1 Introduction
The affine Brauer category 
, introduced in [Reference Rui and SongRS19], and the affine Kauffman category 
, introduced in [Reference Gao, Rui and SongGRS22], are diagrammatic monoidal categories related to the invariant theory of the orthogonal and symplectic groups (or, more generally, the orthosymplectic supergroups) and their quantum analogues. Both categories are generated by a single object, denoted
${\mathsf {B}}$
or
${\mathsf {K}}$
, with identity morphism denoted by an unoriented string. The morphisms are generated by a cup, cap, dot and crossing, subject to relations given in (2-1) and (2-2) for 
and in (3-1)–(3-4) for 
. The resulting morphism spaces are spanned by diagrams like
This particular diagram represents a morphism
${\mathsf {B}}^5 \to {\mathsf {B}}^7$
, corresponding to the number of open ends on the bottom and top of the diagram.
The purpose of the current paper is to introduce a generating function formalism for these categories. A similar technique was introduced in [Reference Brundan, Savage and WebsterBSW20] for the Heisenberg category and its quantum analogue, which include the affine oriented Brauer category and the affinization of the HOMFLY-PT skein category as special cases. The generating function approach was used there to develop a precise relationship between those categories and Kac–Moody 2-categories. In the current paper, this approach allows one to simplify many arguments that have appeared in the study of cyclotomic Birman–Murakami–Wenzl (BMW) algebras and their degenerate analogues. Cyclotomic BMW algebras were introduced in [Reference Häring-OldenburgHO01] as quotients of BMW algebras. The degenerate analogues, also known as cyclotomic Nazarov–Wenzl algebras, were introduced in [Reference Ariki, Mathas and RuiAMR06] as quotients of the affine Brauer algebras defined in [Reference NazarovNaz96] (originally called affine Wenzl algebras there). In [Reference Ehrig and StroppelES18], the terminology affine/cyclotomic VW algebras is used for these degenerate versions.
For the purpose of this introduction, we focus on the affine Brauer category, since our treatment of the affine Kauffman category is parallel. As the name suggests, the affine oriented Brauer category is spanned by diagrams much like those in (1-1), but equipped with orientations. This difference arises from the fact that the generating object in the affine Brauer category is self-dual, whereas the generating object in the affine oriented Brauer category is not. Frequently, we can guess how results in the affine oriented and unoriented Brauer categories are related by considering an unoriented strand as a combination of both possible orientations—in this paper, we will not try to make this observation precise, but it is a useful guide to what to expect.
In the affine oriented Brauer category or, more generally, the Heisenberg category, the endomorphisms of the unit object form a polynomial ring in infinitely many variables—the clockwise dotted bubbles and the counterclockwise dotted bubbles are both sets of free generators for this polynomial ring. In fact, it is combinatorially more natural to identify this ring with symmetric functions, sending the clockwise and counterclockwise bubbles to the elementary and complete symmetric functions (up to sign). These are related by the so-called infinite Grassmannian relation, which expresses the clockwise dotted bubbles in terms of the counterclockwise ones, or vice versa.
Following the principle articulated above, one expects that the endomorphism ring of the identity in the affine Brauer category is generated by dotted bubbles, but that these are not algebraically independent, since they are related to both clockwise and counterclockwise bubbles. Indeed, in the affine Brauer category, the bubbles with an even number of dots freely generate the endomorphism algebra as a polynomial ring and the bubbles with odd numbers of dots can be written in terms of them by [Reference Rui and SongRS19, Theorem B and Lemma 3.4].
For those not used to working with degenerate cyclotomic BMW algebras, the algebraic dependencies can appear quite strange. Thus, in the affine Brauer category, we propose a different generating set for the endomorphism ring of the identity functor, which we think of as the coefficients of a power series
(See (2-12) for the precise definition.) Using this set of generators simplifies calculations in the affine Brauer category considerably. In particular, the dependence of odd-degree bubbles on even ones can be expressed simply by
Thus, in any representation where all bubbles act by scalars, there are algebraic restrictions on the scalars by which they can act. In particular, if L is any object in a module category and
$\operatorname {\mathrm {End}}(L)=\Bbbk $
is a field, then 
must act by a power series 
. One of our main results is Theorem 4.2, which states that
where m is the minimal polynomial of the dot. This is equivalent to the ‘admissibility’ conditions that have appeared in the literature on degenerate cyclotomic BMW algebras [Reference Ariki, Mathas and RuiAMR06, Reference GoodmanGoo11]; see (4-15). In particular, it is clear from existing nondegeneracy results in the literature (in particular, [Reference Rui and SongRS19, Theorem C]) that this is the only possible expression for
$\mathbb {O}_L(u)$
; however, we are able to give a simple direct proof using power series, without using any explicit calculation of bases or construction of representations. Theorem 5.6 gives an analogous result for the affine Kauffman category with a similar proof. We feel that Theorems 4.2 and 5.6 give a particularly concise and elegant formulation of the concept of admissibility.
Closely related to the notion of a categorical action is that of a cyclotomic quotient, which is the quotient of the affine Brauer category by a left tensor ideal that specializes the dotted bubbles to scalars and imposes a polynomial relation on the dot. In [Reference Rui and SongRS19, Theorem C], these quotients are studied and a nondegeneracy theorem is proved in the case of
$\mathbf {u}$
-admissible parameters; see Section 4.2 for a more detailed discussion of admissibility conditions for parameters. However, we consider these quotients from a slightly different perspective than the existing literature—rather than fixing the polynomial relation on the dot first and then analysing compatible choices of the bubble parameters, we fix the bubble parameters first and then study the cyclotomic quotient for different choices of the polynomial relation. In particular, in Theorem 6.10, we give an explicit description of the minimal polynomial of the dot in the quotient categories without any assumption of admissibility, and thus identify each cyclotomic Brauer category with a standard one, where the bubble scalars are determined by the minimal polynomial; see Corollary 6.4. This extends [Reference Rui and SongRS19, Theorem C] to give a basis for the morphism spaces of the cyclotomic quotient in all cases. In Section 7, we perform a similar analysis of the cyclotomic Kauffman categories of [Reference Gao, Rui and SongGRS22].
In addition to providing a useful formalism for studying the affine Brauer and Kauffman categories, we also expect the generating function approach developed in the current paper to be a key ingredient in relating these categories to categorified iquantum groups, including those defined in [Reference Bao, Shan, Wang and WebsterBSWW18, Reference Brundan, Wang and WebsterBWW24], in a manner analogous to the connection between the Heisenberg category and Kac–Moody 2-categories developed in [Reference Brundan, Savage and WebsterBSW20]. This is one of the main motivations for developing such a generating function approach.
2 The affine Brauer category
In this section, we recall the definition of the affine Brauer category and introduce the generating function formalism. We use the generating function approach to recover several important results about the category. We let
$\Bbbk $
be an arbitrary commutative ring in which
$2$
is invertible. Throughout this paper, all categories are
$\Bbbk $
-linear. We also adopt the convention that
$0 \in \mathbb {N}$
.
Definition 2.1 [Reference Rui and SongRS19, Definition 1.2].
The affine Brauer category 
is the strict
$\Bbbk $
-linear monoidal category generated by an object
${\mathsf {B}}$
and morphisms
subject to the relations
We call the morphism 
a dot.
One can easily verify from these relations that there is no nontrivial grading by any abelian group on the morphisms in this category in which the dot, crossing, cup and cap morphisms are homogeneous.
Remark 2.2. The affine Brauer category of [Reference Rui and SongRS19, Definition 1.2] is actually the reverse of ours. Precisely, let 
be the category of [Reference Rui and SongRS19, Definition 1.2]. Then, comparing presentations, one sees that we have an isomorphism of
$\Bbbk $
-linear monoidal categories 
defined on generating morphisms by
where 
denotes the reversed category of 
, with tensor product reversed. Intuitively, this isomorphism flips diagrams in a vertical line. Combined with the isomorphism (2-4) below, this implies that the category of [Reference Rui and SongRS19, Definition 1.2] is also isomorphic to the category defined in Definition 2.1. However, it is more useful to think of the two categories as the reverses of each other, since we work with left tensor ideals, whereas [Reference Rui and SongRS19] uses right tensor ideals.
The definition [Reference Rui and SongRS19, Definition 1.2] includes some additional defining relations: the first and second relations in (2-3) below. However, these follow from (2-1) and (2-2) using adjunction.
Proposition 2.3. The following relations hold in 
:
Proof. The proofs all involve rotating the relations (2-1) and (2-2) using cups and caps. We give the proof of the third relation in (2-3), since the others are analogous. First note that, using the fourth and sixth equalities in (2-1),
Thus,
All rotations of the relations (2-1) and (2-2) by
$180^{\circ }$
also hold in 
. This shows that there is a functor
$\mathbf {D} \colon \mathcal {C} \to \mathcal {C}^{\mathrm {op}}$
given by rotating diagrams through
$180^{\circ }$
. In terms of the theory of monoidal categories, this is the duality functor, and the fact that
$\mathbf {D}$
is strictly monoidal and that
$\mathbf {D}^2 = \operatorname {\mathrm {id}}_{\mathcal {C}}$
shows that the category 
is strict pivotal—this is precisely the definition of a strictly pivotal category. Note, this does not mean that we can freely isotope diagrams, since the final relation of (2-2) shows that we pick up signs when dots move past minima or maxima.
Although we do not use them in the current paper, we point out two useful symmetries of the affine Brauer category. It follows from Proposition 2.3 that reflecting in a horizontal line gives an isomorphism of monoidal categories
where 
denotes the opposite category of 
. On the other hand, reflecting diagrams in a vertical line and then multiplying all dots by
$-1$
gives an isomorphism of monoidal categories
We define
to denote the n-fold vertical composition of 
with itself. To simplify algebraic manipulations with sums of powers, we also use the notation
We similarly interpret dots labelled by polynomials in x by extending this notation linearly. We view the power series
as a generating function for the different possible numbers of dots on a string. Since this power series appears so frequently, we introduce the notation
Then, for example,
It is also useful to note that
For a Laurent series
$p(u) \in \Bbbk (\! ( u^{-1} )\! )$
, we let
$[p(u)]_{u^r}$
denote its
$u^r$
-coefficient and we define
$$ \begin{align} [p(u)]_{u^{\ge 0}} = [p]_{\ge 0}(u) := \sum_{r \ge 0} [p(u)]_{u^r} u^r \in \Bbbk[u]. \end{align} $$
For any polynomial
$p(u) \in \Bbbk [u]$
,
More generally, if
$p(u) \in \Bbbk (\! ( u^{-1} )\! )$
, then
Lemma 2.4. The following relations hold in 
:
Proof. By (2-2),
Composing with
$(u-x)^{-1}$
on the top-left string and bottom-right string then yields the first relation in (2-10). The second equality in (2-10) is proved similarly. The equalities in (2-11) are obtained from those in (2-10) by replacing u by
$-u$
and then multiplying all diagrams by
$-1$
. Note that we cannot replace x by
$-x$
in general, since the defining relations of 
are not homogeneous in the number of dots. Alternatively, the relations (2-11) can be obtained by rotating the relations (2-10) using cups and caps.
We define
We interpret these equations by Laurent expansion in
$u^{-1}$
at
$u=\infty $
; that is,
Proposition 2.5. The following relations hold in 
:
Proof. To see the first equality of (2-13), we note that
Next,
Thus,
which implies that
Multiplying both sides by
${4u^2}/{(1+2u)(1-2u)}$
then gives the second equality of (2-13).
To prove the first equality in (2-14), we compute
The proofs of the second equality in (2-14) and both equalities in (2-15) are analogous.
To prove the first equality in (2-16), we compute
Thus,
Adding 
to both sides, then multiplying both sides by
${2u}/({2u-1})$
, gives
Since
the result follows. Finally, to obtain the second equality in (2-16), we multiply both sides of the first equality in (2-16) on the left and right by 
, use (2-13), compose both sides with
and then multiply both sides by
$-1$
.
Remark 2.6. The power series 
was considered in [Reference Gao, Rui and SongGRS23, Section 4], and (2-13) and (2-16) correspond to Lemmas 4.1 and 4.2 there. When the bubbles 
,
$r \in \mathbb {N}$
, are evaluated at scalars, the second relation in (2-13) corresponds to [Reference Ariki, Mathas and RuiAMR06, Remark 2.11], particularly when written in the form (2-18). Similarly, (2-16) corresponds to [Reference Ariki, Mathas and RuiAMR06, Lemma 4.15]. We discuss these connections in more detail in Section 4.2.
The relations (2-18) are equivalent to the following lemma, which shows how the bubbles with an odd number of dots can be written in terms of bubbles with fewer dots.
, 
Proof. Although (2-19) is proved in [Reference Rui and SongRS19, Lemma 3.4], we give here a proof showing how it follows from (2-13). As explained in the proof of Proposition 2.5, the second equation in (2-13) is equivalent to (2-17). From (2-17) and the second relation in (2-2),
Equating coefficients of
$u^{-m-1}$
,
$m \in \mathbb {N}$
, gives
When m is even, both sides of (2-20) are equal to zero. On the other hand, when
$m = 2r+1$
,
$r \in \mathbb {N}$
, (2-20) yields (2-19).
3 The affine Kauffman category
In this section, we recall the definition of the affine Kauffman category of [Reference Gao, Rui and SongGRS22]. As for the affine Brauer category, we use a generating function approach to deduce some important relations that hold in this category. In this section,
$\Bbbk $
is an arbitrary commutative ring.
Definition 3.1 [Reference Gao, Rui and SongGRS22, Definition 1.3].
Suppose
$z,t \in \Bbbk ^\times $
. The affine Kauffman category 
is the strict
$\Bbbk $
-linear monoidal category generated by an object
${\mathsf {K}}$
and morphisms
subject to the relations
The first relation in (3-3) is called the Kauffman skein relation. We call the morphism 
a dot, or a Kauffman dot, when we want to emphasize that we are working in 
, as opposed to 
.
Remark 3.2. The category of [Reference Gao, Rui and SongGRS22, Definition 1.3] is actually the reverse of ours. To be precise, let 
be the category defined in [Reference Gao, Rui and SongGRS22, Definition 1.3]. Then, comparing the presentations, one sees that we have an isomorphism of
$\Bbbk $
-linear monoidal categories 
defined on generating morphisms by
Lemma 3.3. The following relations hold in 
:
Proof. Since these relations follow from standard arguments, we give only a sketch of the proof. The first relation in (3-5) follows from the third equality in (3-2) by attaching 
to the bottom of both diagrams, then using the first equality in (3-1). To prove the first equality in (3-6), we compute
The remaining relations follow from rotation using cups and caps.
We define
We then adopt the same generating function conventions as in Section 2, using x to denote the dot. The difference here is that the dot is now invertible. Noting that
it is convenient to define
It follows that
Using the identity
$$ \begin{align*} \frac{u^2}{(u-x)(u-x^{-1})} = \frac{u^2}{u^2-1} \left( \frac{u}{u-x} + \frac{u}{u-x^{-1}} - 1 \right), \end{align*} $$
we see that
For any polynomial
$p(u) \in \Bbbk [u]$
,
Just as in (2-9), we can generalize to the case
$p(u) \in \Bbbk (\! ( u^{-1} )\! )$
using the truncation defined in (2-7); we find that
Lemma 3.4. The following relations hold in 
:
Proof. By the first equalities of (3-3) and (3-4),
The first equality in (3-11) follows from multiplication by 
at the bottom left and top right on both sides of the equation.
To obtain the second equality in (3-11), we then attach a cap to the top of the rightmost strand, attach a cup to the bottom of the leftmost strand and then use (3-7).
Define
Lemma 3.5. There is an isomorphism of
$\Bbbk $
-linear monoidal categories
given by flipping crossings and taking inverses of dots. More precisely,
$\beta $
sends
${\mathsf {K}}$
to
${\mathsf {K}}$
and acts on the generating morphisms by
We also have
Proof. This is a straightforward verification of the defining relations.
We call
$\beta $
the bar involution.
Proposition 3.6. The following relation holds in 
:
Proof. To simplify notation, we drop
, identifying a scalar
$c \in \Bbbk $
with 
in what follows. We have
Thus,
which implies that
Multiplying both sides by
${z^2(u^2-1)^2}/({z^2u^2-(u^2-1)^2})$
and using (3-12) then gives (3-13).
Remark 3.7. When the bubbles 
,
$r \in \mathbb {Z}$
, are evaluated at scalars, the relation (3-13) corresponds to [Reference Rui and XuRX09, (2.30)], noting that
$\delta $
and
$\varrho $
of [Reference Rui and XuRX09] correspond to our z and
$t^{-1}$
, respectively. We discuss these connections in more detail in Section 5.2.
Proposition 3.8. The following relations hold in 
:
Proof. We have
The second relation then follows after applying the bar involution
$\beta $
.
Proposition 3.9. The following relations hold in 
:
Proof. To prove the first equality in (3-15), we compute
Thus,
Multiplying both sides by
${z(u^2-1)}/({u^2-zu-1})$
and adding 
to both sides gives
The first equation in (3-15) then follows from the fact that
$$ \begin{align*} 1 + z^2 \left( \frac{u}{u-x} - \frac{u^2}{(u-x)^2} \right) = 1 - \frac{z^2ux}{(u-x)^2}. \end{align*} $$
To obtain the second equality in (3-15), we multiply both sides of the first equality in (3-15) on the left and right by 
, then use (3-13). Alternatively, we can apply the bar involution
$\beta $
.
Note that, when
$z=q-q^{-1}$
for some
$q \in \Bbbk ^\times $
, the rational function appearing in (3-15) can be factored:
$$ \begin{align*} \frac{1-z^2ux^{-1}(u-x^{-1})^{-2}}{1-z^2ux(u-x)^{-2}} = \frac{(u-x)^2 (u-q^2x^{-1}) (u-q^{-2}x^{-1})}{(u-x^{-1})^2 (u-q^2x) (u-q^{-2}x)}. \end{align*} $$
4 Module categories over the affine Brauer category
A module category over a strict
$\Bbbk $
-linear monoidal category
$\mathcal {A}$
is a
$\Bbbk $
-linear category
$\mathcal {R}$
together with a strict
$\Bbbk $
-linear monoidal functor 
, where 
denotes the strict
$\Bbbk $
-linear monoidal category whose objects are
$\Bbbk $
-linear endofunctors of
$\mathcal {R}$
and morphisms are natural transformations. We usually suppress the functor
$\mathbf {R}$
, using the same notation
$f \colon X \to Y$
both for a morphism in
$\mathcal {A}$
and for the natural transformation between endofunctors of
$\mathcal {R}$
that is its image under
$\mathbf {R}$
. For a morphism
$f \colon X \to Y$
in
$\mathcal {A}$
, we represent the evaluation
$f_V \colon XV \to YV$
of this natural transformation on an object
$V \in \mathcal {R}$
diagrammatically by drawing a line labelled by V on the right-hand side of the usual string diagram for f:
We wish to study module categories over the affine Brauer category 
in the case where
$\Bbbk $
is a field of characteristic different from
$2$
, which we assume through the end of Section 4.1. We also want to assume some finiteness conditions on the module category. As in [Reference Brundan, Savage and WebsterBSW20], we consider an abelian category
$\mathcal {R}$
that is either locally finite abelian (that is, all objects are finite length and all morphism spaces are finite dimensional as
$\Bbbk $
-vector spaces) or schurian (that is, equivalent to the category of locally finite-dimensional modules over a locally finite-dimensional locally unital algebra)—see [Reference Brundan, Savage and WebsterBSW20, Section 2.2] for a more detailed discussion of these notions.
4.1 Analysis of minimal polynomials
Let
$L\in \mathcal {R}$
be an object which is a brick, that is, an object such that
$\operatorname {\mathrm {End}}_{\mathcal {R}}(L)=\Bbbk $
. In this case, the coefficients of the series 
must act by scalars in
$\Bbbk $
. Define
On the other hand, we can consider the image
${\mathsf {B}} L$
of the generating functor under the action. This carries an action of 
, which we can think of as defining a ring homomorphism
$\Bbbk [u]\to \operatorname {\mathrm {End}}_{\mathcal {R}}({\mathsf {B}} L)$
. By our finiteness assumptions on
$\mathcal {R}$
, together with the fact that
${\mathsf {B}}$
is self-adjoint, we know that
$\operatorname {\mathrm {End}}_{\mathcal {R}}({\mathsf {B}} L)$
is finite dimensional over
$\Bbbk $
. Thus, the algebra homomorphism
has nontrivial kernel
$I_L$
generated by a unique monic polynomial
$m_L \in \Bbbk [u]$
. Therefore,
$m_L$
is the unique monic polynomial such that, for
$f \in \Bbbk [u]$
,
Let
$$ \begin{align*} d_L = \deg m_L. \end{align*} $$
Lemma 4.1. If
$g \in I_L$
, then
$$ \begin{align} \hat{g}(u) := \big( -u-\tfrac{1}{2} \big) g(-u) \mathbb{O}_L(-u) \in I_L. \end{align} $$
Proof. Suppose
$g \in I_L$
. A priori,
$\hat {g}(u) \in \Bbbk (\! ( u^{-1} )\! )$
. However, since 
, we have, for
$r> 0$
,
where the equality
$\dagger $
holds because
$u^{r}(1-{1}/{2u})$
is a polynomial for
$r> 0$
. From this, we can conclude that
$\hat {g}(u) \in \Bbbk [u]$
.
Next, note that
Therefore,
$\hat {g} - \tfrac 12g \in I_L$
and so
$\hat {g} \in I_L$
, as desired.
For
$f \in \Bbbk [u]$
, define
Theorem 4.2. We have
$\mathbb {O}_L = \mathbb {O}_{m_L}$
.
Proof. By (2-13), we have
$\mathbb {O}_L(-u) = \mathbb {O}_L(u)^{-1}$
. Thus, by (4-2),
$$ \begin{align} \frac{\hat{m}_L(-u)}{\big( u-\frac{1}{2} \big) m_L(u)} = \mathbb{O}_L = \frac{\big( -u-\frac{1}{2} \big) m_L(-u)}{\hat{m}_L(u)}, \end{align} $$
and so
$$ \begin{align} \hat{m}_L(u) \hat{m}_L(-u) = \big( u - \tfrac{1}{2} \big) \big( -u - \tfrac{1}{2} \big) m_L(u) m_L(-u). \end{align} $$
By Lemma 4.1,
$\hat {m}_L(u) \in I_L$
, and so
$m_L(u)$
divides
$\hat {m}_L(u)$
. Combined with (4-7) and the fact that
$\hat {m}_L(-u)$
is monic, this implies that
$$ \begin{align} \hat{m}_L(-u) = \left( (-1)^{d_L}u + \frac{\epsilon}{2} \right)\ m_L(-u) \end{align} $$
for some
$\epsilon \in \{\pm 1\}$
.
Now note that
where
$$ \begin{align*} g(u) = (-1)^{d_L} m_L(u) - \frac{1+\epsilon}{2u} m_L(u). \end{align*} $$
Therefore, the polynomial
$$ \begin{align*} [g(u)]_{u^{\ge 0}} = (-1)^{d_L} m_L(u) - \left[ \frac{1+\epsilon}{2u} m_L(u) \right]_{u^{\ge 0}} \end{align*} $$
must be divisible by
$m_L(u)$
, which is only possible if
$[ \frac {1+\epsilon }{2u} m_L(u) ]_{u^{\ge 0}}=0$
. Since
$m_L(u)$
has positive degree, this in turn can only hold if
$\epsilon =-1$
. Thus,
$$ \begin{align*} \hat{m}_L(-u) = \begin{cases} (u-\frac{1}{2}) m_L(-u) & \text{if } d_L \text{ is even}, \\ (-u-\frac{1}{2}) m_L(-u) & \text{if } d_L \text{ is odd}. \end{cases} \end{align*} $$
The result follows.
4.2 Admissibility in the Brauer case
Restrictions on the scalars by which bubbles can act have played an important role in the study of degenerate cyclotomic BMW algebras. In this section, we show how such restrictions can be easily deduced from the results of this paper. We assume throughout this section that
$\Bbbk $
is a commutative ring in which
$2$
is invertible.
Fix an 
-module category 
. Then,
$\mathbf {R}$
maps elements of 
to natural transformations of the identity functor on
$\mathcal {R}$
. Evaluation on an element V of
$\mathcal {R}$
then defines a ring homomorphism from 
to the centre
$Z_V$
of the endomorphism algebra of V. If V is a brick, then
$Z_V = \Bbbk $
. However, it is sometimes useful to consider the more general situation. In the literature on degenerate cyclotomic BMW algebras, the scalar by which 
acts is usually denoted
$\omega _n$
. For V an object in an 
-module category
$\mathcal {R}$
, let
Lemma 2.7 places a restriction on the possible sequences
$\Omega _V$
. A sequence
$\Omega = (\omega _r)_{r \in \mathbb {N}}$
in a commutative ring satisfying
$$ \begin{align} \omega_{2r+1} = \frac{1}{2} \left( -\omega_{2r} + \sum_{n=0}^{2r} (-1)^n \omega_n \omega_{2r-n} \right) \end{align} $$
is called admissible in [Reference Ariki, Mathas and RuiAMR06, Definition 2.10]. It follows from Lemma 2.7 that admissibility is simply a consequence of the relations of the category 
. We think that the analogue (2-13) of the infinite Grassmannian relation is an elegant statement of this relation.
Suppose
$m(u) \in \Bbbk [u]$
is monic and factors completely as
$$ \begin{align*} m(u) = (u-a_1)(u-a_2) \dotsm (u-a_{d}). \end{align*} $$
In [Reference GoodmanGoo11, Definition 2.5], a sequence
$\Omega = (\omega _r)_{r \in \mathbb {N}}$
in
$\Bbbk $
is called weakly admissible for m if it satisfies (4-10) and
$$ \begin{align} \sum_{j=0}^{d} (-1)^j e_j(a_1,\dotsc,a_{d}) \omega_{j+n} = 0 \quad \text{for all } n \in \mathbb{N}, \end{align} $$
where
$e_j$
is the jth elementary symmetric polynomial. For
$\Omega = \Omega _V$
as in (4-9), the condition (4-11) is equivalent to the condition
In particular,
Note the slightly confusing fact that ‘weakly admissible’ is a stronger condition than ‘admissible’.
Lemma 4.3. The sequence
$\Omega _V$
is weakly admissible for m if and only if 
acts on V by a rational function whose denominator divides m. Equivalently,
$\Omega _V$
is weakly admissible for m if and only if
Proof. For
$n \in \mathbb {N}$
,
The result follows.
We now turn our attention to another notion of admissibility that has played an important role in the literature. For a finite sequence
$\mathbf {a} = (a_1,a_2,\dotsc ,a_{d})$
in
$\Bbbk $
, a sequence
$\Omega = (\omega _r)_{r \in \mathbb {N}}$
in
$\Bbbk $
is called
$\mathbf {a}$
-admissible if
$$ \begin{align} \Omega(u) := \sum_{r=0}^\infty \omega_r u^{-r} = -u + \frac{1}{2} + \left( u - \frac{(-1)^{d}}{2} \right) \prod_{i=1}^{d} \frac{u+a_i}{u-a_i}. \end{align} $$
If we define
then
$$ \begin{align*} \Omega \text{ is admissible} \iff \mathbb{O}_\Omega(u) \mathbb{O}_\Omega(-u) = 1. \end{align*} $$
Furthermore, recalling (4-5),
$$ \begin{align} \Omega \text{ is }\mathbf{a}\text{-admissible} \iff \mathbb{O}_\Omega = \mathbb{O}_m \quad \text{for } m(u) = \prod_{i=1}^{d}(u-a_i). \end{align} $$
The term
$\mathbf {a}$
-admissible was introduced in [Reference Ariki, Mathas and RuiAMR06, Definition 3.6], although here we use the equivalent formulation from [Reference Ariki, Mathas and RuiAMR06, Lemma 3.8].
We can see the significance of these conditions by computing the action of the bubbles 
on L.
Corollary 4.4. If
$\Bbbk $
is a field, then we have
Thus, if
$m_L(u) =\prod _{i=1}^{d}(u-a_i)$
, then
$\mathbb {O}_L(u) = \mathbb {O}_{\Omega }$
for
$\Omega $
the
$\mathbf {a}$
-admissible sequence defined by (4-13).
acts on L by 
Thus, if
$\Bbbk $
is a field and
$m_L(u) = (u-a_1)(u-a_2) \dotsm (u-a_d)$
is a product of linear factors in the algebraic closure of
$\Bbbk $
, then it follows from Corollary 4.4 that
5 Module categories over the affine Kauffman category
In this section, we study module categories over the affine Kauffman category 
. Our treatment is parallel to that of Section 4. As in that section, we let
$\mathcal {R}$
be a module category over 
that is either locally finite abelian or schurian. We assume throughout this section that
$\Bbbk $
is a field.
5.1 Analysis of the minimal polynomial
Let
$L \in \mathcal {R}$
be a brick and let
$J_L$
be the kernel of the algebra homomorphism
Since now the dot is invertible, for all
$k \in \mathbb {Z}$
and
$g \in \Bbbk [u,u^{-1}]$
, the element
$u^k g(u)$
generates the same ideal as g. Let
$m_L$
be the unique generator of
$J_L$
that lies in
$\Bbbk [u]$
, is monic and satisfies
$M := m_L(0) \neq 0$
. As in the previous section, we let
$d_L = \deg m_L$
. Note that we can equally well find the corresponding minimal polynomial for the inverse dot, which has the same degree. For a monic polynomial
$f(u) \in \Bbbk [u]$
with
$f(0) \ne 0$
, define
$$ \begin{align} \check{f}(u) = f(0)^{-1} u^{\deg f} f(u^{-1}), \end{align} $$
which is another monic polynomial with nonzero constant term
$\check {f}(0) = f(0)^{-1}$
. Then,
$\check {m}_L(u)$
is the minimal polynomial of the inverse dot.
The coefficients of the series 
must act by scalars in
$\Bbbk $
. Define
Note that, unlike in the degenerate case, there is no analogue of the first relation in (2-13), which is why we have two series
$\mathbf {O}_L$
and
$\widetilde {\mathbf {O}}_L$
, compared with the single series
$\mathbb {O}_L$
in the degenerate case. However, we show below that, in some module categories, there is a parallel relation between these series; see Proposition 5.4.
We have the following analogue of Lemma 4.1.
Lemma 5.1. If
$g \in J_L\cap \Bbbk [u]$
has nonzero constant term, then
$$ \begin{align} \hat{g}(u) := ( 1 - zu - u^2 ) g(u^{-1})u^{\deg g} \mathbf{O}_L(u^{-1}) \end{align} $$
is a polynomial of degree
$\deg g+2$
with nonzero constant term and
$\hat {g} \in J_L$
.
Proof. Suppose
$g \in J_L \cap \Bbbk [u]$
has nonzero constant term. A priori, 
with nonzero constant term. However, since 
, we have, for
$s< 0$
,
where, in the second equality, we replaced u by
$u^{-1}$
, and the equality
$\dagger $
holds because
$s<0$
and
$g(u) ( ( t^{-1}-z ) u^2 - t^{-1} )$
is a polynomial. From this, we can conclude that
$\hat {g} \in \Bbbk [u]$
. On the other hand, if
$s=0$
, a similar argument shows that
$[ \hat {g}(u) ]_{u^{\deg g+2}}=-t^{-1}g(0)$
and so the degree of
$\hat {g}$
is precisely
$\deg g+2$
.
Next, note that
Therefore,
$\hat {g}(u)u^{-\deg g-2} - zu^2 g(u) \in J_L$
and so
$\hat {g}(u) \in J_L$
, as desired.
Note that, rearranging (5-4), we obtain the formula
$$ \begin{align*} \mathbf{O}_L = \frac{\hat{g}(u^{-1})u^{\deg g+2}}{( u^2 - zu -1 ) g(u)}, \end{align*} $$
showing that
$\mathbf {O}_L$
is a rational function. Furthermore, applying Lemma 5.1 with
$g=m_L$
, we see that
$m_L$
must divide
$\hat {m}_L$
and so we can factor
$\hat {m}_L(u)=h(u)m_L(u)$
, where
$h(u)$
is a quadratic polynomial. Thus, we find that
$$ \begin{align*} \mathbf{O}_L = \frac{h(u^{-1})m_L(u^{-1})u^{d_L+2}}{( u^2 - zu -1 ) m_L(u)}. \end{align*} $$
It follows from (5-3) and (5-4) that
$h(0) = tM^{-1}$
. Thus, we can write
$u^2h(u^{-1}) = t M^{-1}(u^2-1)+au+b$
for some
$a,b \in \Bbbk $
. Using (5-2), then
$$ \begin{align} \mathbf{O}_L = \frac{u^2-1}{u^2-zu-1} \left( t + \frac{(au+b)M}{u^2-1} \right) \frac{\check{m}_L(u)}{m_L(u)}. \end{align} $$
Lemma 5.2. Suppose that
$f \in \Bbbk [u]$
and that
$r \in \mathbb {N}$
is odd and satisfies
$r \ge \deg f$
. Then,
Proof. By linearity, it suffices to consider the case
$f = u^n$
,
$n \in N$
. In this case, letting x denote the dot, the left-hand side of (5-6) is
$$ \begin{align*} &\left[ \frac{u^{n-1}}{1-u^{-2}} \left( \frac{1}{1-xu^{-1}} - 1 \right) + \frac{u^{r-n-2}}{1-u^{-2}} \frac{1}{1-x^{-1}u^{-1}} \right]_{u^0} \\&\quad \quad = \left[ u^{n-1}(1+u^{-2} + u^{-4} + \dotsb)(xu^{-1} + x^2 u^{-2} + \dotsb) \right. \\& \quad \quad \quad \quad \left. + u^{r-n-2} (1 + u^{-2} + u^{-4} + \dotsb) (1 + x^{-1}u^{-1} + x^{-2}u^{-2} + \dotsb) \right]_{u^0} \\ &\quad \quad = x^{n-1} + x^{n-3} + \dotsb + x^{n+2-r}, \end{align*} $$
as desired.
Corollary 5.3. Suppose
$f \in \Bbbk [u] \cap J_L$
,
$r \in \mathbb {N}$
,
$r \ge \deg f$
, and r is odd. Then,
For monic
$f \in \Bbbk [u]$
with
$f(0) \ne 0$
, define
Proposition 5.4. For monic
$f \in \Bbbk [u]$
with
$f(0) \ne 0$
, the following statements are equivalent:
-
(a) we have an equality of rational functions
$\mathbf {O}_f(u^{-1}) = \widetilde {\mathbf {O}}_f(u)$
; -
(b) we have
$z = t^{-1} f(0) - t f(0)^{-1}$
if
$\deg f$
is even and
$f(0) = \pm t$
is
$\deg f$
is odd; -
(c) we have
$\mathbf {O}_f(u) \widetilde {\mathbf {O}}_f(u) = 1$
.
Proof. Let
$d = \deg f$
. We first show that statements (a) and (b) are equivalent. First, suppose that d is even. Then, using (5-2),
$$ \begin{align*} \mathbf{O}_f(u) &= \frac{tf(0)^{-1}u^2 - z - tf(0)^{-1}}{u^2-zu-1} \frac{f(u^{-1})}{f(u)} u^d,\\ \widetilde{\mathbf{O}}_f(u) &= \frac{t^{-1}f(0)u^2 + z - t^{-1}f(0)}{u^2+zu-1} \frac{f(u)}{f(u^{-1})} u^{-d}, \end{align*} $$
and the result follows easily. Similarly, when d is odd,
$$ \begin{align*} \mathbf{O}_f(u) &= \frac{tf(0)^{-1}u^2 - zu - tf(0)^{-1}}{u^2-zu-1} \frac{f(u^{-1})}{f(u)} u^d,\\\widetilde{\mathbf{O}}_f(u) &= \frac{t^{-1}f(0)u^2 + z - t^{-1}f(0)}{u^2+zu-1} \frac{f(u)}{f(u^{-1})} u^{-d}, \end{align*} $$
and so
$$ \begin{align*} \mathbf{O}_f(u^{-1}) = \widetilde{\mathbf{O}}_f(u) \iff t f(0)^{-1} = t^{-1} f(0) \iff f(0) = \pm t, \end{align*} $$
as desired.
Now we show that statements (b) and (c) are equivalent. We have
$$ \begin{align*} \mathbf{O}_f(u)\widetilde{\mathbf{O}}_f(u) = \begin{cases} \displaystyle \frac{(u^2-1)^2 + z(u^2-1)(tf(0)^{-1}-t^{-1}f(0)) - z^2}{(u^2-1)^2-z^2u^2} & \text{if }d\text{ is even}, \\ \displaystyle \frac{(u^2-1)^2 + z(u^2-1)(tf(0)^{-1}-t^{-1}f(0))u - z^2u^2}{(u^2-1)^2-z^2u^2} & \text{if }d\text{ is odd}, \end{cases} \end{align*} $$
and the result follows from a straightforward computation.
Remark 5.5. Note that, for
$q \in \Bbbk ^\times $
, we have
$z = q - q^{-1}$
if and only if q and
$-q^{-1}$
are the two roots of the quadratic polynomial
$u^2-zu-1$
. If
$f \in \Bbbk [u]$
is even degree and satisfies the conditions of Proposition 5.4, then we must have
$z = q - q^{-1}$
for some
$q \in \Bbbk ^\times $
. In this case, by the uniqueness of roots of the quadratic polynomial, we must have
$f(0) = qt$
or
$f(0) = -q^{-1}t$
.
Recall that
$M = m_L(0)$
and that
$d_L = \deg m_L$
.
Theorem 5.6. We have
$$ \begin{align*} \mathbf{O}_L(u) = \mathbf{O}_{m_L}(u) \quad \text{and} \quad \widetilde{\mathbf{O}}_L(u) = \widetilde{\mathbf{O}}_{m_L}(u) = \mathbf{O}_{m_L}(u^{-1}). \end{align*} $$
If
$d_L$
is even, then
$z = t^{-1} M - t M^{-1}$
. If
$d_L$
is odd, then
$M = \pm t$
.
Proof. The statements
$\mathbf {O}_L = \mathbf {O}_{m_L}$
and
$\widetilde {\mathbf {O}}_L = \widetilde {\mathbf {O}}_{m_L}$
are equivalent by Lemma 3.5. The other assertions of the theorem follow from Proposition 5.4. Therefore, we need only prove that
$\mathbf {O}_L = \mathbf {O}_{m_L}$
.
Note that
where, in the last equality, we use the fact that
$\check {m}_L(u)$
is the minimal polynomial of the inverse dot.
First, consider the case where
$d_L$
is even. Then, we can apply Corollary 5.3 with
$f(u)=am_L(u)$
,
$r=d_L+1$
, and with
$f(u) = bu m_L(u)$
,
$r=d_L+1$
, to get
where
$g(u) = -[ {(zu^2+bu^2+au)m_L(u)}/({u^2-1}) ]_{u^{\ge 1}}$
. We take the strictly positive powers of u, instead of the nonnegative powers of u, because of the subtraction of the identity strand in the penultimate expression above. Note that
$\deg g \leq d_L$
and its constant term is zero. Since (5-9) implies that
$g \in J_L$
, it follows that
$u^{-1}g(u) \in \Bbbk [u] \cap J_L$
. This contradicts the definition of
$m_L$
unless
$g=0$
. Therefore,
Since
$d_L$
is even, we must have
$d_L \geq 2$
, and so the vanishing of the
$u^{d_L}$
and
$u^{d_L-1}$
terms of (5-10) requires that
$b=-z$
,
$a=0$
, since the entire series vanishes. This implies that
$\mathbf {O}_L = \mathbf {O}_{m_L}$
when
$d_L$
is even.
On the other hand, if
$d_L$
is odd, we can apply Corollary 5.3 with
$f(u)=bm_L(u)$
,
$r=d_L$
, and with
$f(u)= a u m_L(u)$
,
$r=d_L+2$
, giving us
where
$g(u) = -[ {(zu^2+au^2+bu)m_L(u)}/({u^2-1}) ]_{u^{\ge 1}} $
. An argument analogous to (5-10) shows that we must have
$a=-z$
and, if
$d_L>1$
, that
$b=0$
. This implies that
$\mathbf {O}_L = \mathbf {O}_{m_L}$
when
$d_L> 1$
and
$d_L$
is odd.
It remains to consider the case
$d_L = 1$
, when we have
$m_L(u) = u - M$
. First, note that
Thus,
$M = \pm t$
. We can then compute directly
which matches
$\mathbf {O}_{m_L}$
.
Corollary 5.7. We have
and
Fix q such that
$z=q-q^{-1}$
(extending
$\Bbbk $
if needed). Since
$z \neq 0$
, we have
$q\neq \pm 1$
. For a polynomial
$f \in \Bbbk [u]$
satisfying the conditions of Proposition 5.4, define
$$ \begin{align} (\epsilon_1(f),\epsilon_2(f)) = \begin{cases} (1,1) & \text{if } \deg f \text{ is even and } f(0) = qt, \\ (-1,-1) & \text{if } \deg f \text{ is even and } f(0) = -q^{-1}t, \\ (1,-1) & \text{if } \deg f \text{ is odd and } f(0) = t, \\ (-1,1) & \text{if } \deg f \text{ is odd and } f(0) = -t. \\ \end{cases} \end{align} $$
One of these cases must hold by Remark 5.5. Note that, in each of these cases,
$$ \begin{align} \deg f\equiv\frac{\epsilon_1(f)+\epsilon_2(f)}{2}+1\, (\mathrm{mod}\ 2) \quad f(0) = \epsilon_1(f) q^{{\epsilon_1(f)+\epsilon_2(f)}{2}}t. \end{align} $$
Since
$q \ne \pm 1$
, the pair
$(\epsilon _1(f),\epsilon _2(f))$
is uniquely determined by the second equation in (5-14). We can then rewrite (5-7) and (5-8) as
$$ \begin{align} \mathbf{O}_f(u) = t \frac{( u-q^{\epsilon_1(f)} ) ( u+q^{\epsilon_2(f)} )}{(u-q)(u+q^{-1})}\frac{\check{f}(u)}{f(u)}, \quad \widetilde{\mathbf{O}}_f(u) = t^{-1}\frac{( u-q^{-\epsilon_1(f)} ) ( u+q^{-\epsilon_2(f)} )}{(u+q)(u-q^{-1})}\frac{f(u)}{\check{f}(u)}. \end{align} $$
Note that the equality
$\mathbf {O}_{f}(u^{-1}) = \widetilde {\mathbf {O}}_{f}(u)$
means that when we expand
$\mathbf {O}_{f}(u^{-1})$
as a Taylor series at
$u=\infty $
, the constant term is t, whereas at
$u=0$
, it is
$t^{-1}$
.
5.2 Admissibility in the Kauffman case
In this section, we explain how admissibility conditions for cyclotomic BMW algebras can be deduced from our results. This analysis is very similar to that from Section 4.2, which considered the degenerate case. Therefore, we will be brief here. Although different notions of admissibility were originally introduced by Rui and Xu [Reference Rui and XuRX09] and by Wilcox and Yu [Reference Wilcox and YuWY11], these were shown to be equivalent in [Reference GoodmanGoo10, Theorem 4.4]. We therefore restrict our attention to the conditions from [Reference Rui and XuRX09], which are also used in [Reference Gao, Rui and SongGRS22]. Throughout, we use Remark 3.2 to move between our definition of the affine Kauffman category and that in [Reference Gao, Rui and SongGRS22].
Fix an 
-module category 
. As explained in Section 4.2,
$\mathcal {R}$
defines a ring homomorphism from 
to the centre
$Z_V$
of the endomorphism algebra of V. For V an object in an 
-module category
$\mathcal {R}$
, let
For a sequence
$\Omega = (\omega _r)_{r \in \mathbb {Z}}$
in
$\Bbbk $
with
$\omega _0 = (({t-t^{-1}})/{z})+1$
, define
$$ \begin{align*} \Omega^{\ge 0} = ( \omega_r )_{r \ge 0} \quad \text{and} \quad \Omega^{\le 0} = ( \omega_r )_{r \le 0}. \end{align*} $$
Then, inspired by (3-12), we define
Suppose
$m \in \Bbbk [u]$
is monic, has nonzero constant term and
$$ \begin{align*} m(u) = \prod_{a=1}^d (u-a_i) \end{align*} $$
in the algebraic closure of
$\Bbbk $
.
In [Reference Rui and XuRX09, Definition 2.19] and [Reference Gao, Rui and SongGRS22, Definition 1.7], the data
$(m,\Omega )$
are called admissible if they satisfy two conditions:
-
(a) The condition [Reference Gao, Rui and SongGRS22, Definition 1.7(1)] is equivalent to the statement that
(5-16)which holds for the sequence
$$ \begin{align} \mathbf{O}_\Omega \widetilde{\mathbf{O}}_\Omega = 1, \end{align} $$
$\Omega _V$
by (3-13).
-
(b) When
$\Omega = \Omega _V$
, the condition [Reference Gao, Rui and SongGRS22, Definition 1.7(2)] is equivalent to the statement This holds automatically if
The condition [Reference Gao, Rui and SongGRS22, Assumption 1.9] (see also [Reference Rui and XuRX09, Definition 2.20(2)]) on the sequence
$\mathbf {a} = (a_1,a_2,\dotsc ,a_{d})$
(denoted
$\mathbf {u}$
in [Reference Gao, Rui and SongGRS22]) is, in our language,
$$ \begin{align} m(0) = \prod_{i=1}^{d}(-a_i) = \begin{cases} \pm t & \text{if } \deg m \text{ is odd}, \\ qt \text{ or } -q^{-1}t & \text{if } \deg m \text{ is even}, \end{cases} \end{align} $$
where, in the second case,
$z = q - q^{-1}$
. By Proposition 5.4 and Remark 5.5, this must hold if condition (a) holds for
$\mathbf {O}_{m}$
,
$\widetilde {\mathbf {O}}_m$
. Assuming m satisfies (5-17), the notion of
$\mathbf {a}$
-admissibility for
$\Omega $
is defined in [Reference Gao, Rui and SongGRS22, Definition 1.10]; see also [Reference Rui and XuRX09, Lemma 2.28]. In our language, it follows as in the proof of Corollary 5.7 that
$$ \begin{align} \Omega \text{ is }\mathbf{a}\text{-admissible} \iff \mathbf{O}_\Omega = \mathbf{O}_m \text{ and } \widetilde{\mathbf{O}}_\Omega = \widetilde{\mathbf{O}}_m \text{ for } m(u) = \prod_{i=1}^{d}(u-a_i). \end{align} $$
Note that, if (5-16) and (5-17) are satisfied, then
$\mathbf {O}_\Omega = \mathbf {O}_m$
if and only if
$\widetilde {\mathbf {O}}_\Omega = \widetilde {\mathbf {O}}_m$
, by Proposition 5.4.
6 Cyclotomic Brauer categories
In this section, we consider the cyclotomic Brauer categories introduced in [Reference Rui and SongRS19]. These are a categorical analogue of the degenerate cyclotomic BMW algebras; as Theorem 6.3 below shows, the endomorphism algebras in these categories are degenerate cyclotomic BMW algebras, though the parameters that appear are slightly subtle. Throughout this section, we assume that
$\Bbbk $
is a field whose characteristic is not
$2$
.
6.1 Definition and first properties
Fix a monic polynomial
$p(u) \in \Bbbk [u]$
and a power series
Let
$\mathcal {I}(p,\mathbb {O})$
be the left tensor ideal of 
generated by
The corresponding cyclotomic Brauer category is the quotient
It follows from (2-13) that 
is the zero category unless
$$ \begin{align} \mathbb{O}(u) \mathbb{O}(-u) = 1. \end{align} $$
Therefore, we assume for the remainder of this section that (6-2) is satisfied.
Remark 6.1. The above definition coincides with the (specialized) cyclotomic Brauer category of [Reference Rui and SongRS19, Definition 1.7], except that some additional conditions are placed on the parameters there. Note also that [Reference Rui and SongRS19] works with right tensor ideals instead of left tensor ideals. This agrees with the fact that we consider the reverse of their affine Brauer category; see Remark 2.2. We omit the word ‘specialized’ in the terminology since we do not use the other cyclotomic Brauer category of [Reference Rui and SongRS19, Definition 1.5], where they take the quotient by 
.
Let
$L(p,\mathbb {O})$
denote the image of 
in the quotient 
. The category 
is no longer monoidal, but it is a left module category over 
and it is generated under this action by
$L(p,\mathbb {O})$
. By [Reference Rui and SongRS19, Theorem B], all the endomorphisms of
$L(p,\mathbb {O})$
are polynomials in the bubbles, which have all been specialized to scalars. This shows that the element
$L(p,\mathbb {O})$
is a brick if it is nonzero. Recall, from (4-1), that
$m_{L(p,\mathbb {O})}$
is the monic minimal polynomial of the action of the dot on
$1_{{\mathsf {B}} L(p,\mathbb {O})}$
. Recall also the definition (4-5) of
$\mathbb {O}_f$
.
Proposition 6.2 [Reference Rui and SongRS19, Theorem C].
Suppose
$f \in \Bbbk [u]$
is monic. The endomorphism ring
$\operatorname {\mathrm {End}}({\mathsf {B}} L(f,\mathbb {O}_f))$
in the category 
is 
.
In fact, [Reference Rui and SongRS19, Theorem C] describes a basis for all morphism spaces in 
, but for our purposes, we only need to know the result for endomorphisms of
${\mathsf {B}} L(f,\mathbb {O}_f)$
.
Proof. The category denoted 
in [Reference Rui and SongRS19] is denoted 
in our language, where
$\Omega = \omega = (\omega _k)_{k \in \mathbb {N}}$
is a sequence in
$\Bbbk $
and
$\mathbb {O}_\Omega $
is defined in (4-14). As we explain below in (4-15), if the d-tuple
$\mathbf {a}$
is the root of f with multiplicity, then
$\mathbb {O}_\Omega $
is
$\mathbf {a}$
-admissible, in the language of [Reference Rui and SongRS19], if
$\mathbb {O}_\Omega = \mathbb {O}_f$
. Thus, the result follows from [Reference Rui and SongRS19, Theorem C].
Theorem 6.3. The cyclotomic Brauer category 
is not the zero category if and only if
$\mathbb {O} = \mathbb {O}_f$
for some positive-degree polynomial f dividing p. If this category is nonzero, then
$\mathbb {O} = \mathbb {O}_{m_{L(p,\mathbb {O})}}$
, and
$m_{L(p,\mathbb {O})}$
is divisible by any polynomial f that divides p and satisfies
$\mathbb {O} = \mathbb {O}_f$
.
Proof. First, suppose that
$\mathbb {O} = \mathbb {O}_f$
for some positive-degree polynomial f dividing p. Then, we have the quotient functor 
. Proposition 6.2 implies that 
is not the zero category and hence 
is not the zero category. For the other direction, suppose that 
is not the zero category. Then
$m_{L(p,\mathbb {O})}$
has positive degree and Theorem 4.2 implies that
$\mathbb {O} = \mathbb {O}_{m_{L(p,\mathbb {O})}}$
.
Now, let f be a monic polynomial dividing p and satisfying
$\mathbb {O} = \mathbb {O}_f$
. Then, we have the quotient functor 
. It follows that
$m_{L(f,\mathbb {O})}$
divides
$m_{L(p,\mathbb {O})}$
. On the other hand, Proposition 6.2 implies that the endomorphisms 
,
$0 \le n \le \deg f - 1$
, are linearly independent in 
. It follows that
$f = m_{L(f,\mathbb {O})}$
and so f divides
$m_{L(p,\mathbb {O})}$
, as claimed.
Corollary 6.4. If 
is not the zero category, then it is isomorphic to
If L is a brick in an 
-module category
$\mathcal {R}$
, then it follows from Theorem 4.2 that the action of 
on L factors through 
. Thus, we have the following proposition.
Proposition 6.5. Suppose
$\mathbf {a} = (a_1,a_2,\dotsc ,a_{d})$
is a finite sequence in
$\Bbbk $
. The sequence
$\Omega $
is
$\mathbf {a}$
-admissible if and only if
$\Omega = \Omega _L$
for some brick L in an 
-module category
$\mathcal {R}$
satisfying
$m_L(u) = \prod _{i=1}^{d} (u-a_i)$
.
Proof. If L is a brick in an 
-module category
$\mathcal {R}$
with
$m_L(u) = \prod _{i=1}^{d} (u-a_i)$
, then
$\Omega _L$
is
$\mathbf {a}$
-admissible by Corollary 4.4.
Conversely, suppose that the sequence
$\Omega $
is
$\mathbf {a}$
-admissible. Define
$m(u) = {\prod _{i=1}^{d} (u-a_i)}$
and let L be the object of the cyclotomic Brauer category 
corresponding to the unit object 
of 
. Then, it follows from Proposition 6.2 that
$m = m_L$
. Thus,
$$ \begin{align*} \mathbb{O}_\Omega \overset{({\scriptstyle 4\text{-}15})}{=} \mathbb{O}_{m_L} \overset{\text{Th.~}4.2}{=} \mathbb{O}_L \ \implies \ \Omega = \Omega_L.\\[-34pt] \end{align*} $$
Corollary 6.6 [Reference Ariki, Mathas and RuiAMR06, Corollary 3.9].
If a sequence
$\Omega $
in
$\Bbbk $
is
$\mathbf {a}$
-admissible for some finite sequence
$\mathbf {a}$
in
$\Bbbk $
, then
$\Omega $
is weakly admissible (and hence also admissible).
Proof. Suppose
$\Omega $
is
$\mathbf {a}$
-admissible, with
$\mathbf {a} = (a_1,a_2,\dotsc ,a_{d})$
. Then, by Proposition 6.5,
$\Omega = \Omega _L$
for some object L in an 
-module category
$\mathcal {R}$
satisfying
$\operatorname {\mathrm {End}}_{\mathcal {R}}(L) = \Bbbk $
and
$m_L(u) = \prod _{i=1}^{d} (u-a_i)$
. It then follows from (4-12) that
$\Omega $
is weakly admissible.
6.2 Calculation of
$m_L$
It follows from Theorem 6.3 that, when 
is not the zero category, then
$m_{L(p,\mathbb {O})}$
is the unique monic polynomial of maximal degree in the set of polynomials f that divide p and satisfy
$\mathbb {O} = \mathbb {O}_f$
. Our next goal is to describe
$m_{L(p,\mathbb {O})}$
explicitly. To simplify notation, set
$$ \begin{align*} L = L(p,\mathbb{O}),\quad m = m_{L(p,\mathbb{O})}\quad d=\deg m. \end{align*} $$
Recall that
$I_L$
is the ideal of
$\Bbbk [u]$
generated by
$m_L$
. The following lemma will be useful.
Lemma 6.7. If elements
$x,y,z,w$
of a principal ideal domain R satisfy
$x/y=z/w$
in the fraction field of R, then
$x/y = a \gcd (x,z)/\gcd (y,w)$
for some unit
$a \in R$
.
Proof. Using
$\sim $
to indicate equality up to multiplication by a unit in R,
$$ \begin{align*} w \gcd(x,z) \sim \gcd(xw,zw) \sim \gcd(yz,zw) \sim z \gcd(y,w), \end{align*} $$
and the result follows.
Convention 6.8. For
$a_1,a_2,\dotsc ,a_n \in \Bbbk [u]$
, the notation
$\gcd (a_1,a_2,\dotsc ,a_n)$
denotes the unique monic gcd of the elements
$a_1,a_2,\dotsc ,a_n$
.
By Lemma 4.1,
$$ \begin{align} \hat{p}(u) := \big(\kern-3pt-u-\tfrac{1}{2}\big)p(-u)\mathbb{O}(-u) \in I_L. \end{align} $$
As in (4-6), by (6-2) and the definition of
$\hat {p}$
,
$$ \begin{align*} \frac{\hat{p}(-u)}{\big(u-\tfrac{1}{2}\big)p(u)} = \mathbb{O} = \frac{\big(\kern-3pt-u-\tfrac{1}{2}\big)p(-u)}{\hat{p}(u)}. \end{align*} $$
Since the leading term of
$\mathbb {O}$
is
$1$
, it follows from Lemma 6.7 that
$$ \begin{align} \mathbb{O}(u) = (-1)^{\deg Q}\frac{Q(-u)}{Q(u)}, \quad \text{where} \quad Q(u)=\gcd \big( \big(u-\tfrac{1}{2}\big)p(u),\hat{p}(u) \big) \in I_L. \end{align} $$
Lemma 6.9. If
$\deg Q$
is odd, then Q is divisible by u and
$({Q(u)}/{u}) \in I_L$
.
Proof. Suppose
$\deg Q$
is odd. Since
$Q \in I_L$
, the polynomial m divides Q. Let
$f=Q/m$
. Since
$$ \begin{align} \frac{( (-1)^{d}u-\tfrac{1}{2} ) m(-u)}{(u-\tfrac{1}{2})m(u)} = \mathbb{O} = -\frac{Q(-u)}{Q(u)} = -\frac{f(-u)m(-u)}{f(u)m(u)}, \end{align} $$
we have
$$ \begin{align*} \big( (-1)^{d}u-\tfrac{1}{2} \big) f(u) = - \big( u-\tfrac{1}{2} \big) f(-u). \end{align*} $$
Thus, if d is even, then
$f(u)$
is an odd function of u. Similarly, if d is odd, we have that
$(u+1/2)f(u)$
is an odd function of u. In both cases,
$u=0$
is a root of f and hence a root of Q. So, Q is divisible by u.
Next, setting
$\bar {Q}(u) = {Q(u)}/{u}$
,
Thus,
$-Q-\bar {Q} \in I_L$
and hence
$\bar {Q} \in I_L$
, completing the proof.
We are now ready to give the explicit expression for m. Recall the definition (6-3) of
$\hat {p}$
.
Theorem 6.10. We have
$$ \begin{align*} m_{L(p,\mathbb{O})}(u) = \begin{cases} \dfrac{\gcd(p(u),\hat{p}(u))}{u} & \text{if }\gcd((u-1/2)p(u),\hat{p}(u))\text{ has odd degree}, \\ \gcd(p(u),\hat{p}(u)) & \text{otherwise}. \end{cases} \end{align*} $$
Proof. It follows from the definition (6-4) of Q that, if Q divides p, then
$Q = \gcd (p,\hat {p})$
. On the other hand, if Q does not divide p, then
$u-\tfrac 12$
divides
$Q(u)$
and
$$ \begin{align*} \frac{Q(u)}{u-\frac{1}{2}} = \gcd(p(u),\hat{p}(u)). \end{align*} $$
Note that
$\gcd (p,\hat {p}) \in I_L$
by Lemma 4.1. We break the proof into four cases.
Case 1: Q has even degree and divides p. In this case, it follows from Theorem 6.3 and (6-4) that Q divides m. Since Q are both monic, we have
$m=Q$
, as desired.
Case 2: Q has odd degree and divides p. By Lemma 6.9,
$Q(u)$
is divisible by u, and so
$$ \begin{align*} \mathbb{O}(u) \overset{({\scriptstyle 6\text{-}4})}{=} -\dfrac{Q(-u)}{Q(u)} = \frac{\dfrac{Q(-u)}{-u}}{\dfrac{Q(u)}{u}}. \end{align*} $$
Then, it follows from Theorem 6.3 that
${Q(u)}/{u}$
divides
$m(u)$
. The polynomial
$m(u)$
also divides
${Q(u)}/{u}$
, by Lemma 6.9. Since both polynomials are monic, we have
$m(u)= {Q(u)}/{u}$
, as desired.
Case 3: Q has even degree and does not divide p. We have
$$ \begin{align*} \mathbb{O}(u) \overset{({\scriptstyle 6\text{-}4})}{=} \dfrac{Q(-u)}{Q(u)} = \dfrac{(-u-\frac{1}{2}) \dfrac{Q(-u)}{-u-\frac{1}{2}}}{(u-\frac{1}{2}) \dfrac{Q(u)}{u-\frac{1}{2}}}. \end{align*} $$
Thus, by Theorem 6.3,
${Q(u)}/{u-\tfrac 12} = \gcd (p(u),\hat {p}(u))$
divides
$m(u)$
. Since
${Q(u)}/{u-\tfrac 12}$
is monic, the result follows.
Case 4: Q has odd degree and does not divide p. We have
$$ \begin{align*} \mathbb{O}(u) \overset{({\scriptstyle 6\text{-}4})}{=} - \dfrac{Q(-u)}{Q(u)} = \dfrac{(-u-\frac{1}{2}) \dfrac{Q(-u)}{(-u)(-u-\frac{1}{2})}}{(u-\frac{1}{2}) \dfrac{Q(u)}{u(u-\frac{1}{2})}}, \end{align*} $$
and the result again follows.
Many of the relations (6-1) are redundant. We now give a more efficient presentation of cyclotomic Brauer categories. By Corollary 6.4, it suffices to consider the categories 
for
$m \in \Bbbk [u]$
of positive degree.
Proposition 6.11. For
$m \in \Bbbk [u]$
, the category 
is isomorphic to the quotient of 
by the left tensor ideal generated by
Proof. Let 
be the quotient of 
by the left tensor ideal generated by (6-6). It suffices to show that 
is a scalar multiple of 
in 
for all
$r \in \mathbb {N}$
. Since
it follows from (6-6) that 
is a scalar multiple of
in 
for
$r \in 2\mathbb {N}$
,
$0 \le r < \deg m$
. Furthermore, by Lemma 2.7, for
$r \in 2\mathbb {N}+1$
, the bubble 
can be expressed in terms of 
for
$s<r$
. Therefore, 
is a scalar multiple of 
in 
for all
$0 \le r < \deg m$
.
Now, since 
in 
, any bubble 
for
$r \ge \deg m$
can be written as a linear combination of 
for
$s < r$
. Therefore, 
is a scalar multiple of 
in 
for all
$r \in \mathbb {N}$
.
6.3 Cyclotomic BMW algebras
Since we have worked exclusively in the affine Brauer category, it might not be clear to the reader whether our arguments yield any results about the degenerate cyclotomic BMW algebras, which is the context of the earlier admissibility results we discussed. In the remainder of this section, we explain how our arguments do, in fact, yield such results.
For a polynomial
$p \in \Bbbk [u]$
that factors completely in
$\Bbbk $
(we can always pass to the algebraic closure of
$\Bbbk $
to ensure that this is the case),
$\Omega = (\omega _r)_{r \in \mathbb {N}}$
a sequence of elements of
$\Bbbk $
and
$n \in \mathbb {N}$
, we denote by
$W_n(p,\Omega )$
the corresponding n-strand degenerate cyclotomic BMW algebra. (We do not assume any admissibility conditions on these data.) The algebra
$W_n(p,\Omega )$
is denoted
$W_{d,n}(a_1,\dotsc ,a_{d})$
in [Reference GoodmanGoo11, Definition 2.2], where
$d = \deg p$
and
$a_1,\dotsc ,a_{d}$
are the roots of p, with multiplicity. This algebra is generated by elements
$e_i$
,
$s_i$
,
$x_j$
,
$1 \le i \le n-1$
,
$1 \le j \le n$
. It is useful to use string diagrams to represent elements in
$W_n(p,\Omega )$
in the usual way, numbering strands from right to left. We use thick (blue) strings when drawing diagrams representing elements of
$W_n(p,\Omega )$
in this way. Thus, we represent the generators of
$W_2(p,\Omega )$
as follows:
Define
$I(p,\Omega )$
to be the kernel of the map
$$ \begin{align*} \Bbbk[u] \to W_2(p,\Omega),\quad g(u) \mapsto g(x_1)e_1, \end{align*} $$
and let
$f_{p,\Omega }$
be the unique monic generator of this ideal. Equivalently,
$f_{p,\Omega }$
is the unique monic polynomial of minimal degree such that
$f_{p,\Omega }(x_1)e_1=0$
. Since
$p(x_1)e_1=0$
by the definition of
$W_2(p,\Omega )$
, we know that
$p \in I(p,\Omega )$
and so
$f_{p,\Omega }$
divides p.
In 
, the adjunction relations (the third and fourth equalities in (2-1)) imply, for
$n \in \mathbb {N}$
,
This shows that for any object L in an 
-module category and any polynomial f,
However, in the algebra
$W_2(p,\Omega )$
, the implication (6-8) may not hold, since the equality (6-7) does not make sense in this algebra.
The closest analogue to Theorem 4.2 and Corollary 4.4 that we have found in the literature is [Reference GoodmanGoo11, Theorem 5.2], more specifically, the implication
$(1) \Rightarrow (4)$
, the ‘only if’ direction of Proposition 6.12 below. (Note that the proof in the published version of [Reference GoodmanGoo11, Theorem 5.2] suggests that there was a misprint and the points labelled 1. and 2. should have been (3) and (4). We use the latter notation, which matches the proof. Note also that the implication
$(1) \Leftarrow (4)$
, the ‘if’ direction of Proposition 6.12, was already shown in [Reference Ariki, Mathas and RuiAMR06, Theorem 5.5].) In our language, the implication
$(1) \Leftrightarrow (4)$
of [Reference GoodmanGoo11, Theorem 5.2] is the following result.
Proposition 6.12 [Reference GoodmanGoo11, Theorem 5.2].
We have
$f_{p,\Omega }=p$
if and only if
$\mathbb {O}_\Omega = \mathbb {O}_p$
, where
$\mathbb {O}_p$
is defined in (4-5).
Let us first show how one can deduce Theorem 4.2 and Corollary 4.4 from Proposition 6.12. Suppose L is a brick in an 
-module category
$\mathcal {R}$
, and define
$\mathbb {O}_L$
and
$m_L$
as in Section 4. Let
$m_L(u) = (u-a_1)(u-a_2) \dotsm (u-a_{d_L})$
as a product of linear factors in the algebraic closure of
$\Bbbk $
and let
$\mathbf {a} = (a_1,a_2,\dotsc ,a_{d_L})$
. It is straightforward to verify that we have a homomorphism of associative algebras from
$W_2(m_L,\Omega _L)$
to
$\operatorname {\mathrm {End}}_{\mathcal {R}}({\mathsf {B}}^{\otimes 2} L)$
given by
By the definition of
$m_L(u)$
as the minimal polynomial of the action of
$x_1$
on L, the elements
are linearly independent. By (6-8), this implies that the elements
are also linearly independent. Since these are the images under the homomorphism (6-9) of
$x_1^n e_1 \in W_2(m_L,\Omega _L)$
,
$0 \le n < d_L$
, the latter elements are linearly independent as well. Therefore, by the ‘only if’ direction of Proposition 6.12,
$\Omega _L$
is
$\mathbf {a}$
-admissible, which is precisely the conclusion of Theorem 4.2 and Corollary 4.4.
To illustrate the opposite direction, we prove a stronger result than the ‘only if’ direction of Proposition 6.12, describing precisely the polynomial
$f_{p,\Omega }$
. In fact, we show in Proposition 6.15 that it is the same as the polynomial
$m_{p,\mathbb {O}_{\Omega }}$
calculated in Theorem 6.10. One of the key steps in establishing this relationship is the following result, which shows that the ideal
$I(p,\Omega )$
satisfies a similar closure relation to Lemma 4.1. Note that the proof is strongly analogous to Lemma 4.1, but to carry out this proof in the context of cyclotomic BMW algebras, we have to rotate all our diagrams so that the loop which before joined the bottom and top of the diagram now has both ends at the top, and we must add a cap at the bottom.
Lemma 6.13. Suppose
$e_1 \ne 0$
. If
$g \in I(p,\Omega )$
, then
$$ \begin{align*} \hat{g}(u) := \big( -u-\tfrac{1}{2} \big) g(-u) \mathbb{O}_\Omega(-u) \in I(p,\Omega). \end{align*} $$
Proof.
A priori,
$\hat {g}(u) \in \Bbbk (\! ( u^{-1} )\! )$
. However, by the argument of (4-3),
Since
$e_1 \neq 0$
, the series
$\hat {g}(u)$
must be a polynomial in
$\Bbbk [u]$
. Then, applying the argument of (4-4), we find that
and so
$\hat {g}- \tfrac 12{g}\in I(p,\Omega )$
. Therefore,
$\hat {g} =(\hat {g}- \tfrac 12{g})+ \tfrac 12{g}\in I(p,\Omega )$
.
We can now give a characterization of
$f_{p,\Omega }$
; compare with Theorem 6.3.
Theorem 6.14. We have that
$e_1 \neq 0$
in
$W_2(p,\Omega )$
if and only if
$\mathbb {O}_\Omega = \mathbb {O}_g$
for some positive-degree polynomial g dividing p. If this condition is satisfied, then
$\mathbb {O}_\Omega = \mathbb {O}_{f_{p,\Omega }}$
, and
$f_{p,\Omega }$
is divisible by any polynomial g that divides p and satisfies
$\mathbb {O}_\Omega = \mathbb {O}_g$
.
Proof. First, suppose
$e_1 \ne 0$
. We claim that
$\mathbb {O}_\Omega = \mathbb {O}_f$
for
$f = f_{p,\Omega }$
. Indeed, by Lemma 6.13, f divides
$\hat {f}$
. Then an argument analogous to that in the proof of Theorem 4.2 shows that
$\hat {f}(u) = ((-1)^{1+\deg f}u - \tfrac 12)f(u)$
and that
$\mathbb {O}_\Omega = \mathbb {O}_f$
.
It remains to show that f is divisible by any polynomial g that divides p and satisfies
$\mathbb {O}_\Omega = \mathbb {O}_g$
. Consider the cyclotomic Brauer category 
. We have a natural homomorphism 
. Since the image of
$f(x_1)e_1$
under this map is zero,
On the other hand, by Proposition 6.2, in 
, the morphism 
has minimal polynomial g. Thus, we must have that f is divisible by g, as desired.
We can now give an explicit description of
$f_{p,\Omega }$
.
Proposition 6.15. We have
$$ \begin{align*} f_{p,\Omega}(u)=m_{p,\mathbb{O}_{\Omega}}(u) = \begin{cases} \dfrac{\gcd(p(u),\hat{p}(u))}{u} & \text{if }\gcd((u-1/2)p(u),\hat{p}(u))\text{ has odd degree}, \\ \gcd(p(u),\hat{p}(u)) & \text{otherwise}. \end{cases} \end{align*} $$
Proof. Theorems 6.3 and 6.14 show that
$m_{p,\mathbb {O}_{\Omega }}$
and
$f_{p,\Omega }$
are uniquely characterized by the same property, proving the result.
The polynomial
$f_{p,\Omega }$
is also studied by Goodman [Reference GoodmanGoo12, Section 5]; in that paper, it is denoted
$p_0=(u-u_1)\cdots (u-u_d)$
. The fact that
$\mathbb {O}_\Omega = \mathbb {O}_{f_{p,\Omega }}$
when
$\deg f_{p,\Omega }> 0$
is shown in [Reference GoodmanGoo12, Lemma 5.5(2)]. In [Reference GoodmanGoo12], the condition that
$\deg f_{p,\Omega }>0$
is called semi-admissibility of the data
$(p,\Omega )$
. However, to the best of our knowledge, no explicit description of
$f_{p,\Omega }$
has appeared in the literature.
Another connection between the results of Section 6 and degenerate cyclotomic BMW algebras can be found by considering the natural map
Proposition 6.16. The map
$\alpha _p$
is surjective, with kernel generated, as a two-sided ideal, by
$m_{p,\Omega }(x_1)$
. In particular,
$\alpha _p$
is an isomorphism if and only if
$\mathbb {O}_p = \mathbb {O}_\Omega $
.
Proof. The map
$\alpha _p$
factors through the canonical map
$\pi _W \colon W_n(p,\Omega ) \to W_n(m,\Omega )$
for
$m=m_{p,\Omega }$
, since
by the definition of
$m_{p,\Omega }$
. In fact, we have the following commutative diagram:
By [Reference Rui and SongRS19, Theorem C], the map
$\alpha _m$
is an isomorphism, as is 
, by Corollary 6.4. Thus, the homomorphisms
$\alpha _p$
and
$\pi _W$
are intertwined by the induced isomorphism 
. Therefore, the result follows from the fact that
$\pi _W$
is surjective, with kernel generated by
$m_{p,\Omega }(x_1)$
.
We can understand the map
$\alpha _p$
better by considering its interaction with the two-sided ideal
$E_{p,\Omega }$
of
$W_n(p,\Omega )$
generated by
$e_1$
. Since
$s_is_{i+1}e_is_{i+1}s_i=e_{i+1}$
, the ideal
$E_{p,\Omega }$
is also generated by
$e_i$
for any i, or equivalently by the set of all
$e_i$
. In terms of diagrams,
$E_{p,\Omega }$
is the ideal spanned by all diagrams that factor through
${\mathsf {B}}^{\otimes r}$
for some
$r<n$
; this set is manifestly a two-sided ideal. By [Reference Ariki, Mathas and RuiAMR06, Proposition 7.2], applied with
$f=0$
, the quotient
$W_n(p,\Omega )/E_{p,\Omega }$
is isomorphic to
$H_n^p$
, the degenerate cyclotomic Hecke algebra for p. Thus, for the polynomials p and
$m=m_{p,\Omega }$
, we have short exact sequences compatible with projection:
By [Reference GoodmanGoo12, Proposition 5.11], the map
$\pi _E$
is an isomorphism, so the kernel of
$\pi _W$
projects isomorphically to the kernel of
$ \pi _H$
.
7 Cyclotomic Kauffman categories
In this section, we consider the cyclotomic Kauffman categories introduced in [Reference Gao, Rui and SongGRS22]. These are a categorical analogue of the (nondegenerate) cyclotomic BMW algebras, in the same way that the cyclotomic Brauer categories are a categorical analogue of the degenerate cyclotomic BMW algebras. Throughout this section, we assume that
$\Bbbk $
is a field. Our discussion in this section is parallel to that of Section 6.
Fix a monic polynomial
$p(u) \in \Bbbk [u]$
with nonzero constant term and a power series
In light of (3-13), define
Let
$\mathcal {J}(p,\mathbf {O})$
be the left tensor ideal of 
generated by
It then follows from (3-13) and (7-1) that
We define the corresponding cyclotomic Kauffman category
Remark 7.1. Recall, from Remark 3.2, the identification of 
with the reverse of the affine Kauffman category of [Reference Gao, Rui and SongGRS22]. Our use of the term cyclotomic Kauffman category is slightly more general than that of [Reference Gao, Rui and SongGRS22, Definition 1.8], since we do not impose an analogue of [Reference Gao, Rui and SongGRS22, Definition 1.7(2)] on our parameters. See Section 5.2 for further discussion of these conditions.
Let
$L(p,\mathbf {O})$
denote the image of 
in the quotient 
. The category 
is a left module category over 
and it is generated under this action by
$L(p,\mathbf {O})$
. The element
$L(p,\mathbf {O})$
is a brick if it is nonzero, since, by [Reference Gao, Rui and SongGRS22, Theorem 1.6], all its endomorphisms are polynomials in the bubbles, which have all been specialized to scalars. Recall, from Section 5, that
$J_{L(p,\mathbf {O})}$
is the kernel of the algebra homomorphism (5-1), and
$m_{L(p,\mathbf {O})}$
is the unique generator of
$J_{L(p,\mathbf {O}))}$
that lies in
$\Bbbk [u]$
, is monic and has nonzero constant term. Recall also the definition (5-7) of
$\mathbf {O}_f$
.
Proposition 7.2 [Reference Gao, Rui and SongGRS22, Theorem 1.12].
Suppose
$f \in \Bbbk [u]$
satisfies the conditions in Proposition 5.4. The endomorphism ring
$\operatorname {\mathrm {End}}({\mathsf {K}} L(f,\mathbf {O}_f))$
in the category 
is 
.
In fact, [Reference Gao, Rui and SongGRS22, Theorem 1.12] describes a basis for all morphism spaces in 
, but for our purposes, we only need to know the result for endomorphisms of
${\mathsf {K}} L(f,\mathbf {O}_f)$
.
Proof. Suppose
$f(u) = (u-a_1)(u-a_2) \dotsm (u-a_{d})$
as a product of linear factors in the algebraic closure of
$\Bbbk $
and let
$\mathbf {a} = (a_1,a_2,\dotsc ,a_{d})$
. The category denoted 
in [Reference Gao, Rui and SongGRS22] is denoted 
in our language, under the assumption that
$\omega $
is
$\mathbf {a}$
-admissible. As we explain in (5-18) below,
$\mathbf {a}$
-admissibility of
$\omega $
corresponds to the fact that we have chosen
$\mathbf {O}_f$
(as opposed to some general
$\mathbf {O}$
). The condition in Proposition 5.4(b) corresponds to [Reference Gao, Rui and SongGRS22, Assumption 1.9]. Thus, the result follows from [Reference Gao, Rui and SongGRS22, Theorem 1.12].
Theorem 7.3. The cyclotomic Kauffman category 
is not the zero category if and only if
$\mathbf {O} = \mathbf {O}_f$
for some positive-degree monic polynomial f dividing p and satisfying the conditions of Proposition 5.4. If such a polynomial exists, then
$\mathbf {O} = \mathbf {O}_{m_{L(p,\mathbf {O})}}$
, and
$m_{L(p,\mathbf {O})}$
is divisible by any polynomial f that divides p, satisfies
$\mathbf {O} = \mathbf {O}_f$
and satisfies the conditions of Proposition 5.4.
Proof. First, suppose that
$\mathbf {O} = \mathbf {O}_f$
for some positive-degree polynomial f dividing p and satisfying the conditions of Proposition 5.4. Then, we have the quotient functor 
. Proposition 7.2 implies that 
is not the zero category and hence 
is not the zero category. For the other direction, suppose that 
is not the zero category. Then,
$m_{L(p,\mathbf {O})}$
has positive degree, and Theorem 5.6 implies that
$\mathbf {O} = \mathbf {O}_{m_{L(p,\mathbb {O})}}$
and that
$m_{L(p,\mathbf {O})}$
satisfies the conditions in Proposition 5.4.
Now, let f be a monic polynomial dividing p, satisfying the conditions in Proposition 5.4 and satisfying
$\mathbf {O} = \mathbf {O}_f$
. Then, we have the quotient functor 
. It follows that
$m_{L(f,\mathbf {O})}$
divides
$m_{L(p,\mathbf {O})}$
. On the other hand, Proposition 7.2 implies that the endomorphisms 
,
$0 \le n \le \deg f - 1$
, are linearly independent in 
. It follows that
$f = m_{L(f,\mathbf {O})}$
and so f divides
$m_{L(p,\mathbf {O})}$
, as claimed.
Corollary 7.4. If 
is not the zero category, then it is isomorphic to
If L is a brick in an 
-module category
$\mathcal {R}$
, then it follows from Theorem 5.6 that the action of 
on L factors through 
.
We assume for the remainder of this section that
It follows from Theorem 7.3 that
$m_{L(p,\mathbf {O})}$
is the unique monic polynomial of maximal degree in the set of polynomials f that divide p and satisfy
$\mathbf {O} = \mathbf {O}_f$
. Our next goal is to describe
$m_{L(p,\mathbf {O})}$
explicitly. To simplify notation, set
$$ \begin{align*} L = L(p,\mathbf{O}),\quad m = m_{L(p,\mathbf{O})}. \end{align*} $$
It follows immediately from the definition of 
that
$\mathbf {O} = \mathbf {O}_L = \mathbf {O}_m$
.
For
$f,g \in \Bbbk [u]$
satisfying the conditions of Proposition 5.4, define
$$ \begin{align} H_{f,g} := \left( u - q^{({\epsilon_1(g)-\epsilon_1(f)})/{2}}\right)^{({|\epsilon_1(g)-\epsilon_1(f)|})/{2}} \left( u + q^{({\epsilon_2(g) - \epsilon_2(f)})/{2}}\right)^{({|\epsilon_2(g)-\epsilon_2(f)|})/{2}}, \end{align} $$
where
$\epsilon _1$
and
$\epsilon _2$
are as in (5-13). Note that
$$ \begin{align} \frac{\check{H}_{f,g}}{H_{f,g}} = \frac{( u-q^{\epsilon_1(f)} ) ( u + q^{\epsilon_2(f)} )}{( u - q^{\epsilon_1(g)} ) ( u + q^{\epsilon_2(g)} )}. \end{align} $$
Lemma 7.5. Suppose that
$f,g \in \Bbbk [u]$
satisfy the conditions of Proposition 5.4, and that g is divisible by f. Then,
$$ \begin{align*} \mathbf{O}_g = \mathbf{O}_f \iff g = f H_{f,g} \gamma \end{align*} $$
for some
$\gamma \in \Bbbk [u]$
of even degree satisfying
$\check {\gamma } = \gamma $
and
$\gamma (0)=1$
.
Proof. We have
$$ \begin{align} \mathbf{O}_g = \mathbf{O}_f \overset{({\scriptstyle 5\text{-}15})}{\iff} \frac{g(u)}{f(u)} \frac{( u-q^{\epsilon_1(f)} ) ( u + q^{\epsilon_2(f)} )}{( u - q^{\epsilon_1(g)} ) ( u + q^{\epsilon_2(g)} )} = \frac{\check{g}(u)}{\check{f}(u)} \overset{({\scriptstyle 7\text{-}3})}{\iff} \frac{g}{f H_{f,g}} = \frac{\check{g}}{\check{f} \check{H}_{f,g}}. \end{align} $$
Since
$$ \begin{align*} \frac{\check{g}}{\check{f}}(u) = \frac{g(u^{-1}) u^{\deg g}}{f(u^{-1}) u^{\deg f}} = \frac{g}{f}(u^{-1}) u^{\deg g - \deg f} \in \Bbbk[u], \end{align*} $$
it follows from (7-3) and (7-4) that, when
$\epsilon _1(f) \ne \epsilon _1(g)$
, the polynomial
$g/f$
is divisible by
$u-q^{\epsilon _1(g)} = u - q^{({\epsilon _1(g)-\epsilon _1(f)})/{2}}$
. Similarly, when
$\epsilon _2(f) \ne \epsilon _2(g)$
, the polynomial
$g/f$
is divisible by
$u + q^{\epsilon _2(g)} = u + q^{({\epsilon _2(g)-\epsilon _2(f)})/{2}}$
. Thus,
$\gamma := g/f H_{f,g}$
is a polynomial satisfying
$\check {\gamma } = \gamma $
and
$g = f H_{f,g} \gamma $
.
It follows from (5-14) and (7-2) that
$\deg \gamma \equiv 0\ (\mathrm{mod}\ 2)$
. Finally, we compute
Recall, from (5-4), the polynomial
$$ \begin{align} \hat{p}(u) = (1 - zu - u^2) p(u^{-1}) u^{\deg p} \mathbf{O}(u^{-1}) \in \Bbbk[u]. \end{align} $$
By Lemma 5.1, we have
$ \hat {p}\in J_L$
. Since we are seeking a generator of the ideal
$J_L$
, it is natural whenever we have a pair of elements of
$J_L$
to consider their greatest common denominator, which also lies in
$J_L$
. For certain technical reasons, we want to instead consider
$$ \begin{align*} R = \gcd \big( (u^2-zu-1)p(u),\hat{p}(u) \big)\in J_L. \end{align*} $$
By definition, the minimal polynomial m must divide R. It follows from (5-15) that the numerator and denominator of
$\mathbf {O} = \mathbf {O}_m$
vanish to the same order at
$1$
and at
$-1$
. Therefore,
$$ \begin{align} p, \hat{p} \text{ and }R\text{ vanish to the same order at 1 and vanish to the same order at }{-}1. \end{align} $$
Lemma 7.6. We have
$$ \begin{align} \mathbf{O} = t \frac{\check{R}}{R} \end{align} $$
and
$R(0) = \pm t$
.
Proof. Let
$d = \deg p$
and let
$\sim $
denote the relation on the ring of rational functions differing by a factor of
$\Bbbk ^\times $
. We have
$$ \begin{align*} &\frac{\check{\hat{p}}(u)}{(u^2-zu-1)p(u)} \overset{({\scriptstyle 5\text{-}2})}{\sim} \frac{\hat{p}(u^{-1}) u^{d+2}}{(u^2-zu-1)p(u)} \overset{({\scriptstyle 7\text{-}5})}{=} \mathbf{O}(u) = \mathbf{O}(u^{-1})^{-1} \\ &\qquad\overset{({\scriptstyle 7\text{-}5})}{=} \frac{(1-zu-u^2)p(u^{-1})u^d}{\hat{p}(u)} \overset{({\scriptstyle 5\text{-}2})}{\sim} \frac{(u^2+zu-1) \check{p}(u)}{\hat{p}(u)}. \end{align*} $$
By Lemma 6.7,
$$ \begin{align*} \mathbf{O}(u) \sim \frac{\gcd ( \check{\hat{p}}(u), (u^2+zu-1) \check{p}(u) )}{\gcd ( \hat{p}(u), (u^2-zu-1) p(u) )}. \end{align*} $$
For monic
$a,b,c \in \Bbbk [u]$
with nonzero constant terms, we see that if a divides b and c, then
$\check {a}$
divides
$\check {b}$
and
$\check {c}$
. It follows that the map
$a \mapsto \check {a}$
commutes with taking the
$\gcd $
. Therefore,
$$ \begin{align*} \gcd ( \check{\hat{p}}(u), (u^2+zu-1) \check{p}(u) ) = \check{R}(u). \end{align*} $$
Thus,
$\mathbf {O} \sim \check {R}/R$
. Comparing leading terms, we see that (7-7) holds.
For the final assertion, note that
$$ \begin{align*} \mathbf{O}(0) = \mathbf{O}_m(0) \overset{({\scriptstyle 5\text{-}15})}{=} t q^{\epsilon_1(m)+\epsilon_2(m)} \frac{\check{m}(0)}{m(0)} = \frac{t q^{\epsilon_1(m)+\epsilon_2(m)}}{m(0)^2} \overset{({\scriptstyle 5\text{-}14})}{=} t^{-1}. \end{align*} $$
Thus,
$$ \begin{align*} t^{-1} = t \frac{\check{R}(0)}{R(0)} = \frac{t}{R(0)^2} \implies R(0)^2 = t^2 \implies R(0) = \pm t. \end{align*} $$
Theorem 7.7. If 
is not the zero category, then
$$ \begin{align*} m_{L(p,\mathbf{O})} = \begin{cases} \gcd(p,\hat{p}) & \text{if }\deg R\text{ is odd and }R(0)=t, \\[3pt] \displaystyle\frac{\gcd(p,\hat{p})}{u^2-1} & \text{if }\deg R\text{ is odd and }R(0)=-t, \\[6pt] \displaystyle\frac{\gcd(p,\hat{p})}{u+1} & \text{if }\deg R\text{ is even and }R(0)=t, \\[6pt] \displaystyle\frac{\gcd(p,\hat{p})}{u-1} & \text{if }\deg R\text{ is even and }R(0)=-t. \end{cases} \end{align*} $$
Proof. Let
$$ \begin{align} R_1(u) = \begin{cases} R(u) & \text{if }\deg R\text{ is odd and }R(0)=t, \\[3pt] \displaystyle\frac{R(u)}{u^2-1} & \text{if }\deg R\text{ is odd and }R(0)=-t, \\[6pt] \displaystyle\frac{R(u)}{u+1} & \text{if }\deg R\text{ is even and }R(0)=t, \\[6pt] \displaystyle\frac{R(u)}{u-1} & \text{if }\deg R\text{ is even and }R(0)=-t. \end{cases} \end{align} $$
Since m divides p and
$\hat {p}$
, we have that m divides R and thus m also divides
$R_2 := R_1(u)(u^2-1)^2$
. Note that
$R_2$
is monic, has odd degree and
$R_2(0) = R_1(0) = t$
. Thus,
$R_2$
satisfies the conditions of Proposition 5.4 and, by (5-13), we have
$\epsilon _1(R_2)=1$
,
${\epsilon _2(R_2)=-1}$
. Since
$$ \begin{align*} \mathbf{O}_{R_2} = t \frac{\check{R}_2}{R_2} = t \frac{\check{R}}{R} \overset{({\scriptstyle 7\text{-}7})}{=} \mathbf{O} = \mathbf{O}_m, \end{align*} $$
Lemma 7.5 implies that
$R_2 = m H_{m,R_2} \gamma $
, where
$\gamma \in \Bbbk [u]$
has even degree,
$\check {\gamma } = \gamma $
and
$\gamma (0)=1$
.
It follows from (7-6) that
$R_2$
vanishes at strictly higher order at
$\pm 1$
than p and
$\hat {p}$
. Since m divides p and
$\hat {p}$
, the vanishing order of m at
$\pm 1$
is less than or equal to the vanishing order of p and
$\hat {p}$
. Since
$H_{m,R_2}$
does not vanish at
$\pm 1$
, it follows that
$\gamma $
vanishes at
$\pm 1$
, and thus is divisible by
$(u^2-1)^2$
, since
$\check {\gamma } = \gamma $
. This shows that
$R_1 = {R_2}/{(u^2-1)^2} = m H_{m,R_2} ({\gamma }/{(u^2-1)^2})$
is a polynomial divisible by m.
The polynomial
$R_1$
is monic, has odd degree and
$R_1(0) = t$
. Thus,
$R_1$
satisfies the conditions of Proposition 5.4 and, by (5-13), we have
$\epsilon _1(R)=1$
,
$\epsilon _2(R)=-1$
. Therefore,
$$ \begin{align*} \mathbf{O} \overset{({\scriptstyle 7\text{-}7})}{=} t \frac{\check{R}}{R} = t \frac{\check{R}_1}{R_1} \overset{({\scriptstyle 5\text{-}15})}{=} \mathbf{O}_{R_1}. \end{align*} $$
Consider
$R_3=\gcd (R_1,p)$
, which is divisible by m. Let
$a \in \{1,u^2-1,u+1,u-1\}$
be the denominator appearing in (7-8). Then,
$$ \begin{align*} R_1 = \gcd \left( (u^2-zu-1) \frac{p}{a}, \frac{\hat{p}}{a} \right) = \frac{1}{a} \gcd ( (u^2-zu-1) p, \hat{p} ). \end{align*} $$
Thus,
$$ \begin{align*} R_3 = \gcd \left(p, (u^2-zu-1) \frac{p}{a}, \frac{\hat{p}}{a} \right) = \frac{1}{a} \gcd ( p, \hat{p} ), \end{align*} $$
where the last equality follows from the fact that
$u^2-zu-1$
is not divisible by a. Therefore,
$$ \begin{align*} \frac{R_1}{R_3} = \frac{\gcd ( (u^2-zu-1) p, \check{p} )}{\gcd ( p, \check{p} )}, \end{align*} $$
which clearly divides
$u^2-zu-1$
. Equivalently,
$R_1/R_3 \in \{1, u-q, u+q^{-1}, {u^2-uz-1}\}$
. In fact, one can easily confirm that
$R_1/R_3=H_{R_3,R_1}$
. It thus follows immediately from Lemma 7.5 that
$\mathbf {O}_{R_3} = \mathbf {O}$
.
Since
$R_3$
divides p, Theorem 7.3 implies that m is divisible by
$R_3$
. As both polynomials are monic, it follows that
$m=R_3$
, completing the proof.
Corollary 7.8. The category 
is not the zero category if and only if all three of the following conditions are satisfied:
Proof. The ‘only if’ direction follows from the proof of Theorem 7.7. For the ‘if’ direction, we note that the given conditions imply that the conditions of Theorem 7.3 are satisfied with
$f=R_1$
.
Following the methods of Section 4.2, one can translate the results of the current section from statements about cyclotomic Kauffman categories to statements about cyclotomic BMW algebras. Since the treatment is parallel and we consider the category point of view to be more natural, we do not give the details of this translation here.
Acknowledgements
The authors are grateful for the support and hospitality of the Sydney Mathematical Research Institute (SMRI). They would also like to thank Jon Brundan, Mengmeng Gao, Hebing Rui and Linliang Song for helpful comments on an earlier draft of this paper.
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