Mathematics

We describe L ∞ -algebras governing homotopy relative Rota-Baxter Lie algebras and triangular L ∞ -bialgebras, and establish a map between them. Our formulas are based on a functorial approach to Voronov's higher derived brackets construction which is of independent interest.

We construct new examples of left bialgebroids and Hopf algebroids, arising from noncommutative geometry. Given a first order differential calculus Ω on an algebra A , with the space of left vector fields X , we construct a left A -bialgeroid BX , whose category of left modules is isomorphic to the category of left bimodule connections over the calculus. When Ω is a pivotal bimodule, we construct a Hopf algebroid HX over A , by restricting to a subcategory of bimodule connections which intertwine with both Ω and X in a compatible manner. Assuming the space of 2-forms Ω 2 is pivotal as well, we construct the corresponding Hopf algebroid DX for flat bimodule connections, and recover Lie-Rinehart Hopf algebroids as a quotient of our construction in the commutative case. We use these constructions to provide explicit examples of Hopf algebroids over noncommutative bases.

Our first collection of results parametrize (filtered) actions of a quantum Borel U q (b)⊂ U q ( sl 2 ) on the path algebra of an arbitrary (finite) quiver. When q is a root of unity, we give necessary and sufficient conditions for these actions to factor through corresponding finite-dimensional quotients, generalized Taft algebras T(r,n) and small quantum groups U q ( sl 2 ) . In the second part of the paper, we shift to the language of tensor categories. Here we consider a quiver path algebra equipped with an action of a Hopf algebra H to be a tensor algebra in the tensor category of representations H . Such a tensor algebra is generated by an algebra and bimodule in this tensor category. Our second collection of results describe the corresponding bimodule categories via an equivalence with categories of representations of certain explicitly described quivers with relations.

A family of algebra maps H→ A i whose common domain is a Hopf algebra is said to be jointly inner faithful if it does not factor simultaneously through a proper Hopf quotient of H . We show that tensor and free products of jointly inner faithful maps of Hopf algebras are again jointly inner faithful, generalizing a number of results in the literature on torus generation of compact quantum groups.

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In addition, we show that this construction is unique up to an invertible scalar. Due to this fact, Hopf-Frobenius algebras provide two canonical notions of duality, and give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. We use this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double, but has a much simpler presentation.

As is known to all, Hopf-Galois objects have a significant research value for analyzing tensor categories of comodules and classification questions of pointed Hopf algebras, and are natural generalizations of Hopf algebras with a Galois-theoretic flavour. In this paper, we mainly prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, and introduce the conception of Poisson Hopf-Galois algebras, and establish the relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. Moreover, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and mainly give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.

Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity (x∗y)∗(x∗z)=(y∗y)∗(y∗z) . Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups (X,∗) are obtained by twisting the left division operation in groups (that is, they are of the form x∗y=ψ( x −1 y) for a group (X,⋅) and its automorphism ψ ), and they correspond to idempotent latin solutions. We solve the isomorphism problem for idempotent latin solutions.

Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern-Simons theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in Chern-Simons theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.

Employing the algebraic structure of the left brace and the dynamical extensions of cycle sets, we investigate a class of indecomposable involutive set-theoretic solutions of the Yang-Baxter equation having specific imprimitivity blocks. Moreover, we study one-generator left braces of multipermutation level 2.

We prove a general theorem for constructing integral quantum cluster algebras over Z[ q ±1/2 ] , namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[ q ±1/2 ] . Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra g and Weyl group element w , the dual canonical form A q ( n + (w) ) Z[ q ±1 ] of the corresponding quantum unipotent cell has the property that A q ( n + (w) ) Z[ q ±1 ] ⊗ Z[ q ±1 ] Z[ q ±1/2 ] is isomorphic to a quantum cluster algebra over Z[ q ±1/2 ] and to the corresponding upper quantum cluster algebra over Z[ q ±1/2 ] .