Mathematics

Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the A n groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric R -matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for A 3 and A 4 in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for A n via sequences of cluster mutations in the special A n -quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.

The paper explores the relation between noncommutative power series and topological theories of one-dimensional cobordisms decorated by labelled zero-dimensional submanifolds. These topological theories give rise to several types of tensor envelopes of noncommutative recognizable power series, including the categories built from the syntactic algebra and syntactic ideals of the series and the analogue of the Deligne category.

We construct a Kitaev model, consisting of a Hamiltonian which is the sum of commuting local projectors, for surfaces with boundaries and defects of dimension 0 and 1. More specifically, we show that one can consider cell decompositions of surfaces whose 2-cells are labeled by semisimple Hopf algebras and 1-cells are labeled by semisimple bicomodule algebras. We introduce an algebra whose representations label the 0-cells and which reduces to the Drinfeld double of a Hopf algebra in the absence of defects. In this way we generalize the algebraic structure underlying the standard Kitaev model without defects or boundaries, where all 1-cells and 2-cells are labeled by a single Hopf algebra and where point defects are labeled by representations of its Drinfeld double. In the standard case, commuting local projectors are constructed using the Haar integral for semisimple Hopf algebras. A central insight we gain in this paper is that in the presence of defects and boundaries, the suitable generalization of the Haar integral is given by the unique symmetric separability idempotent for a semisimple (bi-)comodule algebra.

The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra.

We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we formulate a generalization of the Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing Matsumoto's formula for Jack functions.

We determine the \emph{ L ∞ -algebra} that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the {\em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a \emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to \emph{pre-Lie ∞ -algebras}.

The deformed W algebras of type A have a uniform description in terms of the quantum toroidal gl 1 algebra E . We introduce a comodule algebra K over E which gives a uniform construction of basic deformed W currents and screening operators in types B,C,D including twisted and supersymmetric cases. We show that a completion of algebra K contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except D (2) ℓ+1 . We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.

Let A=kQ/I be the path algebra of any finite quiver Q modulo any two-sided ideal I of relations. We develop a method to give a concrete and complete description of the deformation theory of A via the combinatorics of reduction systems. We introduce a new natural notion of equivalence for reduction systems and obtain a one-to-one correspondence between formal deformations of A and formal deformations of any reduction system for A , up to equivalence. Moreover, we give criteria for the existence of algebraizations of formal deformations and give a wide range of applications in algebra and geometry.

This is a continuation of a previous study initiated by one of us on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right H -comodule nonlocal vertex algebra for a nonlocal vertex bialgebra H and give a construction of deformations of vertex algebras with a right H -comodule nonlocal vertex algebra structure and a compatible H -module nonlocal vertex algebra structure. We also give a construction of ? -coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.

The q -difference equation, the shift and the contiguity relations of the Askey-Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras ska 3 and ska 4 . It is shown that the q -para Racah polynomials corresponding to a non-conventional truncation of the Askey-Wilson polynomials form a basis for a finite-dimensional representation of ska 4 . The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses ska 4 . The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun-Askey-Wilson operator.