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Conformal blocks of q, t-deformed Virasoro and W -algebras are important special functions in representation theory with applications in geometry and physics. In the Nekrasov–Shatashvili limit t → 1 , whenever one of the representations is degenerate then conformal block satisfies a difference equation with respect to the coordinate associated with that degenerate representation. This is a stationary Schrodinger equation for an appropriate relativistic quantum integrable system. It is expected that generalization to generic t ≠ 1 is a non-stationary Schrodinger equation where t parametrizes shift in time. In this paper we make the non-stationary equation explicit for the q, t-Virasoro block with one degenerate and four generic Verma modules and prove it when three modules out of five are degenerate, using occasional relation to Macdonald polynomials. [ABSTRACT FROM AUTHOR]

We extend Willerton’s [24] graphical calculus for bimonads to comodule monads, a monadic interpretation of module categories over a monoidal category. As an application, we prove a version of Tannaka–Krein duality for these structures. [ABSTRACT FROM AUTHOR]

In this paper, all string modules of one class of basic Hopf algebras of tame type, which are denoted by H 2 are studied. Firstly, the isomorphism classes of all string modules of H 2 are classified. Then the projective class ring of H 2 is described. Finally, the McKay matrix and the corresponding McKay quiver of H 2 are discussed. [ABSTRACT FROM AUTHOR]

We describe the structure of the quotient G / H of a formal supergroup G by its formal sub-supergroup H . This is a consequence which arises as a continuation of the authors' work (partly with M. Hoshi) on algebraic/analytic supergoups. The results are presented and proved in terms of super-cocommutative Hopf superalgebras. The notion of co-free super-coalgebras plays a role, in particular. [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional Hopf algebra over an algebraically closed field k with the dual Chevalley property. We prove that H is of finite corepresentation type if and only if it is coNakayama, if and only if the link quiver Q (H) of H is a disjoint union of basic cycles, if and only if the link-indecomposable component H (1) containing k 1 is a pointed Hopf algebra and the link quiver of H (1) is a basic cycle. [ABSTRACT FROM AUTHOR]

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects. [ABSTRACT FROM AUTHOR]

AbstractThe present article is devoted to introduce the notion of Hopf bracoid in the braided monoidal framework as the quantum version of skew bracoids, which have been presented by Martin-Lyons and Truman. Taking into account that Hopf braces are particular examples of Hopf bracoids, in this paper we generalize many properties of Hopf braces to the Hopf bracoid setting, we study the relationships between Hopf bracoids and skew bracoids and, finally, we obtain a categorical isomorphism between certain full subcategories of the Hopf bracoids category and the 1-cocycles category. [ABSTRACT FROM AUTHOR]

In this paper, we give the notion of (lax or weak) Hom-corings, which leads to the introduction of partial coaction of a Hom-Hopf algebra on a Hom-algebra. We construct several pairs of adjoint functors, and define Galois Hom-corings, as a special case of which Galois Hom-coalgebras are introduced. [ABSTRACT FROM AUTHOR]

Let K be the smallest dimensional non-pointed basic Hopf algebra. We determine all finite-dimensional Nichols algebras in and compute the liftings. The result shows that there are some new Nichols algebras of non-diagonal type and new finite-dimensional Hopf algebras without the dual Chevalley property. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of curved O -operator systems as a generalization of T. Brzeziński's (curved) Rota-Baxter systems, and then investigate their relations with O -operator systems, (tri)dendriform systems, pre-Lie algebras, associative Yang-Baxter pairs and quasitriangular covariant bialgebras. [ABSTRACT FROM AUTHOR]

In this paper, we mainly research Rota-Baxter co-operators on commutative Hopf algebras. We get a new Hopf algebra and some Hopf co-braces via Rota-Baxter co-operators on commutative Hopf algebras. Finally, we generalize Rota-Baxter co-operators to relative Rota-Baxter co-operators of Hopf algebras, construct a Hopf cobrace by a relative Rota-Baxter co-operator of commutative Hopf algebra, and establish the relation of relative Rota-Baxter co-operators and bijective 1-cocycles. [ABSTRACT FROM AUTHOR]

Given a Hopf algebra H and a counital 2-cocycle μ on H, Drinfeld introduced a notion of twist which deforms an H-module algebra A into a new algebra A μ . We show that when A is a quadratic algebra, and H acts on A by degree-preserving endomorphisms, then the twist A μ is also quadratic. Furthermore, if A is a Koszul algebra, then A μ is a Koszul algebra. As an application, we prove that the twist of the q-quantum plane by the quasitriangular structure of the quantum enveloping algebra U q (s l 2) is a quadratic algebra equal to the q − 1 -quantum plane. [ABSTRACT FROM AUTHOR]

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216. [ABSTRACT FROM AUTHOR]

Thanks to the topological Hopf algebra of renormalization of Green’s functions in a gauge field theory, we associate a bi-Heyting algebra to each combinatorial Dyson–Schwinger equation. This setting leads us to characterize subsystems generated by the solution spaces of quantum motions. In addition, we apply the c 2 -invariant of Feynman diagrams to build a new Heyting algebra of multiplicative groups which encodes a more general class of subsystems of the physical theory. [ABSTRACT FROM AUTHOR]

We associate to a projective n-dimensional toric variety X Δ a pair of co-commutative (but generally non-commutative) Hopf algebras H X α , H X T . These arise as Hall algebras of certain categories Coh α (X) , Coh T (X) of coherent sheaves on X Δ viewed as a monoid scheme—i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When X Δ is smooth, the category Coh T (X) has an explicit combinatorial description as sheaves whose restriction to each A n corresponding to a maximal cone σ ∈ Δ is determined by an n-dimensional generalized skew shape. The (non-additive) categories Coh α (X) , Coh T (X) are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff–Kapranov. The Hall algebras H X α , H X T are graded and connected, and so enveloping algebras H X α ≃ U (n X α) , H X T ≃ U (n X T) , where the Lie algebras n X α , n X T are spanned by the indecomposable coherent sheaves in their respective categories. We explicitly work out several examples, and in some cases are able to relate n X T to known Lie algebras. In particular, when X = P 1 , n X T is isomorphic to a non-standard Borel in gl 2 [ t , t - 1 ] . When X is the second infinitesimal neighborhood of the origin inside A 2 , n X T is isomorphic to a subalgebra of gl 2 [ t ] . We also consider the case X = P 2 , where we give a basis for n X T by describing all indecomposable sheaves in Coh T (X) . [ABSTRACT FROM AUTHOR]

As a double generalization of biFrobenius algebras and weak Hopf algebra, we introduce and study weak biFrobenius algebras (or briefly wbF algebras). A wbF algebra is a Frobenius algebra and a Frobenius coalgebra and satisfies some compatibility conditions. We discuss more properties of wbF algebras keeping as close as possible to the classical theory of weak Hopf algebras, and give conditions for finite dimensional algebras and coalgebras to be wbF algebras. Then we studies substructures in wbF algebras. Finally, we prove that our wbF algebras can give rise to Frobenius double algebras in sense of Szlachányi. [ABSTRACT FROM AUTHOR]

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg–Tran–Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane. [ABSTRACT FROM AUTHOR]

We propose a quantization algebra of the Loday-Ronco Hopf algebra k [ Y ∞ ] , based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra k [ Y ∞ ] h is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion A TopRec h is a subalgebra of a quotient algebra A Reg h obtained from k [ Y ∞ ] h that nevertheless doesn't inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of A TopRec h in low degree. [ABSTRACT FROM AUTHOR]

The notions of a post-group and a pre-group are introduced as a unification and enrichment of several group structures appearing in diverse areas from numerical integration to the Yang–Baxter equation. First the Butcher group from numerical integration on Euclidean spaces and the -group of an operad naturally admit a pre-group structure. Next a relative Rota–Baxter operator on a group naturally splits the group structure to a post-group structure. Conversely, a post-group gives rise to a relative Rota–Baxter operator on the sub-adjacent group. Further a post-group gives a braided group and a solution of the Yang–Baxter equation. Indeed the category of post-groups is isomorphic to the category of braided groups and the category of skew-left braces. Moreover a post-Lie group differentiates to a post-Lie algebra structure on the vector space of left invariant vector fields, showing that post-Lie groups are the integral objects of post-Lie algebras. Finally, post-Hopf algebras and post-Lie Magnus expansions are utilized to study the formal integration of post-Lie algebras. As a byproduct, a post-group structure is explicitly determined on the Lie–Butcher group from numerical integration on manifolds. [ABSTRACT FROM AUTHOR]

We study Yetter-Drinfeld modules over Nichols systems to generalize results about Verma modules in the represation theory of Lie algebras to Nichols algebras of group type. We construct and study reflections of such objects and obtain geometric properties of their support. In particular we study a subcategory obtained by inducing comodules of Nichols systems. Finally we study the maximal subobject and irreducibility of such Yetter-Drinfeld modules. To do so we introduce a special morphism that we name Shapovalov morphism, to generalize a polynomial called Shapovalov determinant. We will study its properties and behavior under reflections. [ABSTRACT FROM AUTHOR]

For a split quasireductive supergroup G defined over a field, we study structure and representation of Frobenius kernels G r of G and we give a necessary and sufficient condition for G r to be unimodular in terms of the root system of G . We also establish Steinberg's tensor product theorem for G under some natural assumptions. [ABSTRACT FROM AUTHOR]

We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension k ≤ k ′ is a right adjoint on the category of k -coalgebras). [ABSTRACT FROM AUTHOR]

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over D ∞ , the infinite dihedral group.We find all the irreducible Yetter-Drinfeld modules V over D ∞ , and determine which Nichols algebras B (V) of V are finite GK-dimensional. [ABSTRACT FROM AUTHOR]

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid M ⋊ H , we construct a locally convex bialgebroid Dirac (M ⋊ H) with an antipode over C c ∞ (M) , from which the groupoid M ⋊ H can be reconstructed as its spectral action Lie groupoid A G sp (Dirac (M ⋊ H)) . [ABSTRACT FROM AUTHOR]

We construct a class of quantum doubles D (H D n ) of pointed Hopf algebras of rank one H D . We describe the algebra structures of D (H D n ) by generators with relations. Moreover, we give the comultiplication ΔD, counit εD and the antipode SD, respectively. [ABSTRACT FROM AUTHOR]

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques. [ABSTRACT FROM AUTHOR]