iquantum Brauer Category Overview

• iquantum Brauer category is defined as a Brauer-type diagrammatic framework attached to quantum symmetric pairs for orthosymplectic Lie superalgebras, featuring generators like cups, caps, and thick crossings.
• It functions as a strict right module category over the framed HOMFLYPT skein category, establishing an equivalence with the disoriented skein category for coherent tensor-module representations.
• Its structure is governed by explicit skein, curl, bubble, and braid relations, with a basis theorem linking it to classical Brauer algebras upon specialization at q=1.

Searching arXiv for papers directly related to “iquantum Brauer category,” plus closely related Brauer-type, $q$-Brauer, and $\imath$-quantum categorification work. The iquantum Brauer category is a Brauer-type diagram category attached to quantum symmetric pairs for the orthosymplectic Lie superalgebras inside the general linear Lie superalgebras. In the form developed in "The disoriented skein and iquantum Brauer categories" (Salmasian et al., 16 Jul 2025), it is the $\Bbbk$-linear category $B(q,t)$ with objects $B_r$ for $r\in\mathbb N$, generated by cups, caps, and thick positive and negative crossings, subject to skein, curl, bubble, braid, and commutation relations. It is not presented as a monoidal category; rather, its natural structure is that of a strict right module category over the framed HOMFLYPT skein category. The category is equivalent, as such a module category, to the disoriented skein category, and it admits full incarnation functors to tensor-module categories for the relevant $\imath$-quantum enveloping superalgebras. In this sense it is an interpolating diagram category for the representation theory of the quantum symmetric pair $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$ (Salmasian et al., 16 Jul 2025).

1. Terminology and mathematical position

The terminology is specific. In this context, “iquantum” refers to the $\imath$-quantum or quantum-symmetric-pair setting, not merely to an arbitrary $q$-deformation of Brauer diagrams. The category $\imath$0 is therefore distinct from several other “quantum Brauer” objects in the literature. In the broader taxonomy of Brauer-type categories, the BWM-category is the deformation of the ordinary Brauer category, and the periplectic $\imath$1-Brauer category is the deformation of the periplectic Brauer category (Barbier, 2024). By contrast, the iquantum Brauer category of (Salmasian et al., 16 Jul 2025) is organized around a coideal subalgebra $\imath$2 for a quantum symmetric pair, and its diagrammatics are built to model restriction from $\imath$3 to the corresponding orthosymplectic $\imath$4-side.

This specificity matters because the phrase “Brauer category” already has a classical meaning. In the classical category-theoretic formulation, the Brauer category has objects $\imath$5, generating morphisms $\imath$6, $\imath$7, $\imath$8, and $\imath$9, and a complete presentation by seven relations; its endomorphism algebras are the classical Brauer algebras (Lehrer et al., 2012). The iquantum Brauer category is a deformation away from that symmetric-monoidal world, but not by the BMW route alone. It is instead adapted to a skein-theoretic and coideal-algebraic setting in which the two oriented tensor generators $\Bbbk$0 and $\Bbbk$1 become isomorphic after restriction to the $\Bbbk$2-quantum side (Salmasian et al., 16 Jul 2025).

2. Definition of $\Bbbk$3

The ground data are a commutative ring $\Bbbk$4 and invertible parameters $\Bbbk$5 such that $\Bbbk$6 is divisible by $\Bbbk$7. The scalar

$\Bbbk$8

therefore lies in $\Bbbk$9. The paper also assumes a ring automorphism $B(q,t)$0 with

$B(q,t)$1

used in antilinear symmetries (Salmasian et al., 16 Jul 2025).

The objects are

$B(q,t)$2

The generating morphisms are a cup

$B(q,t)$3

a cap

$B(q,t)$4

and positive and negative thick crossings on adjacent thick strands. A fundamental convention is the thick-strand notation: horizontal juxtaposition of thick identity strands is encoded by addition of labels, so a block of adjacent thick identity strands is written as a single thick strand labeled by the sum of the widths (Salmasian et al., 16 Jul 2025).

The defining relations include braid and inverse relations for the thick crossings, together with the skein relation

$B(q,t)$5

There are also curl and bubble relations. A closed bubble evaluates to

$B(q,t)$6

times the relevant thick identity strand, while the left and right curls evaluate to $B(q,t)$7 and $B(q,t)$8, respectively. Crossed cup-cap reductions occur with coefficients $B(q,t)$9 and $B_r$0, and the category further satisfies the “humps” relations and a commutation relation expressing that generating morphisms commute past crossings in the specified diagrammatic sense (Salmasian et al., 16 Jul 2025).

Structurally, $B_r$1 is $B_r$2-linear but not monoidal. Horizontal concatenation is not freely available as a categorical tensor product; it is only used in the restricted thick-strand conventions built into the presentation. This is one of its defining differences from both the ordinary Brauer category and the framed HOMFLYPT skein category (Salmasian et al., 16 Jul 2025).

The category carries two notable antilinear symmetries. Horizontal reflection defines an antilinear isomorphism

$B_r$3

sending cups to caps and positive crossings to negative crossings. There is also a bar involution

$B_r$4

fixing cups and caps and interchanging positive and negative crossings (Salmasian et al., 16 Jul 2025).

3. Module-category structure and equivalence with the disoriented skein category

The ambient monoidal category is the framed HOMFLYPT skein category $B_r$5, a strict monoidal $B_r$6-linear category generated by two oriented objects $B_r$7 and $B_r$8, together with oriented crossings, cups, and caps subject to oriented HOMFLYPT skein relations. The iquantum Brauer category is not monoidal on its own, but it becomes a strict right module category over $B_r$9 (Salmasian et al., 16 Jul 2025).

This module structure is encoded by a strict monoidal functor

$r\in\mathbb N$0

On objects, both $r\in\mathbb N$1 and $r\in\mathbb N$2 act by adding one strand: $r\in\mathbb N$3 On morphisms, the action is given by explicit natural transformations obtained by adjoining crossings or the appropriate cup/cap diagrams on the right. This reflects the representation-theoretic fact that the natural and dual quantum $r\in\mathbb N$4-modules become isomorphic after restriction to the $r\in\mathbb N$5-side (Salmasian et al., 16 Jul 2025).

The comparison object is the disoriented skein category $r\in\mathbb N$6, defined as a right $r\in\mathbb N$7-module category generated by two mutually inverse toggles

$r\in\mathbb N$8

subject to inverse, curl, and twisted reflection-type relations. These toggles encode the passage between the two orientations in the $r\in\mathbb N$9-setting (Salmasian et al., 16 Jul 2025).

A central theorem establishes a strict equivalence of right $\imath$0-module categories

$\imath$1

There is also a $\imath$2-linear quasi-inverse

$\imath$3

and an explicit natural isomorphism $\imath$4. The equivalence identifies the iquantum Brauer category with a more flexible skein-theoretic model. The paper stresses that $\imath$5 has advantages: cups and caps may occur in arbitrary positions, it has duality structure, and the incarnation functors become strict morphisms of module categories there, whereas the corresponding functor from $\imath$6 is only module-functorial up to natural isomorphism (Salmasian et al., 16 Jul 2025).

4. Representation-theoretic incarnation

The representation-theoretic background is the quantum symmetric pair attached to

$\imath$7

Let $\imath$8, and let $\imath$9 be the corresponding coideal subalgebra; the paper also uses a slightly enlarged algebra $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$0 to obtain fullness statements (Salmasian et al., 16 Jul 2025).

On the full quantum-group side, there are the natural and dual modules $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$1 and $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$2. After restriction to the $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$3-side, they become isomorphic: $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$4 This is the algebraic source of the toggle morphisms in $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$5 and, through the equivalence $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$6, of the single-object-per-degree structure of the iquantum Brauer category (Salmasian et al., 16 Jul 2025).

Three incarnation functors organize the picture. First, there is a full monoidal functor

$(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$7

sending $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$8 to $(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$9, $\imath$0 to $\imath$1, and the oriented skein generators to the corresponding braidings, evaluations, and coevaluations. Second, there is a strict module functor

$\imath$2

sending the toggle generators to $\imath$3 and $\imath$4. Third, the iquantum Brauer incarnation is defined by

$\imath$5

On objects,

$\imath$6

On generators, thick crossings act by the braiding $\imath$7 on adjacent $\imath$8-factors, while cups and caps are realized by composites involving $\imath$9, $q$0, and $q$1 (Salmasian et al., 16 Jul 2025).

The paper treats $q$2, and hence $q$3, as an interpolating category for these tensor-module categories. Its abstract formulation is therefore not merely combinatorial: it is designed to encode the tensor calculus seen by the coideal algebra $q$4 (Salmasian et al., 16 Jul 2025).

5. Bases, diagrammatics, and classical specialization

A major structural result is the explicit basis theorem. For the disoriented skein category, one defines reduced $q$5-diagrams by fixing a matching of boundary points and imposing normal-form conditions: no closed loops, at most one critical point per string, no self-intersections, no pair of strings crossing more than once, and controlled placement of toggles. Choosing one reduced diagram per matching gives a set $q$6, and the theorem states that

$q$7

is a free $q$8-module with basis $q$9. Transporting this basis along the equivalence yields a basis $\imath$00 of

$\imath$01

for every $\imath$02 (Salmasian et al., 16 Jul 2025).

This basis theorem identifies the iquantum Brauer category as a genuine Brauer-type diagram category: its morphism spaces are controlled by pairing combinatorics, but the local calculus is $\imath$03-deformed and adapted to the $\imath$04-setting. The proof combines diagram straightening with representation-theoretic separation arguments under specialization (Salmasian et al., 16 Jul 2025).

The same paper proves the classical limit. After base change to $\imath$05 and specialization $\imath$06, the category $\imath$07 becomes the classical Brauer category $\imath$08 (Salmasian et al., 16 Jul 2025). This connects the iquantum category directly to the standard classical Brauer formalism, in which morphisms are generated by identity, crossing, cup, and cap diagrams and the endomorphism algebras are Brauer algebras (Lehrer et al., 2012).

6. Related constructions and broader context

Several nearby Brauer-type categories clarify what the iquantum Brauer category is, and what it is not. The marked Brauer category is a super/graded generalization of the ordinary Brauer category adapted to homogeneous bilinear forms on $\imath$09-graded vector spaces; it is explicitly not a quantum deformation, although the paper points to a marked analogue of the BMW algebra as a natural future direction (Kujawa et al., 2014). By contrast, the classification of diagram categories of Brauer type shows that the BWM-category is the unique deformation of the ordinary Brauer category in that framework, while the periplectic $\imath$10-Brauer category deforms the periplectic Brauer category (Barbier, 2024). The iquantum Brauer category of (Salmasian et al., 16 Jul 2025) belongs to neither family in a direct sense; its natural home is the theory of quantum symmetric pairs.

On the categorification side, the nil-Brauer category supplies a rank-one Brauer-type model for $\imath$11-quantum groups. It is a strict graded monoidal category with one generating object and four generating morphisms—dot, crossing, cup, and cap—and its split Grothendieck ring is isomorphic to an integral form of the split $\imath$12-quantum group of rank one (Brundan et al., 2023). The companion paper proves that indecomposable graded projective modules correspond to the $\imath$13-canonical basis and that standard modules categorify a new PBW basis (Brundan et al., 2023). In the higher-categorical direction, the 2-categories $\imath$14 introduced to categorify quasi-split $\imath$15-quantum groups contain the nil-Brauer category in rank one and are described as widely expected to be related to affine Brauer categories and affine Brauer algebras in quasi-split AIII-type situations (Brundan et al., 28 May 2025).

Taken together, these results place the iquantum Brauer category at the intersection of skein theory, Brauer-type diagrammatics, and the representation theory of coideal algebras. Its distinctive feature is that it is not merely a $\imath$16-deformed Brauer category with braid-like crossings; it is a module-category model for the $\imath$17-side of quantum symmetric pairs, equivalent to a disoriented skein category and specialized, at $\imath$18, to the ordinary Brauer category (Salmasian et al., 16 Jul 2025).