The disoriented skein and iquantum Brauer categories

We develop a diagrammatic approach to the representation theory of the quantum symmetric pairs corresponding to orthosymplectic Lie superalgebras inside general linear Lie superalgebras. Our approach is based on the disoriented skein category, which we define as a module category over the framed HOMFLYPT skein category. The disoriented skein category admits full incarnation functors to the categories of modules over the iquantum enveloping algebras corresponding to the quantum symmetric pairs, and it can be viewed as an interpolating category for these categories of modules. We define an equivalence of module categories between the disoriented skein category and the iquantum Brauer category (also known as the $q$-Brauer category), after endowing the latter with the structure of a module category over the framed HOMFLYPT skein category. The disoriented skein category has some advantages over the iquantum Brauer category, possessing duality structure and allowing the incarnation functors to be strict morphisms of module categories. Finally, we construct explicit bases for the morphism spaces of the disoriented skein and iquantum Brauer categories.

💡 Research Summary

The paper develops a diagrammatic framework for the representation theory of quantum symmetric pairs associated with the orthosymplectic Lie superalgebras osp(m|2n) inside the general linear Lie superalgebras gl(m|2n). The starting point is the framed HOMFLYPT skein category OS(q,t) (also called the oriented skein category), which is a strict monoidal, pivotal, k‑linear category generated by two objects ↑ and ↓ corresponding to the natural module V⁺ and its dual V⁻. OS encodes the relations of the HOMFLYPT link invariant and admits a full monoidal functor R_OS to the category of U_q(gl(m|2n))‑modules.

Because the iquantum enveloping algebra U^ι (the right coideal subalgebra that quantizes the symmetric pair (gl, osp)) does not inherit a Hopf structure, its module category U^ι‑mod is not monoidal but only a right module category over U_q(gl)‑mod. Consequently, a diagrammatic description of U^ι‑mod should be a right module category over OS(q,t). The authors introduce the disoriented skein category DS(q,t) as precisely such a module category. DS is obtained from OS by adjoining two inverse isomorphisms (the “toggles”) •◦: ↓→↑ and ◦•: ↑→↓, subject to relations (2.17)–(2.18) that encode caps, cups, and a q‑scaling. These toggles reflect the fact that V⁺ and V⁻ become isomorphic after restriction to U^ι, and they give DS a strict OS‑module structure: the action of OS on DS respects the tensor product on the nose.

Independently, the literature contains a q‑Brauer category B(q,t) (which the authors rename the iquantum Brauer category). B is built from a tower of Iwahori–Hecke algebras and also carries a right OS‑module structure, but its caps are forced to appear only on the left side of diagrams, and the associated functor R_B: B→U^ι‑tmod is only a non‑strict module morphism.

The central result (Theorem 4.5) establishes an equivalence of right OS‑module categories between DS(q,t) and B(q,t). Two explicit functors F: DS→B and G: B→DS are constructed; they send the toggles to left‑side caps/cups and vice‑versa. Both functors are mutually inverse up to natural isomorphism and compatible with the respective incarnation functors to U^ι‑tmod. Consequently, the diagram (1.1) in the introduction commutes strictly for DS (top square) and only up to natural isomorphism for B (bottom square). This demonstrates that DS enjoys a cleaner module‑theoretic behavior while still encoding the same representation‑theoretic information as B.

A further major contribution is the basis theorem (Theorem 7.1). Using the flexibility of DS—where caps and cups may appear anywhere—the authors construct explicit bases for all morphism spaces of DS in terms of strings of toggles, crossings, and oriented cups/caps. By transporting this basis via the equivalence, they obtain a corresponding basis for B (Corollary 7.2). These bases give concrete presentations of the endomorphism algebras End_{U^ι}((V⁺)^{⊗r}) and thus provide a diagrammatic handle on the centralizer algebras that appear in Schur–Weyl duality for the iquantum setting.

The authors view both DS and B as interpolating categories in the spirit of Deligne’s categories: they sit between the monoidal category OS(q,t) and the module categories U‑tmod and U^ι‑tmod, allowing one to pass from the full quantum group to its coideal subalgebra while retaining a uniform diagrammatic calculus.

Finally, the paper outlines several promising directions: (i) describing the kernels of R_DS and R_B to obtain full presentations of U^ι‑tmod; (ii) extending the construction to other quantum symmetric pairs where a diagrammatic calculus for the ambient quantum group is known; (iii) developing a “quantum spin Brauer” analogue that captures spin representations; (iv) constructing affine and cyclotomic versions of DS and B; and (v) relating DS to web categories introduced in recent works, potentially via idempotent completions at quantum (anti)symmetrizers.

In summary, the work introduces a new diagrammatic category DS(q,t) that provides a strict, duality‑rich, and computationally convenient model for the representation theory of iquantum symmetric pairs, proves its equivalence with the previously known iquantum Brauer category, and supplies explicit bases for morphism spaces, thereby opening the way for deeper algebraic and categorical investigations of quantum symmetric pairs.