Mathematics

The paper presents a detailed description of duality for braided algebras, coalgebras, bialgebras, Hopf algebras and their modules and comodules in the infinite setting. Assuming that the dual objects exist, it is shown how a given braiding induces compatible braidings for the dual objects, and how actions (resp. coactions) can be turned into coactions (resp. actions) of the dual coalgebra (resp. algebra), with an emphasis on braided bialgebras and their braided (co)module algebras.

We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type sl(1|n),osp(2|2n) . The type A case proves a conjecture of Feigin and Semikhatov. Let ( g 1 , g 2 )=( sl n+1 ,sl(1|n+1)) or ( so 2n+1 ,osp(2|2n)) and let r be the lacity of g 1 . Let k be a complex number and ℓ defined by r(k+ h ∨ 1 )(ℓ+ h ∨ 2 )=1 with h ∨ i the dual Coxeter numbers of the g i . Our first main result is that the Heisenberg cosets C k ( g 1 ) and C ℓ ( g 2 ) of these W-algebras at these dual levels are isomorphic, i.e. C k ( g 1 )≃ C ℓ ( g 2 ) for generic k. We determine the generic levels and furthermore establish analogous results for the cosets of the simple quotients of the W-algebras. Our second result is a novel Kazama-Suzuki type coset construction: We show that a diagonal Heisenberg coset of the subregular W-algebra at level k times the lattice vertex superalgebra V Z is the principal W-superalgebra at the dual level ℓ . Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level ℓ times the lattice vertex superalgebra V −1 √ Z is the subregular W-algebra at the dual level k. Again this is proven for the universal W-algebras as well as for the simple quotients. We show that a consequence of the Kazama-Suzuki type construction is that the simple principal W-superalgebra and its Heisenberg coset at level ℓ are rational and/or C_2-cofinite if the same is true for the simple subregular W-algebra at dual level ℓ . This gives many new C_2-cofiniteness and rationality results.

We introduce the dynamical quantum Pfaffian on the dynamical quantum general linear group and prove its fundamental transformation identity. Hyper quantum dynamical Pfaffian is also introduced and formulas connecting them are given.

We develop a notion of ellipsitomic associators by means of operad theory. We take this opportunity to review the operadic point-of-view on Drinfeld associators and to provide such an operadic approach for elliptic associators too. We then show that ellipsitomic associators do exist, using the monodromy of the universal ellipsitomic KZB connection, that we introduced in a previous work. We finally relate the KZB ellipsitomic associator to certain Eisenstein series associated with congruence subgroups of S L 2 (Z) , and to twisted elliptic multiple zeta values.

We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with U(1)×U(1) -equivariant cohomology of CP 1 . Motivated by the relationship between the Temperley-Lieb algebra and annular Khovanov homology, we also introduce an equivariant analogue of the Temperley-Lieb algebra.

We consider recognizable evaluations for a suitable category of oriented two-dimensional cobordisms with corners between finite unions of intervals. We call such cobordisms thin flat surfaces. An evaluation is given by a power series in two variables. Recognizable evaluations correspond to series that are ratios of a two-variable polynomial by the product of two one-variable polynomials, one for each variable. They are also in a bijection with isomorphism classes of commutative Frobenius algebras on two generators with a nondegenerate trace fixed. The latter algebras of dimension n correspond to points on the dual tautological bundle on the Hilbert scheme of n points on the affine plane, with a certain divisor removed from the bundle. A recognizable evaluation gives rise to a functor from the above cobordism category of thin flat surfaces to the category of finite-dimensional vector spaces. These functors may be non-monoidal in interesting cases. To a recognizable evaluation we also assign an analogue of the Deligne category and of its quotient by the ideal of negligible morphisms.

We determine some Nichols algebras that admit a non-trivial quadratic relation associated to some families of upper triangular solutions of the Yang-Baxter equation of dimension 3.

We present new examples of finite-dimensional Nichols algebra over fields of characteristic 2 starting from braided vector spaces that are not of diagonal type, admit realizations as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of odd characteristic with finite-dimensional Nichols algebras presented in arXiv:1905.03074. As these last ones, they are related to the Nichols algebras of finite Gelfand-Kirillov dimension in characteristic 0 described in arXiv:1606.02521. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of characteristic zero whose Nichols algebras have finite Gelfand-Kirillov dimension described in arXiv:1606.02521. We obtain nex examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.

We prove that, in types E (1) 6,7,8 , F (1) 4 and E (2) 6 , every Kirillov--Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion introduced by Kang this http URL. In order to apply the criterion, we need to prove some statements concerning values of a bilinear form. We achieve this by using the global bases of extremal weight modules.