0, -1)60

with the following element of $B_r(-2n)$

(100, 80)(-5, -35)
(19, 20)(24, 2)(29, 20)
(31, 16)...

(31, 8)$k$
(19, 0)(24, 18)(29, 0)
(31, 4)...
(42, 0)(47, 18)(52, 0)

(5, 20)(0, -1)20
(6, 10)...
(16, 20)(0, -1)20

(5, 20)(0, 1)20
(16, 20)(0, 1)20

(19, 20)(0, 1)20
(29, 20)(0, 1)20
(42, 20)(0, 1)20

(42, 20)(47, -5)(80, 40)

(0, 0)(1, 0)55
(0, 0)(0, -1)20
(55, 0)(0, -1)20
(0, -20)(1, 0)55
(15, -12)${2n+1-p}$
(5, -20)(0, -1)20
(20, -30)...
(50, -20)(0, -1)20

(65, -15)...
(70, -25)$q$

(52, 40)(1, -5)16
(65, 40)(1, -5)16

Note that $D'$ belongs to $\ker{{F}_{r-1}}$.
Thus $D'\in \langle \Phi\rangle_{r-1}$ by the
induction hypothesis and it follows that
$D(p, q; k)\in \langle \Phi\rangle_r$.
This completes the proof.

§ REMARKS CONCERNING THE INVARIANT THEORY OF QUANTUM GROUPS

In this section we give a brief indication as to how the ideas we have discussed may be extended to quantised
enveloping algebras, both in the classical and super cases. General references for the material below are
<cit.>, while some references for applications in our context are <cit.>.

§.§ Quantum groups and $R$-matrices
Let $\fg$ be a complex semi-simple Lie algebra of rank $r>0$.
Associated to $\fg$ we have its universal enveloping algebra $\U(\fg)$ as well as its $q$-deformation $\U_q(\fg)$,
which is an infinite dimensional Hopf algebra over a field $\K$ which is a finite extension of the function field
$\C(q)$, where $q$ is an indeterminate over $\C$. Unlike $\U(\fg)$, $\U_q(\fg)$ is not co-commutative, but
suitably interpreted, we have $\lim_{q\to 1}\U_q(\fg)=\U(\fg)$.

Let $\fh$ be a Cartan subalgebra of $\fg$ and $\fb\supseteq \fh$ a Borel subalgebra, with $\Phi\subset\fh^*$ the root system
og $\fg$ with respect to $\fh$, and $\Pi=\{\alpha_1,\dots\alpha_r\}\subset \Phi$ the set of simple roots of $\Phi$ corresponding to the choice of $\fb$.
If $W$ is the Weyl group, there is a unique inner product $(-,-)$ on $\fh^*$ which is $W$-invariant and satisfies the condition
$( \alpha,\alpha)=2$ for short roots $\alpha\in\Phi$.

The algebra $\U_q(\fg)$ has generators $\{E_i,F_i,K_i^{\pm 1}\mid i=1,\dots, r\}$ and relations which include the following
(see, e.g., <cit.>).

K_iK_j=K_jK_i,  K_iK_i=K_iK_i=1,

K_iE_jK_i=q^(α_i,α_j)E_j,  K_iF_jK_i=q^-(α_i,α_j)F_j,

[E_i,F_j]=δ_ijK_i-K_i/q-q,  and

the quantum Serre relations.

It has the structure of a Hopf algebra, with the coproduct defined by

Δ(K_i)=K_iK_i,   Δ(E_i)=E_iK_i+1E_i,   Δ(F_i)=F_i1+K_iF_i.

Corresponding to each simple Lie superalgebra $\fg$, there exists a similarly defined quantum supergroup $\U_q(\fg)$ <cit.>, which has the structure of a Hopf superalgebra.

It was proved by Drinfeld <cit.> for his version <cit.> of $\U_q(\fg)$ over the formal power series ring that there is an invertible element $R\in{\U_q(\fg)\widehat\ot\U_q(\fg)}$, a suitably defined
“completion” of $\U_q(\fg)\ot\U_q(\fg)$, which satisfies, among others, the following relations

RΔ(u)=   Δ'(u)R,

R_12R_13R_23=   R_23R_13R_12,

where $\Delta'$ is the opposite coproduct. This element is called the universal $R$-matrix, and the second relation above is known as the Yang-Baxter equation.

As explained in <cit.>, when suitably interpreted, the universal $R$ matrix leads to an $R$-matrix for each pair of locally $\U_q(\fb)$-finite modules of type-${\bf 1}$  [8] for the Jimbo version [9] of  $\U_q(\fg)$ over $\K$.
Hereafter we will consider this version of the quantum group and its type-${\bf 1}$ representations only.

If $V$ and $W$ are $\U_q(\fg)$-modules, which are locally finite for the action of $\U_q(\fb)$, then $R$ defines a $\K$-linear map $R_{V, W}: V\ot W\lr V\ot W$.
Moreover if $\tau_{V, W}: V\ot W\lr W\ot V$, $v\ot w\mapsto w\ot v$,  is the functorial linear map and we write $\check R_{V, W}:=\tau_{V, W}\circ R_{V, W}$, the equations (<ref>)
translate into

Let $V_1,V_2$ and $V_3$ be any highest weight $\U_q(\fg)$-modules. Then

(1) $\check R$ defines a $\U_q(\fg)$-isomorphism: $V_1\ot V_2\lr V_2\ot V_1$.

(2) There is the following equality of isomorphisms $V_1\ot V_2\ot V_3\lr V_3\ot V_2\ot V_1$.

\[
\begin{aligned}
&(\check{R}_{V_1, V_2}\ot\id_{V_3}) ( \id_{V_2}\ot \check{R}_{V_1, V_3})(\check{R}_{V_2, V_3}\ot\id_{V_1})\\
&=( \id_{V_1}\ot \check{R}_{V_2, V_3})  (\check{R}_{V_1, V_3}\ot\id_{V_2})  ( \id_{V_1}\ot \check{R}_{V_1, V_2}).
\end{aligned}
\]

§.§ Braid group actions on tensor spaces
Let $\B_n$ be the $n$-string braid group. It is well known that
$\B_n$ has a presentation $\B_n=\langle \sigma_1,\dots,\sigma_{n-1}\mid \sigma_i\sigma_j=\sigma_j\sigma_i\text{ if }
|i-j|\geq 2,\;\; \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\;\;\forall i  \rangle$. This group, also
known as the Artin braid group of type $A_{n-1}$, has a
well known depiction in terms of braid diagrams.

A closely related group is the Artin braid group $\Gamma_n$ of type $B_n$, which has generators $\xi,\sigma_1,\dots,\sigma_{n-1}$.
The relations are those already given for the $\sigma_i$ in the presentation of $\B_n$ with additional relations: $\xi\sigma_1\xi\sigma_1=
\sigma_1\xi\sigma_1\xi$ and $\xi\sigma_i=\sigma_i\xi$ for $i>1$. The group $\Gamma_n$ has a well known depiction in terms of
either “cylindrical braids” <cit.> or “polar braids”, that is, braids which may encircle a “pole” [7].

* The $\sigma_i$, $i=1,\dots,n-1$ generate a subgroup of $\Gamma_n$ which is isomorphic to $\B_n$.

* Let $\B_{n+1}=\langle\sigma_0,\sigma_1,\dots,\sigma_{n-1}  \rangle$ be the $(n+1)$-string braid group.
Then the map $\iota:\B_{n+1}\lr\Gamma_n$ defined by $\sigma_0\mapsto\xi^2, $ and $\sigma_i\mapsto \sigma_i\in\Gamma_n$
for $i=1,2,\dots,n-1$ defines a monomorphism of groups.

Proofs of these statements may be found in <cit.>. The above statements lead to the following invariant theoretic

Let $V,W$ be highest weight $\U_q(\fg)$-modules, and assume that $V$ is finite dimensional.

* There are algebra homomorphisms

η_n:_n__q()(V^n),  n=1,2,…defined by $\sigma_i\mapsto (\id_V)^{\ot(i-1)}\ot \check R_{V, V}\ot(\id_V)^{\ot(n-i-1)}$
for $i=1,2,\dots, n-1$.

* There are algebra homomorphisms

ν_n:Γ_n__q()(WV^n),   n=1,2,…defined by $\xi\mapsto \check R_{V,W}\circ\check R_{W,V}$ and $\sigma_i\mapsto \id_W\ot (\id_V)^{\ot(i-1)}\ot \check R_{V, V}\ot(\id_V)^{\ot(n-i-1)}$
for $i=1,2,\dots, n-1$.

The homomorphisms $\eta_n$ and $\nu_n$ have been the subject of much literature. They play an extremely important role in the area of quantum topology (see, e.g.,  <cit.>).

From the viewpoint of invariant theory, the very first natural questions which arise are:

Question A
For which $\fg, V, W$ are the maps $\mu_n$ and $\nu_n$ surjective?

Question B Which algebras arise as images of $\mu_n$ and $\nu_n$, for simple or affine Kac-Moody Lie algebras and superalgebras $\fg$?

These questions provide a context for the Hecke algebras,
BMW <cit.> algebras and their affine analogues, and for problems with applications in quantum computing.

§.§ Relating classical and quantum invariant theory and roots of unity
In relation to Question A, it is shown in <cit.>
that $\mu_n$ is surjective for all $n$ when $V$ is a strongly multiplicity free module for $\U_q(\fg)$. It is beyond the scope of this survey
to go into details of this result, or even the definition of strongly multiplicity free modules. However we note that the proof
involves a comparison of the quantum and classical cases by means of a “base change” argument.

This involves taking $A$-forms of $\U_q(\fg)$ and of the modules $V$ and $V^{\ot n}$, where $A$ is a suitable subring of $\K$.
One then uses careful deformation arguments to compare the dimensions (over the respective fields $\K$ and $\C$) of the endomorphism
algebras in the classical and quantum cases (cf. <cit.>).

Question A for quantum supergroups <cit.>  is addressed in
<cit.> by using results from deformation quantisation. Quantum analogues of FFTs given in Theorem <ref> and Theorem <ref> are proved, establishing full tensor functors from categories of oriented and non-oriented tangles to categories of tensor representations of the quantum general linear supergroup and quantum orthosymplectic supergroup.  This includes the quantum groups associated with classical Lie algebras as special cases, thus gives an independent proof of the main results of <cit.> discussed above by a different method.  The work <cit.> is also closely related to papers [20, 21]  on the construction of quantum supergroup invariants of knots and $3$-manifolds.

Question B may also be addressed in this way. A particular case when the method works well is if there is a finite dimensional cellular algebra
$\CA_n$ through which $\eta_n$ factors, for example the BMW algebra in the case when $V$ is the natural module for  $\Or_m$ or $\Sp_{2r}$. The cellularity can then be used to explicitly use the representation theory of the relevant algebra  to compare the classical and quantum cases. This is explained in <cit.>

Integral forms of the quantum group and its modules also permit an analysis of the invariants when $q$ is specialised to a root of unity. This is explained in <cit.>.

§.§ Some category equivalences
There are very few known cases when we have an actual equivalence of categories
between a subcategory of the category of representations of a Lie algebra or its quantum analogue on the one hand, and a category
of diagrams on the other. One example we have seen in the classical case, is the full tensor subcategory of representations of the
special orthogonal group $\SO_m(\C)$ generated by the natural representation $V$,
and the enhanced Brauer category $\wt\CB(m)$ discussed in <ref>. It is proved in Theorem <ref> (see also [14]) that these categories are equivalent.

In the quantum case, it is well known (e.g., through Khovanov's theory of categorification for $\U_q(\fsl_2)$)
that there is a quotient category $\TL(q)$ of the tangle category <cit.>, called the Temperley-Lieb category,
such that if $V$ is the two-dimensional Weyl module for $\U_q(\fsl_2)$, and $Rep^0(\fsl_2)$
is the tensor category of $\U_q(\fsl_2)$-representations generated by $V$, then there is an equivalence
of categories: $\CF:\TL(q)\lr Rep^0(\fsl_2)$ (see <cit.>).

The other case where such an equivalence is known is given in the recent work [7], where it is shown that
there is a family of categories $\TLB(q,Q)$, such that $\TL(q)$ is a subcategory of $\TLB(q,Q)$ for each $Q$. If
$M(m)$ is the (projective) Verma module for $\U_q(\fsl_2)$ with highest weight $m$, and the category of representations
with objects $M(m)\ot V^{\ot n}$, $n=0,1,2,\dots$ is denoted $Rep(\fsl_2)$, then there exists an equivalence of categories

\[
\wt\CF: \TLB(q,q^m)\lr Rep(\fsl_2)
\]

whose restriction to $\TL(q)$ is the classical equivalence $\CF$ of the last paragraph.

It would be desirable to have more such equivalences, since they make potentially complex questions about representations
amenable  to diagrammatic methods, which could be essentially combinatorial.

§ AN ALGEBRAIC PROOF OF THE PRESENTATION FOR THE BRAUER CATEGORY

In principle, we can deduce a proof of Theorem <ref> from
<cit.>.  Here  we provide an independent  proof, which is taken from [11].

We first prove (1). The fact that the elementary Brauer diagrams $I,X$, $A$ and $U$
generate all Brauer diagrams under the operations of $\circ$ and $\ot$ may be seen as follows.
Fix the nodes of an arbitrary diagram $D$ from $k$ to $\ell$, and draw all the arcs as
piecewise smooth curves, in such a way
that there are at most two arcs through any point, and that no two crossings
or turning points have the same vertical coordinate. We may now draw a set of horizontal lines
(possibly after a small perturbation of the diagram) such that

(i) each line is not tangent to any of the arcs,

(ii) between successive lines there is precisely one crossing or turning point.

Then the part of the diagram between successive lines may be thought of as the $\ot$-product
of the four generators, all except one being equal to $I$.
Thus we have exhibited $D$ as a word in the generators, of the form
$D=D_1\circ D_2\circ\dots\circ D_n$, where each $D_i$ is of the form

with $Y$ being one of $A,U$ or $X$. Such an expression will be called
a regular expression, and the
factors $D_i$ elementary diagrams.
A product of elementary diagrams
in which $Y=X$ for each factor will be called a permutation diagram.
An example of a particular regular expressions is given in Figure <ref>.

(130, 70)(-10,0)
(0, 70)(0, 30)(0, 0)
(3, 10)$...$

(15,70)(15, 30)(15, 0)

[60](-10, 16)(56, 16)(130, 16)

(30, 70)(30, 30)(30, 15)
(30, 15)(35, -10)(45, 15)
(45, 15)(53, 25)(90, 70)

[60](-10, 33)(56, 33)(130, 33)

(45, 70)(53, 30)(60, 15)
(60, 0)(60, 10)(60, 15)

[60](-10, 50)(56, 50)(130, 50)

(60, 70)(68, 30)(75, 15)
(75, 0)(75, 10)(75, 15)

(75, 70)(83, 30)(90, 15)
(90, 0)(90, 10)(90, 15)

(77, 10)$...$

(105, 70)(105, 30)(105, 0)
(108, 10)$...$
(120, 70)(120, 30)(120, 0)

Regular expression

This completes the proof of (1).

We now turn to the proof that the stated relations form a complete set.
Observe first that any expression for a diagram $D$ as a word in the generators
provides a regular expression for $D$ by repeated use of the relation
(<ref>) and its dual. Accordingly we say that two regular expressions
$\mf{D},\mf{D}'$ are equivalent, and write $\mf D\sim \mf D'$ if
one can be obtained from the other by a sequence of applications of the relations in part (2)
of the Theorem. This is clearly an equivalence relation on regular expressions.

However, a word in the generators does not in general yield a Brauer diagram, but
rather a diagram multiplied by $\delta^k$ for some nonnegative integer $k$,
where $k$ is the number of deleted loops.
For any Brauer diagram $D$ and any $N\in\Z_+$, the above argument shows that
we can always represent $\delta^N D$
as a word in the generators, and hence also as a regular expression. We therefore
need to work with morphisms of the form
$\delta^N D$, where $D$ is a diagram. We refer to such a morphism
as  a scaled Brauer diagram, or simply a scaled diagram.
Every Brauer diagram is clearly a scaled diagram.

The discussion above shows that to prove the theorem, it will suffice to show that

Any two regular expressions for a scaled diagram are equivalent.

We shall extend the notion of equivalence to any expression of the form $D_1\circ\dots\circ D_n$,
where the $D_i$ are diagrams.

The two compositions $D_1\circ\dots\circ D_n$ and $D_1'\circ\dots\circ D_m'$ are said to be equivalent
if one can be obtained from the other using only the relations
in Theorem <ref> (2), and the properties of $\circ$ and $\ot$.

To prove (<ref>) we require some analysis of regular expressions and equivalence.
We shall return to the proof after carrying this out.

* The valency of scaled diagram $D\in B_k^l$ is the pair $(k,l)$.

* If $D=I^{\ot r}\ot Y\ot I^{\ot s}$ is elementary, the abscissa
$a(D)$ of $D$ is $r+1$, while the type $t(D)=Y$ $(=A,U$ or $X)$.

* The length of a regular expression $E_1\circ\dots\circ E_n$ is $n$.

We shall repeatedly apply the following elementary observation, which we refer to
as the “commutation principle”.

* Let $E_1,E_2$ be elementary diagrams such that $E_1\circ E_2$ makes sense.
If $|a(E_1)-a(E_2)|>1$ then $E_1\circ E_2\sim E_1'\circ E_2'$,
where $t(E_1')=t(E_2)$ and $t(E_2')=t(E_1)$.

* If $D,D'$ are scaled diagrams of valency $(k,l)$ and $(k',l')$ respectively,
then $D\ot D'=(I^{\ot l}\ot D')\circ (D\ot I^{\ot k'})=(D\ot I^{\ot l'})\circ (I^{\ot k}\ot D')$.

Part (2) of the Remark states the obvious relations among diagrams depicted in Figure <ref>.

(100, 60)(70,0)

(16, 40)(16, 50)(16, 60)
(29, 40)(29, 50)(29, 60)
(18, 55)$...$

(10, 40)(20, 40)(35, 40)
(10, 25)(20, 25)(35, 25)
(18, 30)$D$
(10, 40)(10, 15)(10, 30)
(35, 40)(35, 15)(35, 30)

(16, 25)(16, 15)(16, 5)
(29, 25)(29, 15)(29, 5)
(18, 10)$...$

(46, 40)(46, 50)(46, 60)
(59, 40)(59, 50)(59, 60)
(48, 55)$...$

(40, 40)(50, 40)(65, 40)
(40, 25)(60, 25)(65, 25)
(48, 30)$D'$
(40, 40)(40, 15)(40, 30)
(65, 40)(65, 15)(65, 30)

(46, 25)(46, 15)(46, 5)
(59, 25)(59, 15)(59, 5)
(48, 10)$...$

(70, 30)$=$

(96, 30)(96, 50)(96, 60)
(109, 30)(109, 50)(109, 60)
(98, 55)$...$

(90, 30)(110, 30)(115, 30)
(90, 15)(110, 15)(115, 15)
(98, 18)$D$
(90, 30)(90, 15)(90, 15)
(115, 30)(115, 15)(115, 15)

(96, 15)(96, 10)(96, 5)
(109, 15)(109, 10)(109, 5)
(98, 10)$...$

(126, 50)(126, 50)(126, 60)
(139, 50)(139, 50)(139, 60)
(128, 55)$...$

(120, 50)(130, 50)(145, 50)
(120, 35)(140, 35)(145, 35)
(128, 40)$D'$
(120, 50)(120, 45)(120, 35)
(145, 50)(145, 45)(145, 35)

(126, 35)(126, 15)(126, 5)
(139, 35)(139, 15)(139, 5)
(128, 10)$...$

(150, 30)$=$

(176, 50)(176, 55)(176, 60)
(189, 50)(189, 55)(189, 60)
(178, 55)$...$

(170, 50)(180, 50)(195, 50)
(170, 35)(180, 35)(195, 35)
(178, 40)$D$
(170, 50)(170, 45)(170, 35)
(195, 50)(195, 45)(195, 35)

(176, 35)(176, 15)(176, 5)
(189, 35)(189, 15)(189, 5)
(178, 10)$...$

(206, 30)(206, 40)(206, 60)
(219, 30)(219, 40)(219, 60)
(208, 55)$...$

(200, 30)(210, 30)(225, 30)
(200, 15)(210, 15)(225, 15)
(208, 20)$D'$
(200, 30)(200, 20)(200, 15)
(225, 30)(225, 20)(225, 15)

(206, 15)(206, 10)(206, 5)
(219, 15)(219, 10)(219, 5)
(208, 10)$...$

This follows from the fact that $(A\ot B)\circ (A'\ot B')\sim (A\circ A')\ot(B\circ B')$
for $A,A',B,B'$ of appropriate valency,
and the relation (<ref>).

The next two results will be used in the reduction of the proof of Theorem <ref>
(2) to a single case.

Let $P,Q$ be permutation diagrams of valency
$(l,l)$ and $(k,k)$ respectively and let $D\in B_k^l$ be a scaled diagram.
If any two regular expressions for ${P}\circ{D}\circ{Q}$
are equivalent, then so are any two regular expressions for ${D}$.

Let $\mf D$, $\mf D'$ be two regular expressions for $D$,
and suppose for the moment that $P$ is an elementary permutation diagram.
Then $P\circ\mathfrak{D}$ and $P\circ\mathfrak{D}'$ are regular expressions
for $P\circ D$, and hence are equivalent by hypothesis. Now
$P\circ P\circ\mathfrak{D}$ is a regular expression, and
it is evident that $P\circ P\circ\mathfrak{D}$ is equivalent
to $P\circ P\circ\mathfrak{D}'$. But from (<ref>), $P\circ P\circ\mathfrak{D}\sim \mf{D}$
and $P\circ P\circ\mathfrak{D}'\sim\mathfrak{D}'$, whence
$\mf D$ and $\mf D'$ are equivalent. This proves the Lemma for elementary $P$ and $Q=\id$.

Applying the above statement repeatedly, we see that for any permutation diagram $P$,
if any two regular expressions for $P\circ D$ are equivalent, the same is true for
$D$. A similar argument applies to prove the corresponding statement for $D\circ Q$,
for any permutation diagram $Q$.

It follows that in proving (<ref>), we may pre- and post-multiply
$D$ by arbitrary permutation diagrams, and replace $D$ by the resulting scaled diagram.

For the second reduction, we require the following definitions.

* Define $R:B_k^l\to B_{k-1}^{l+1}$ (for $k\geq 1$) (the raising operator) by
$R(D)=(D\ot I)\circ (I^{\ot (k-1)}\ot U)$, and (the lowering operator)
$L:B_k^l\to B_{k+1}^{l-1}$ by
$L(D)=(I^{\ot (l-1)}\ot A)\circ(D\ot I)$.

* If $\mf D=D_1\circ D_2\circ\dots\circ D_n$ is a regular expression for the
scaled diagram $D\in B_k^l$, define the regular expression $R(\mf D)$ for $R(D)$ by
$R(\mf D)=(D_1\ot I)\circ (D_2\ot I)\circ\dots\circ (D_n\ot I)\circ (I^{\ot k-1}\ot U)$,
and similarly define the regular expression $L(\mf D)$ for $L(D)$.
Note that if $E$ is elementary, then so is $E\ot I$, so that the above definition makes sense.

* For any regular expression $\mf D$ for a scaled diagram $D\in B_k^l$,
we have $R\circ L(\mf D)\sim \mf D$ and $L\circ R(\mf D)\sim\mf D$.

* Suppose $D$ is a scaled diagram of valence $(k,l)$ with $k\geq 1$.
The regular expressions $\mf D,\mf D'$ for $D$ are equivalent if and only if
$L(\mf D)$ and $L(\mf D')$ (or $R(\mf D)$ and $R(\mf D')$) are equivalent.

To prove (1), let $\mf D=E_1\circ\dots\circ E_n$ be a regular expression for $D\in B_k^l$. Then

\begin{aligned}
R\circ L&(\mf D)=R\left((I^{\ot(l-1)}\ot A)\circ(E_1\ot I)\dots\circ (E_n\ot I)\right)\\
&=(I^{\ot(l-1)}\ot A\ot I)\circ(E_1\ot I\ot I)\dots\circ(E_n\ot I)\circ (I^{\ot k}\ot U)\\
&\sim(I^{\ot(l-1)}\ot A\ot I)\circ(I^{\ot l}\ot U)\circ E_1\circ\dots\circ E_n
\text{ by several applications of }\ref{rem:commute}\\
&\sim I^{\ot l}\circ E_1\circ\dots\circ E_n\text {by }\eqref{eq:straight}\\
&\sim E_1\circ\dots\circ E_n\text{ by }\eqref{eq:identity}\\
&=\mf D.\\
\end{aligned}

This shows that $R\circ L(\mf D)\sim\mf D$, and the proof that $L\circ R(\mf D)\sim\mf D$ is similar.

Now to prove (2), suppose first that $\mf D, \mf D'$ are equivalent regular expressions
for $D$. Then the same sequence of moves using the relations in Theorem
<ref> (2) which convert $\mf D$ into $\mf D'$ may be applied to
$L(\mf D)$ to convert it into $L(\mf D')$. This shows that if $\mf D,\mf D'$ are equivalent
regular expressions for $D$, then $L(\mf D),L(\mf D')$ are equivalent regular expressions
for $L(D)$. A similar argument proves the corresponding statement for $R(D)$.

To prove the converse,
suppose that any two regular expressions for $R(D)$ are equivalent,
and that $\mf D_1$ and $\mf D_2$ are two regular expressions for $D$. Then $R(\mf D_1)$ and
$R(\mf D_2)$ are two regular expressions for $R(D)$, and hence by hypothesis are
equivalent. Hence by the above, $L\circ R(\mf D_1)$ and $L\circ R(\mf D_2)$ are two
equivalent regular expressions for $L\circ R(D)$, which is equal to $D$ by (1).
But by (1), $L\circ R(\mf D_1)\sim \mf D_1$ and $L\circ R(\mf D_2)\sim\mf D_2$, whence
$\mf D_1\sim\mf D_2$.

The following lemma is the key computation involving the relations in
Theorem <ref> (2).

Let $\mf T_s:=E_s\circ E_{s-1}\circ\dots\circ E_0$ be a regular expression, where
$t(E_0)=U$, $a(E_0)=a$, $t(E_i)=X$ and $a(E_i)=a+i$ for $i\geq 1$. The diagram $\mf T_s$
is shown in Figure <ref>.
Let $E$ be an elementary diagram of type $A$ or $X$ which does not `commute with'
$E_s\circ E_{s-1}\circ\dots\circ E_0$, i.e. such that $a-1\leq a(E)\leq a+s+1$.

* If $t(E)=A$, then $E\circ\mf T_s$ is equivalent to a shorter regular expression unless
$s=0$ and $a(E)=a(E_0)$. In the latter case, $E\circ\mf T_s$ is the identity multiplied by $\delta$.

* Suppose $t(E)=X$; then

(i) if $a+1\leq a(E)\leq a+s-1$, then $E\circ\mf T_s\sim \mf T_s\circ E'$ for
an elementary diagram $E'$ of type $X$. (Thus $E$ may be `moved through' $E\circ\mf T_s$).

(ii) if $a(E)=a$ or $a+s$, then $E\circ\mf T_s$ is equivalent to
a shorter regular expression.

(iii) if $a(E)=a-1$ or $a+s+1$ then $E\circ \mf T_s\sim \mf T_{s+1}$.

* Let $\mf T_s$ be as above and let $E$ be elementary of type $A$ or $X$.
Then $E\circ \mf T_s$ is equivalent to a shorter regular expression
(possibly multiplied by $\delta$) or to $\mf T_s\circ E'$
for some elementary $E'$, or to $\mf T_{s+1}$.

Consider first the case where $t(E)=A$.

If $s=0$ and $a(E)=a(E_0)$, the claim follows from the loop removal relation (<ref>).

If $a(E)=a+s+1$, then
applying (<ref>), $E\circ E_s\sim E'\circ E_s'$, where $t(E')=t(E)=A$,
$t(E_s')=t(E_s)=X$, $a(E')=a+s$ and $a(E_s')=a+s+1$. It now follows by repeated application
of Remark <ref> about commutation, that $E\circ\mf T_s\sim E''\circ \mf T_{s-1}\circ E'''$,
where $t(E'')=A$ and $a(E'')=a+s$. Repeating this argument $s$ times, we see that
$E\circ \mf T_s$ is equivalent to a regular expression
of length $s+1$ which includes $F\circ E_0$ as a subexpression, where $t(F)=A$
and $a(F)=a+1$. Applying (<ref>), we see that $F\circ E\sim I^{\ot k}$ for some $k$,
and hence $E\circ\mf T_s$ is equivalent to a regular expression of length $s-1$.

If $a(E)=a+s$, then by (<ref>), $E\circ E_s\sim E$, and we have again shortened $E\circ\mf T_s$.

If $a\leq a(E)\leq a+s-1$, then by commutation, $E\circ\mf T_s$ is equivalent to a regular
expression with a subexpression of the form $E\circ E_i\circ E_{i-1}$, where $t(E_i)=X$
and $a(E)=a(E_i)-1$. Applying (<ref>), this is equivalent
to an expression $E'\circ E_i'\circ E_{i-1}$, where $a(E_i')=a(E_{i-1})$, and $t(E_i')=X$.
Using either (<ref>) (if $i>1$) or the $^*$ of (<ref>), we again reduce the
length to show that $E\circ\mf T_s$ is equivalent to a shorter regular expression.

Finally if $a(E)=a-1$, we use commutation to show that $E\circ\mf T_s$ is equivalent
to a regular expression of length $s+1$ with a subexpression of the form
$E'\circ E_0$, where $t(E')=A$ and $a(E')=a-1=a(E_0)-1$. Applying (<ref>),
we see that $E'\circ E_0\sim I^{\ot k}$ for some $k$, and this completes the proof of (1).

Now consider the case where $t(E)=X$.

If $a+1\leq a(E)\leq a+s-1$, then after applying the commutation rule, $E\circ\mf T_s$
is equivalent to a regular expression of length $s+1$ which has a subexpression of
the form $E\circ E_{a(E)+1}\circ E_{a(E)}$. But using the braid relation (<ref>),
this is equivalent to $E'\circ E_{a(E)}\circ E_{a(E)+1}$, where $E'=E_{a(E)+1}$.
Again using commutation, we may now move the last factor below $E_0$ (since $a(E)+1\geq a+2$).
It follows that $E\circ\mf T_s\sim \mf T_s\circ E'$, where $t(E')=X$. This proves (i).

If $a(E)=a+s+1$ then evidently $E\circ\mf T_s= \mf T_{s+1}$.
If $a(E)=a+s$, the relation $X\circ X=I\ot I$ (<ref>) shows that $E\circ E_s\sim I^{\ot r}$
for some $r$, and hence $E\circ\mf T_s$ is equivalent to a shorter regular expression.
If $a(E)=a-1$, then we may use commutation to see that
$E\circ\mf T_s\sim E_s\circ\dots\circ E_1\circ E\circ E_0$. Using the relation (<ref>)
we see that this is equivalent to $E_s\circ\dots\circ E_1\circ E_1\circ E_0'$, where $t(E_0')=U$.
Applying (<ref>), we see that $E\circ\mf T_s$ is equivalent to a shorter regular expression.
Finally, if $a(E)=a$, we again use commutation to see that $E\circ\mf T_s$ is equivalent to
$E_s\circ E_{s-1}\circ\dots\circ E\circ E_1\circ E_0$. Again applying (<ref>),
we obtain a factor $E\circ E$, and applying (<ref>), we again shorten the regular
expression $E\circ\mf T_s$. This completes the proof of (2).

The statement (3) is a summary of the previous two statements.

It remains to prove (<ref>).
It follows from Lemmas <ref> and <ref> that to complete the
proof of the theorem, it suffices to prove (<ref>) for any scaled diagram which can be
obtained from $D$ by raising or lowering, or multiplication by a permutation diagram.
It follows that we may take $D$ to be the scaled diagram
$D=\delta^N U^{\ot r}$ $(N\in\Z_+)$. Hence we shall be done if we prove
the following result.

Any two regular expressions for $D=\delta^N U^{\ot r}$ are equivalent.

We shall prove (<ref>) by induction on $r$, starting with $r=0$.
For convenience, we adopt the following local convention:

* scaled diagrams will
be simply called “diagrams";

* a regular expression $\mf D$ is said to be “$\delta$-equivalent" to another regular expression
$\mf D'$ if it can be changed to $\delta^k \mf D'$ for some $k\in\Z_+$ by the relations in
Theorem <ref> (2).

Let $r=0$ and suppose $\mf D:=D_1\circ\dots\circ D_n$ is a regular expression for the empty
scaled diagram $\delta^N$ in $B_0^0$. We need to show that $\mf D$ is $\delta$-equivalent to the empty regular
expression; we do this by showing that every non-empty regular expression for the
empty scaled diagram is $\delta$-equivalent to one of shorter length.

Now by valency considerations, we must have $D_1=A$ and $D_n=U$. Let $i$ be the least
integer such that $t(D_i)=U$; then for all $j<i$, $t(D_j)=A$ or $X$. Applying
Lemma <ref> repeatedly, we see that since at least one of the $D_j$
for $j<i$ is of type $A$, $\mf D$ is $\delta$-equivalent to a shorter regular expression.
This proves the result for $r=0$

Now take $r>0$ and let $\mf D=D_1\circ\dots\circ D_n$ be a regular
expression for $D$. Then since at least $r$ of the $D_i$ must have type $U$, we
have $n\geq r$. Moreover if $n=r$, which happen only if $N=0$,
then the $D_i$ are all of type $U$, and have odd
abscissa, and any such regular expression represents $D$.
Any two such regular expressions (which will be called minimal) are equivalent
by the commutation rule (see Remark <ref>).

It therefore suffices to show that if $n>r$, then $\mf D$ is $\delta$-equivalent to a shorter
regular expression.

Clearly we have $t(D_n)=U$; if $t(D_1)=U$ then $\mf D':=D_2\circ\dots\circ D_n$ is a
regular expression for $U^{\ot (r-1)}$, and we conclude by induction on $r$ that $\mf D'$
is $\delta$-equivalent to a shorter regular expression. Thus we are finished.
Let $p=p(\mf D)$ be the least index such that $D_p$ is of type $U$. We have seen that if $p=1$
then we are finished by induction. It will therefore suffice to show that
$\mf D$ is either equivalent to a regular expression $\mf D'$ with $p(\mf D')<p(\mf D)$,
or is $\delta$-equivalent to a shorter regular expression $\mf D'$.

Thus we take $p>1$; then $t(D_p)=U$, and $t(D_i)=A$ or $X$ for $i<p$.
We now apply Lemma <ref> to conclude that either
we may commute one of the $D_i$ ($i<p$) past $D_p$, or $D_1\circ\dots\circ D_p
\sim \mf T_{p-1}$ or at least one of the $D_i$ ($i<p$) is of type $A$.
In the first case, we obtain a regular expression with small $p$-value;
in the second case, in the diagram $D_1\circ\dots\circ D_n$ if the nodes are
numbered $1,2,\dots,2r$ from left to right, node $a(D_p)$ would be joined to
node $a(D_p)+p$. Hence $p=1$, which has been excluded.

In the third case, suppose $i$ is the largest index such that $1\leq i\leq p-1$
and $D_i$ is of type $A$. Then either some $D_j$ ($i\leq j\leq p-1$)
can be commuted past $D_p$ by application of Remark (<ref>), or else
we are in the situation of Lemma <ref> (1). In the former case, we have reduced
$p$; in the latter, by loc. cit. $D_i\circ\dots\circ D_p$ is $\delta$-equivalent to a shorter
regular expression.

We have now shown that either $\mf D$ is $\delta$-equivalent to a shorter regular expression,
or equivalent to a regular expression which has the same length as $\mf D$ but a smaller $p$ value.

This completes the proof of (<ref>), and hence of Theorem <ref>.

We note that to prove part (2) of the theorem, we could
have proceeded by regarding $\CB(\delta)$ as a quotient category of
the category of (unoriented) tangles (see Remark <ref>) and deduce
the relations among the generators of Brauer diagrams
from a complete set of relations among the generators of tangles
given in <cit.> (suppressing information about orientation).
This way we obtain all relations except the one which
enforces the removal of free loops and multiplication by powers of $\delta$,
i.e., (<ref>).

[ALZ] ALZ Andersen, Henning H.; Lehrer, Gustav I.; Zhang, Ruibin   “Cellularity of certain quantum endomorphism algebras”, Pacific J. Math. 279 (2015), no. 1-2, 11–35.

[BW] BW Birman, Joan;  Wenzl, Hans   “Braids,
link polynomials and a new algebra”, Trans. Amer. Math. Soc.
313 (1989), 249–273.

[B] B Brauer,  Richard  “On algebras which are connected
with the semisimple continuous groups”, Ann. of Math. (2)
38  (1937), 857–872.

[BGZ] BGZ
Bracken, A.J.;  Gould, M.D.; Zhang, R.B.
“Quantum supergroups and solutions of the Yang-Baxter equation”,
Modern Physics Lett. A5(1990) 831–840.

[BR] BR Berele, A;  Regev, A.  “Hook Young diagrams with applications
to combinatorics and to representations of Lie superalgebras",  Adv.  Math. 64 (1987), 118-175.

[2] Brundan, Jonathan; Stroppel,  Catharina,  “Gradings on walled Brauer algebras and Khovanov's arc algebra”,
Adv. Math. 231 (2012), no. 2, 709–773.

[3] Comes, Jonathan; Wilson, Benjamin “Deligne's category $Rep(GL_\delta)$
and representations of general linear supergroups”, Represent. Theory 16, (2012), 568–609.

[C] Ck
Coulembier, Kevin,  “Tensor ideals, Deligne categories and invariant theory.” Selecta Math. (N.S.) 24 (2018), no. 5, 4659–4710.

[CZ] CZ Coulembier, Kevin; Zhang, Ruibin “Borelic pairs for stratified algebras”, Adv. Math. 345 (2019), 53–115.

[4] Deligne, P.  “La catégorie des représentations du groupe symétrique $S_t$, lorsque t n'est pas un entier naturel”. In Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., pages 209–273. Tata Inst. Fund. Res., Mumbai, 2007.

[DLZ] DLZ
Deligne, P.; Lehrer, G. I.; Zhang, R. B. “The first fundamental theorem of invariant theory for the orthosymplectic super group”, Adv. Math. 327 (2018), 4–24.

[Dr] D Drinfel'd, V. G. “Quantum groups".  Proceedings of the
International Congress of Mathematicians, Vol. 1, 2 (Berkeley,
Calif., 1986),  798–820, Amer. Math. Soc., Providence, RI, 1987.

[5] Du, Jie; Parshall, B.;   Scott, L. “Quantum Weyl reciprocity and tilting modules."
Commun. Math. Phys. 195 (1998), 321–352.

[ES] ES  Ehrig, Michael; Stroppel, Catharina,  “Koszul gradings on Brauer algebras", Int. Math. Res. Not. IMRN 2016, no. 13, 3970–4011.

[FY] FY
Freyd, Peter; Yetter, David N.
“Braided compact closed categories with applications to low-dimensional topology."
Adv. Math. 77 (1989), 156–182.

[GW] GW Goodman, R.;  Wallach, N. R.
“Representations and Invariants of the Classical Groups."
Cambridge University Press, third corrected printing, 2003.

[GL96] GL96 Graham, J. J.;  Lehrer, G. I. “Cellular algebras”,
Inventiones Math. 123 (1996), 1–34.

[6] Graham, J. J.;   Lehrer, G. I. “The representation theory of affine Temperley-Lieb algebras”,
Enseign. Math. (2) 44 (1998), no. 3-4, 173–218.

[GL03] GL03 Graham, J. J.;   Lehrer, G. I.  “Diagram algebras,
Hecke algebras and decomposition numbers at roots of unity” Ann. Sci. École Norm. Sup. 36  (2003), 479–524.

[GL04] GL04 Graham, J. J.;  Lehrer, G. I.
“Cellular algebras and diagram algebras in representation theory”,
Representation theory of algebraic groups and quantum groups,
Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo, (2004),

[HX] HX Hu, Jun;  Xiao, Zhankui, “On tensor spaces for Birman-Murakami-Wenzl algebras."
J. Algebra 324 (2010) 2893–2922.

[7] Iohara, K.; Lehrer, G. I.; Zhang, R. B. “Equivalence of a tangle category and a category of infinite dimensional Uq(sl2)-modules”. Represent. Theory 25 (2021), 265–299.

[8] Jantzen, Jens Carsten,  “Lectures on quantum groups”,
Graduate Studies in Mathematics, 6, American Mathematical Society, Providence, RI, 1996.

[9] Jimbo, M.  “A $q$-difference analog of $\U(g)$ and the Yang-Baxter equation”.
Lett.Math.Phys. 10 (1985) 63–69.

[Jo] Jo Jones, V. F. R.   “Hecke algebra representations of braid groups and link polynomials."
Annals of Mathematics 126 (1987), 335–388.

[JS] JS
Joyal, A.; Street, R. “Braided Tensor Categories." Adv. Math. 102
(1993), 20–78.

[10] Kassel, C. Quantum groups,
Springer-Verlag, Berlin (1994).

[LZ1] LZ1 Lehrer, G. I.; Zhang, R. B. “Strongly
multiplicity free modules for Lie algebras and quantum groups”,
J. of Alg. 306 (2006), 138–174.

[LZ2] LZ2 Lehrer, G. I.; Zhang, R. B.
“A Temperley-Lieb analogue for the BMW algebra."
Representation theory of algebraic groups and quantum groups, 155–190,
Progr. Math., 284, Birkhäuser/Springer, New York, 2010.

[LZ3] LZ3 Lehrer, G. I.; Zhang, R. B.
“On endomorphisms of quantum tensor space", Lett. Math. Phys. 86 (2008),  209–227.

[LZ4] LZ4 Lehrer, G. I.; Zhang, R. B.  “The second fundamental theorem of
invariant theory for the orthogonal group",
Ann. of Math. (2) 176 (2012), no. 3, 2031–2054.

[11] Lehrer, G. I.; Zhang, R. B.   “The Brauer Category and Invariant Theory",  J. European Math Society, 17 (2015) 2311 – 2351.

[12]
Lehrer, G. I.; Zhang, R. B. “The first fundamental theorem of invariant theory for the orthosymplectic supergroup”, Comm. Math. Phys. 349 (2017), no. 2, 661 – 702.

[13]
Lehrer, G. I.; Zhang, R. B. “Invariants of the orthosymplectic Lie superalgebra and super Pfaffians”,  Math. Z. 286 (2017), no. 3–4, 893–917.

[14]
Lehrer, Gustav I.; Zhang, Ruibin,  “Invariants of the special orthogonal group and an enhanced Brauer category”,  Enseign. Math. 63 (2017), no. 1–2, 181–200.

[15] Lehrer, G. I.; Zhang, R. B.   “The second fundamental theorem of invariant theory for the orthosymplectic supergroup”,
Nagoya Journal of Mathematics, accepted for publication.  arXiv:1407.1058.

[LZZ1] LZZ1 Lehrer, G.I.; Zhang, Hechun; Zhang, R.B.   “A quantum analogue of
the first fundamental theorem of invariant theory."  Commun. Math. Phys.
301 (2011), 131–174.

[LZZ2] LZZ2
Lehrer, Gustav I.;  Zhang, Hechun;  Zhang, Ruibin,  “First fundamental theorems of invariant
theory for quantum supergroups”,
European Journal of Mathematics 3,  (2020), 928–976.

[L] L Lusztig, G. “Introduction to quantum groups". Progress in
Mathematics, 110. Birkhäuser Boston, Inc., Boston, MA, 1993.

[M] MacL Mac Lane, S.
“Categories for the working mathematician."
Graduate Texts in Mathematics, 5. Springer-Verlag, New York-Berlin, 1971.

[16] Martin, Paul,  “On diagram categories, representation theory and statistical mechanics”,
Noncommutative rings, group rings, diagram algebras and their applications, Contemp. Math.,456,
Amer. Math. Soc., Providence, RI, (2008),  99–136.

[RT1] RT Reshetikhin, N.; Turaev, V. G.
“Ribbon graphs and their invariants derived from quantum groups."
Comm. Math. Phys. 127 (1990), 1–26.

[17]  Reshetikhin, N.; Turaev, V. G. “Invariants of
$3$-manifolds via link polynomials and quantum groups”.
Invent. Math. 103 (1991), no. 3, 547–597.

[RS] RS Rui, Hebing ; Si, Mei, “A criterion on the
semisimple Brauer algebras. II.” J. Combin. Theory Ser. A
113 (2006), 1199–1203.

[Sch] Sch
Schur, Issai,  Gesammelte Abhandlungen. Band I. (German) Herausgegeben von Alfred Brauer und Hans Rohrbach. Springer-Verlag, Berlin-New York, 1973.

[T1] T1  Turaev, V.G.
“Operator invariants of tangles, and $R$-matrices."
Math. USSR-Izv. 35 (1990), 411–444.

[T2] T2 Turaev, V.G.
“Quantum invariants of knots and 3-manifolds."
de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 1994.

[19] Wenzl, Hans,  “On the structure of Brauer's centralizer algebras."  Ann. of Math. (2) 128 (1988), no. 1, 173–193.

[W] W  Weyl, Hermann, “The Classical Groups. Their Invariants and Representations."
Princeton University Press, Princeton, N.J., 1939.

[Wi] Wi Witten, Edward,  “Quantum field theory and the Jones polynomial.” Comm. Math. Phys. 121 (1989), no. 3, 351–399.

[20]  Zhang, R. B. “Finite dimensional representations of the quantum supergroup
$\rm{U}_{q}(gl(m/n))$”, J. Math. Physics 34(1993) 1236–1254.

[21]  Zhang, R. B.
“Quantum supergroups and topological invariants of three-manifolds”,
Rev. Math. Phys. 7 (1995), no. 5, 809–831.

[22]  Zhang, R. B. “Structure and representations of the quantum general linear
supergroup”,       Commun. in Math. Physics
195 (1998) 525–547.

[ZGB] ZGB Zhang, R. B.; Gould, M. D.; Bracken, A. J.
“Quantum group invariants and link polynomials."
Commun. Math. Phys. 137 (1991), 13–27.

[23]Zhang, Yang,   “On the second fundamental theorem of invariant theory for the orthosymplectic supergroup.”  J. Algebra 501 (2018), 394–434.

[ALZ] ALZ Andersen, Henning H.; Lehrer, Gustav I.; Zhang, Ruibin “Cellularity of certain quantum endomorphism algebras”, Pacific J. Math. 279 (2015), no. 1-2, 11–35.

[BW] BW Birman, Joan and Wenzl, Hans,  “Braids,
link polynomials and a new algebra”, Trans. Amer. Math. Soc.
313 (1989), 249–273.

[B] B Richard Brauer, “On algebras which are connected
with the semisimple continuous groups”, Ann. of Math. (2)
38  (1937), 857–872.

[BGZ] BGZ
A.J. Bracken, M.D. Gould and R.B. Zhang,
“Quantum supergroups and solutions of the Yang-Baxter equation”,
Modern Physics Lett. A5(1990) 831–840.

[BR] BR A. Berele and A. Regev, “Hook Young diagrams with applications
to combinatorics and to representations of Lie superalgebras",  Adv.  Math. 64 (1987), 118-175.

[2] Jonathan Brundan and Catharina  Stroppel, “Gradings on walled Brauer algebras and Khovanov's arc algebra”,
Adv. Math. 231 (2012), no. 2, 709–773.

[3] Comes, Jonathan; Wilson, Benjamin “Deligne's category $Rep(GL_\delta)$
and representations of general linear supergroups”, Represent. Theory 16, (2012), 568–609.

[CZ] CZ Coulembier, Kevin; Zhang, Ruibin “Borelic pairs for stratified algebras”, Adv. Math. 345 (2019), 53–115.

[4] P. Deligne, “La catégorie des représentations du groupe symétrique $S_t$, lorsque t n'est pas un entier naturel”. In Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., pages 209–273. Tata Inst. Fund. Res., Mumbai, 2007.

[DLZ] DLZ
Deligne, P.; Lehrer, G. I.; Zhang, R. B. “The first fundamental theorem of invariant theory for the orthosymplectic super group”, Adv. Math. 327 (2018), 4–24.

[5] Jie Du, B. Parshall and L. Scott, “Quantum Weyl reciprocity and tilting modules."
Commun. Math. Phys. 195 (1998), 321–352.

[D] D Drinfel'd, V. G. “Quantum groups".  Proceedings of the
International Congress of Mathematicians, Vol. 1, 2 (Berkeley,
Calif., 1986),  798–820, Amer. Math. Soc., Providence, RI, 1987.

[ES] ES  Ehrig, Michael and Stroppel, Catharina,  “Koszul gradings on Brauer algebras", Int. Math. Res. Not. IMRN 2016, no. 13, 3970–4011.

[FY] FY
Freyd, Peter; Yetter, David N.
“Braided compact closed categories with applications to low-dimensional topology."
Adv. Math. 77 (1989), 156–182.

[GW] GW R. Goodman and N.R. Wallach,
“Representations and Invariants of the Classical Groups."
Cambridge University Press, third corrected printing, 2003.

[GL96] GL96 J.J. Graham and G.I. Lehrer, “Cellular algebras”,
Inventiones Math. 123 (1996), 1–34.

[6] J. J. Graham and G. I.  Lehrer, “The representation theory of affine Temperley-Lieb algebras”,
Enseign. Math. (2) 44 (1998), no. 3-4, 173–218.

[GL03] GL03 J.J. Graham and G.I. Lehrer, “Diagram algebras,
Hecke algebras and decomposition numbers at roots of unity” Ann. Sci. École Norm. Sup. 36  (2003), 479–524.

[GL04] GL04 J.J. Graham and G.I. Lehrer,
“Cellular algebras and diagram algebras in representation theory”,
Representation theory of algebraic groups and quantum groups,
Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo, (2004),

[HX] HX Hu, Jun and Xiao, Zhankui, “On tensor spaces for Birman-Murakami-Wenzl algebras."
J. Algebra 324 (2010) 2893–2922.

[7] Kenji Iohara, Gus Lehrer and Ruibin Zhang, “Schur-Weyl duality for certain infinite dimensional $\U_q(sl_2)$-modules”,

[8] Jantzen, Jens Carsten,  “Lectures on quantum groups”,
Graduate Studies in Mathematics, 6, American Mathematical Society, Providence, RI, 1996.

[9] Jimbo, M. A $q$-difference analog of $\U(g)$ and the Yang-Baxter equation.
Lett.Math.Phys. 10 (1985) 63–69.

[Jo] Jo V.F.R. Jones, “Hecke algebra representations of braid groups and link polynomials."
Annals of Mathematics 126 (1987), 335–388.

[JS] JS
Joyal, A.; Street, R. “Braided Tensor Categories." Adv. Math. 102
(1993), 20–78.

[10] C. Kassel, Quantum groups,
Springer-Verlag, Berlin (1994).

[LZ1] LZ1 G. I. Lehrer and R. B. Zhang, “Strongly
multiplicity free modules for Lie algebras and quantum groups”,
J. of Alg. 306 (2006), 138–174.

[LZ2] LZ2 G. I. Lehrer and R. B. Zhang,
“A Temperley-Lieb analogue for the BMW algebra."
Representation theory of algebraic groups and quantum groups, 155–190,
Progr. Math., 284, Birkhäuser/Springer, New York, 2010.

[LZ3] LZ3 G.I. Lehrer and R.B. Zhang,
“On endomorphisms of quantum tensor space", Lett. Math. Phys. 86 (2008),  209–227.

[LZ4] LZ4 G. I. Lehrer and R. B. Zhang, “The second fundamental theorem of
invariant theory for the orthogonal group",
Ann. of Math. (2) 176 (2012), no. 3, 2031–2054.

[11] G. I. Lehrer and R. B. Zhang, “The Brauer Category and Invariant Theory",  J. European Math Society, 17 (2015) 2311 – 2351.

[12]
Lehrer, G. I.; Zhang, R. B. “The first fundamental theorem of invariant theory for the orthosymplectic supergroup”, Comm. Math. Phys. 349 (2017), no. 2, 661 – 702.

[13]
Lehrer, G. I.; Zhang, R. B. “Invariants of the orthosymplectic Lie superalgebra and super Pfaffians”,  Math. Z. 286 (2017), no. 3–4, 893–917.

[14]
Lehrer, Gustav I.; Zhang, Ruibin “Invariants of the special orthogonal group and an enhanced Brauer category”,  Enseign. Math. 63 (2017), no. 1–2, 181–200.

[15] G.I. Lehrer; R.B. Zhang,  “The second fundamental theorem of invariant theory for the orthosymplectic supergroup”,
Nagoya Journal of Mathematics, accepted for publication.  arXiv:1407.1058.

[LZZ1] LZZ1 G.I. Lehrer, Hechun Zhang and R.B. Zhang, “A quantum analogue of
the first fundamental theorem of invariant theory."  Commun. Math. Phys.
301 (2011), 131–174.

[LZZ2] LZZ2
Gustav I. Lehrer, Hechun Zhang; Ruibin Zhang, “First fundamental theorems of invariant
theory for quantum supergroups”,
European Journal of Mathematics 3,  (2020), 928–976.

[L] L Lusztig, G. “Introduction to quantum groups". Progress in
Mathematics, 110. Birkhäuser Boston, Inc., Boston, MA, 1993.

[M] MacL S. Mac Lane,
“Categories for the working mathematician."
Graduate Texts in Mathematics, 5. Springer-Verlag, New York-Berlin, 1971.

[16] Paul Martin, “On diagram categories, representation theory and statistical mechanics”,
Noncommutative rings, group rings, diagram algebras and their applications, Contemp. Math.,456,
Amer. Math. Soc., Providence, RI, (2008),  99–136.

[RT] RT N. Yu. Reshetikhin; Turaev, V. G.
“Ribbon graphs and their invariants derived from quantum groups."
Comm. Math. Phys. 127 (1990), 1–26.

[17]  Reshetikhin, N.; Turaev, V. G. “Invariants of
$3$-manifolds via link polynomials and quantum groups”.
Invent. Math. 103 (1991), no. 3, 547–597.

[RS] RS Hebing Rui and Mei Si, “A criterion on the
semisimple Brauer algebras. II.” J. Combin. Theory Ser. A
113 (2006), 1199–1203.

[Sch] Sch
Schur, Issai,  Gesammelte Abhandlungen. Band I. (German) Herausgegeben von Alfred Brauer und Hans Rohrbach. Springer-Verlag, Berlin-New York, 1973.

[T1] T1 V.G. Turaev,
“Operator invariants of tangles, and $R$-matrices."
Math. USSR-Izv. 35 (1990), 411–444.

[T2] T2 V.G. Turaev,
“Quantum invariants of knots and 3-manifolds."
de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 1994.

[W] W Hermann Weyl, “The Classical Groups. Their Invariants and Representations."
Princeton University Press, Princeton, N.J., 1939.

[19] Wenzl, Hans,  “On the structure of Brauer's centralizer algebras", Ann. of Math. (2) 128 (1988), no. 1, 173–193.

[Wi] Wi Witten, Edward,  “Quantum field theory and the Jones polynomial”, Comm. Math. Phys. 121 (1989), no. 3, 351–399.

[20]  Zhang, R. B. “Finite dimensional representations of the quantum supergroup
$\rm{U}_{q}(gl(m/n))$”, J. Math. Physics 34(1993) 1236–1254.

[21]  Zhang, R. B.
“Quantum supergroups and topological invariants of three-manifolds”,
Rev. Math. Phys. 7 (1995), no. 5, 809–831.

[22]  Zhang, R. B. “Structure and representations of the quantum general linear
supergroup”,       Commun. in Math. Physics
195 (1998) 525–547.

[ZGB] ZGB R.B. Zhang, M.D. Gould and A.J. Bracken,
“Quantum group invariants and link polynomials."
Commun. Math. Phys. 137 (1991), 13–27.

[23]Zhang, Yang, “On the second fundamental theorem of invariant theory for the orthosymplectic supergroup”,  J. Algebra 501 (2018), 394–434.