We introduce a super analogue of \mathfrak{gl}_{2} -foams and use it to define an invariant of oriented tangles, shown to coincide with odd Khovanov homology when restricted to links. We then define a supercategorification of the q -Schur algebra of level 2 and realize our construction as a certain super-2-representation. This gives a representation-theoretic construction of odd Khovanov homology, where signs naturally arise as a byproduct of the super-2-categorical structure. In the process, we define a tensor product for chain complexes in super-2-categories, suitably compatible with homotopies. This could be of independent interest.
In this paper, we study the indecomposable module categories over \mathcal{C}(\mathfrak{sl}_{N}, k) , the category of integrable level- k representations of affine Kac–Moody \mathfrak{sl}_{N} . Our first main result classifies these module categories in the case of generic k ; i.e., k is sufficiently large relative to N . As \mathcal{C}(\mathfrak{sl}_{N}, k) is a braided tensor category, there is a relative tensor product structure on its category of module categories. In the generic setting, we obtain a formula for the relative tensor product rules between the indecomposable module categories. Our second main result classifies the indecomposable module categories over \mathcal{C}(\mathfrak{sl}_{N}, k) for N\leq 7 , with no restrictions on k . In this non-generic setting, exceptional module categories are obtained. This work relies heavily on previous results by the two authors. In previous literature, module category classification results were known only for \mathfrak{sl}_{2} and \mathfrak{sl}_{3} .
This paper proves that certain monotone Lagrangians in the standard symplectic vector space cannot be displaced by a Hamiltonian isotopy which commutes with the antipodal map. The method of proof is to develop a Borel equivariant version of the quantum cohomology of Biran and Cornea and prove that it is sensitive to equivariant displacements. The Floer–Euler class of Biran and Khanevsky appears as a term in the equivariant differential in certain cases.
We present a new solution to the formality problem for the framed Goldman-Turaev Lie bialgebra in genus zero, constructing Goldman-Turaev homomorphic expansions (formality isomorphisms) from the Kontsevich integral. Our proof uses a three-dimensional derivation of the Goldman-Turaev Lie bialgebra arising from a low-degree Vassiliev quotient - the emergent quotient - of tangles in a thickened punctured disk modulo a Conway skein relation. This is in contrast to Massuyeau's 2018 proof using braids. A feature of our approach is a general conceptual framework which is applied to prove the compatibility of the homomorphic expansion with both the Goldman bracket and the technically challenging Turaev cobracket.
For each braided category \mathcal{C} , we show that, under mild hypotheses, there is an associated category of “half braided algebras” and their bimodules internal to \mathcal{C} which is not only monoidal but even braided and balanced. We use this in the case where \mathcal{C} is the category of modules over a ribbon Hopf algebra to interpret stated skeins as a TQFT, namely, a braided balanced functor from a category of cobordisms to this category of algebras and their bimodules. Although our construction works in full generality, we relate in the special case of finite-dimensional ribbon factorizable Hopf algebras the stated skein functor to the Kerler–Lyubashenko TQFT by interpreting the former as the “endomorphisms” of the latter.
