A basis theorem for the affine Kauffman category and its cyclotomic quotients (Q2161170)

Let \(\mathcal{K}\) denote the Kauffman category of tangle diagrams over an integral domain \(k\) as introduced by \textit{V. Turaev} [Math. USSR, Izv. 35, No. 2, 411--444 (1990; Zbl 0707.57003)]. Turaev showed that \(\mathcal{K}\) is a strict \(k\)-linear monoidal category generated by a single object and four elementary morphisms (subject to a set of 9 relations) and identified a basis for homomorphism spaces in \(\mathcal{K}\). The goal of this work is to introduce and study an affine Kauffman category \(\mathcal{AK}\) along with certain cyclotomic quotients in a manner that parallels earlier work of the second and third authors [Math. Z. 293, Nos. 1--2, 503--550 (2019; Zbl 1461.18013)], where they constructed an affine Brauer category. The category \(\mathcal{AK}\) is again a strict \(k\)-linear monoidal category generated by a single object with two additional elementary morphisms (and three additional relations). \par With the objects in \(\mathcal{K}\) or \(\mathcal{AK}\) identified with the natural numbers \({\mathbb N}\), given \(m, s \in {\mathbb N}\), the first main result is an identification of \(\Hom_{\mathcal{AK}}(m,s)\). As for \(\mathcal{K}\), this Hom-space is zero if \(m + s\) is odd. When \(m + s\) is even, an infinite \(k\)-basis is given in terms of certain equivalence classes of normally ordered tangle diagrams. The authors then introduce a family of cyclotomic quotients of \(\mathcal{AK}\) determined by two functions (or, equivalently, collections of elements in \(k\)): \({\mathcal C}(\omega,{\mathbf u})\) for \(\omega : {\mathbb Z} \to k\) and \({\mathbf u} : \{1, 2, \dots, a\} \to k^{\times}\) (for some \(a\)). That is, \(\omega\) represents an infinite sequence of elements in \(k\), while \({\mathbf u}\) represents a finite set of units in \(k\). Further, one needs certain ``admissibility'' conditions on \(\omega\). When working with cyclotomic categories, the authors also add the assumption that \(k\) contains a unit \(q\) for which \(q - q^{-1}\) is also a unit. The second main result is an identification of a basis (this time finite) for the homomorphisms \(\Hom_{\mathcal{CK}}(\omega,\mathbf{u})(m,s)\). To obtain the main results, the authors first identify an appropriate spanning set in the \(\mathcal{AK}\)-case. The more challenging task is showing linear independence. The problem is first reduced to identifying a basis for \(\Hom_{\mathcal{AK}}(2m,0)\) (and similarly for the cyclotomic case). To deal with this case, the authors make use of symplectic and orthogonal quantum groups. More precisely, one considers a Lusztig quantum group \(\mathbf{U}_v(\mathfrak{g})\) over a an extension of the complex numbers for an indeterminate \(v\) and simple complex symplectic or special orthogonal Lie algebra \(\mathfrak{g}\). Much of the paper is devoted to computations in \({\mathbf U}_v(\mathfrak{g})\) and its action on certain modules and tensor products. For the cyclotomic case, one also needs to consider the BGG category \(\mathcal{O}\) inside the category of \(\mathbf{U}_v(\mathfrak{g})\)-modules. Ultimately, the authors construct monoidal functors from \(\mathcal{K}\) to \(\mathbf{U}_v(\mathfrak{g})\)-mod and then from \(\mathcal{AK}\) to \(\mathrm{End}(\mathbf{U}_v(\mathfrak{g})\text{-mod})\) (the category of endofunctors). This allows them to translate the problem to one involving an action of \(\mathcal{AK}\) on certain \(U_v(\mathfrak{g})\)-modules. \par As consequences of their main results, it is shown that endomorphism algebras for affine and cyclotomic Kauffman categories are isomorphic (as \(k\)-algebras) to affine and cyclotomic Birman-Murakami-Wenzl algebras, respectively. Such a connection was one of the motivations for this work, with results to be applied in more recent work of the authors [``Representations of weakly triangulated categories'', J. Algebra 614, 481--534 (2023)] in the realm of categorification.