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The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Schürmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behavior of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Schürmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of non-crossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Schürmann's aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively, Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra A N 2 n + 1 μ λ and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type A 1 , A 1 × A 1 , A 2 , Super type A 2 (q ; 2) and the Nichols algebra (8). And the involved finite-dimensional Nichols algebras of non-diagonal type are 1 2 , 4 m and m 2 -dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over  A 1 3 + λ . [ABSTRACT FROM AUTHOR]

A class of algebras D(m, d, ξ) introduced by [22] were not pointed and generated by the coradical of D(m, d, ξ). Let D be the quotient of D(m, d, ξ) module the principle ideal (gm - 1). First, we describe all simple left modules of D. Then, according to Radford's method, we construct the Yetter-Drinfeld module over D by the tensor product of a simple module of D and D itself. Hence, we find some simple left Yetter-Drinfeld modules over D, and the relevant braidings are of a triangular type. [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]

Let H be the Radford Hopf algebra of dimension twelve over a field k . In this paper, we investigate the automorphism groups of the Green ring r(H) and the Green algebra ℂ(H) ≔ ℂ ⊗ℤr(H) over the complex number field ℂ. The ring automorphisms of r(H) and the algebra automorphisms of ℂ(H) are all described. It is shown that the ring automorphism group of r(H) is isomorphic to the Klein group K4, and the algebra automorphism group of ℂ(H) is isomorphic to the direct product ℂx × S4, where ℂx is the multiplicative group of all nonzero complex numbers and S4 is the symmetric group of degree 4. [ABSTRACT FROM AUTHOR]

An extended derivation (endomorphism) of a (restricted) Lie algebra L is an assignment of a derivation (respectively) of L' for any (restricted) Lie morphism f:L\to L', functorial in f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L' to every f; and (b) if L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. [ABSTRACT FROM AUTHOR]

Let D (R m , n (q)) be the Drinfeld double of Radford Hopf algebra R m , n (q) and D (H s , t) be the Drinfeld double of generalized Taft algebra H s , t . Both D (R m , n (q)) and D (H s , t) have very symmetric structures. We calculate all Hopf automorphisms of D (R m , n (q)) and D (H s , t) , respectively. Furthermore, we prove that the Hopf automorphism group A u t H o p f (D (R m , n (q))) is isomorphic to the direct sum Z n ⨁ Z m of cyclic groups Z m and Z n , the Hopf automorphism group A u t H o p f (D (H s , t)) is isomorphic to the semi-direct products k * ⋊ Z d of multiplicative group k * and cyclic group Z d , where s = t d , k * = k \ { 0 } , and k is an algebraically closed field with char (k) ∤ t . [ABSTRACT FROM AUTHOR]

In this paper, we use elementary linear algebra methods to explore possible Hopf algebra structures within the generalized quaternion algebra. The sufficient and necessary conditions that make the generalized quaternion algebra a Hopf algebra are given. It is proven that not all of the generalized quaternion algebras have Hopf algebraic structures. When the generalized quaternion algebras have Hopf algebraic structures, we describe all the Hopf algebra structures. Finally, we shall prove that all the Hopf algebra structures on the generalized quaternion algebras are isomorphic to Sweedler Hopf algebra, which is consistent with the classification of 4-dimensional Hopf algebras. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the homomorphisms of modules over groups and Lie algebras as being morphisms in the category of (non-symmetric) Yetter–Drinfel'd modules. These module homomorphisms play a key role in the conjecture of Yau. [ABSTRACT FROM AUTHOR]

In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and the fact that there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions to the Yang-Baxter equation. Then, we introduce the notion of relative Rota-Baxter operator on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Finally, we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions to the Yang-Baxter equation in certain cocommutative Hopf algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce in a braided setting the notion of left module for an invertible 1-cocycle and we prove some categorical equivalences between categories of modules associated to an invertible 1-cocycle and categories of modules associated to Hopf braces. [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]

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]

From the connections between permutations and labeled simple graphs, we generalized the super-shuffle product and the cut-box coproduct on permutations to labeled simple graphs. We then proved that the vector space spanned by labeled simple graphs is a Hopf algebra with these two operations. [ABSTRACT FROM AUTHOR]

In this paper, we present the complete list of U q (s l 2) -symmetries on quantum polynomial algebra k q [ x ± 1 , y ] in the case that the action of the generator K of U q (s l 2) is a non-toric automorphism. The conditions for the isomorphism of such structures are explored as well. [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group G. In this paper, we investigate the McKay matrix W V of H for tensoring with the 2-dimensional indecomposable H -module V := M (2 , 0). It turns out that the characteristic polynomial, eigenvalues and eigenvectors of W V are related to the character table of the finite group G and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for W V by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of W V and give all eigenvectors of each eigenvalue for W V when G is a dihedral group of order 4 N + 2. [ABSTRACT FROM AUTHOR]

In this paper, we mainly give some equivalent characterisations of Hopf cobraces, show that the full subcategory HCB(A) of Hopf co-braces is equivalent to the full subcategory C (A) of bijective 1-cocycles, and prove that the full subcategory HCB(A) is also equivalent to the category M(A) of Hopf matched pairs. Moreover, we construct many Hopf co-braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel'd doubles of finite dimensional Hopf algebras. And we also give a sufficient and necessary condition for a given bicrossed coproduct A t H to be a Hopf co-brace if A or H is a Hopf co-brace. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Nichols algebra B (W X , r) associated to any non-degenerate involutive solution (X, r) of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension nm with m, n ∈ Z ≥ 2 . It turns out that the Nichols algebra B (W X , r) has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal's triangle. [ABSTRACT FROM AUTHOR]

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen–Fliess series and an additive static output feedback is applied. The first step is to consider the so-called Wiener-Fliess connection consisting of a Chen–Fliess series followed by a memoryless function. To explicitly compute the generating series, two Hopf algebras are needed: the existing output feedback Hopf algebra used to describe dynamic output feedback and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener–Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant. The convergence of the Wiener–Fliess composition product and the additive static feedback product are characterised. [ABSTRACT FROM AUTHOR]

We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact \Bbbk \text {-}linear monoidal functor can be lifted to a tensor equivalence under some certain conditions. [ABSTRACT FROM AUTHOR]

This paper is a first step in constructing the category of braided sets and its closest relative, the category of Yang–Baxter sets. Our main emphasis is on the construction of morphisms and extensions of Yang–Baxter sets. This problem is important for the possible constructions of new solutions of the Yang–Baxter equation and the braid equation. Our main result is the description of a family of solutions of the Yang–Baxter equation on and on , given two linear (set-theoretic) solutions and of the Yang–Baxter equation. [ABSTRACT FROM AUTHOR]

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category M , we introduce the notion of a Σ-twisted trace on the class Proj (M) of projective objects of M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on Proj (M) and the set of natural transformations from Σ to the Nakayama functor of M . Non-degeneracy and compatibility with the module structure (when M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular. [ABSTRACT FROM AUTHOR]

In this work, we introduce a nonstandard algebra of q-fermions where q is a nonzero complex deformation parameter for the algebra of the commuting fermions. In order to show that q-fermions provides a proper generalization of the algebra of usual commuting fermions, we prove that there is an inhomogeneous quantum structure associated with q-fermions for a complex number q with |q| = 1. [ABSTRACT FROM AUTHOR]

We define an algebra U 0 using a simplified set of generators for the quantum toroidal algebra U q (s l n + 1 , tor) and show that there exists an epimorphism from U 0 to U q (s l n + 1 , tor) . We derive a closed formula of the comultiplication on the generators of U 0 that extends that of the quantum affine algebra U q ( s l ̂ n + 1) . As a consequence, we show that U 0 is a Hopf algebra for n = 1, 2 and give conjectural formulas in the general case. We further show that U 0 is isomorphic to a double algebra. [ABSTRACT FROM AUTHOR]

This paper is an attempt to construct a special kind of Hopf pairing ⟨-,-⟩:H∙⊗H→k. Specifically, H∙ and H should be both affine, noetherian and of the same GK-dimension. In addition, some properties of them would be dual to each other. We test the ideas in two steps for all the affine prime regular Hopf algebras H of GK-dimension one: 1) We compute the finite duals H∘ of them, which are given by generators and relations; 2) the Hopf pairings desired are determined by choosing certain Hopf subalgebras H∙ of H∘, where ⟨-,-⟩ becomes the evaluation. [ABSTRACT FROM AUTHOR]

We establish a new and accessible notion of a Nichols system, thus providing a foundation to effectively work with them. We study reflections of Nichols systems without constructing necessary functors explicitly, but rather only using how these functors change gradings. We will obtain properties about the geometry of the support of Nichols systems, by looking at iterated reflections. [ABSTRACT FROM AUTHOR]

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of A, called Ω -algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study V-universal measuring coalgebras and V-universal comeasuring algebras between Ω -algebras A and B, relative to a fixed subspace V of V e c t (A , B). By considering the case A = B , we derive the notion of a V-universal (co)acting bialgebra (and Hopf algebra) for a given algebra A. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the V-universal acting bi/Hopf algebra and the finite dual of the V-universal coacting bi/Hopf algebra under certain conditions on V in terms of the finite topology on E n d F (A). [ABSTRACT FROM AUTHOR]

In this paper, for an arbitrary Kac-Moody Lie algebra {\mathfrak g} and a diagram automorphism \mu of {\mathfrak g} satisfying certain natural linking conditions, we introduce and study a \mu-twisted quantum affinization algebra {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) of {\mathfrak g}. When {\mathfrak g} is of finite type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When \mu =\mathrm {id} and {\mathfrak g} in affine type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Second, we give a simple characterization of the affine quantum Serre relations on restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules in terms of "normal order products". Third, we prove that the category of restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Last, we study the classical limit of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the \hbar-deformation of all nullity 2 extended affine Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we prove the fundamental theorem of Hopf modules associated with a Hopf brace. [ABSTRACT FROM AUTHOR]

Loday's notoriously elusive "coquecigrues" are meant to relate to Leibniz algebras in the same various ways that groups relate to Lie algebras. However, with the current approaches based on digroups, deadlock has been reached at the analogues of Lie's Third Theorem. Here, adjoint representations appear in the places where regular representations should be expected. The present work, intended as a stimulus to new approaches to the problem, proposes more symmetrical versions of the algebras involved. The fundamental guiding principle is to maintain both left and right actions on a completely equal footing. A coherent and cumulative series of Cayley theorems gives concrete representations of abstract split versions of semigroups, monoids, and groups, based upon the Galois theory of "symmetries of symmetries". Interpreted within monoidal categories, the new group-like objects we present provide a complete left/right split of Hopf algebra structure. The Cayley embedding appears intrinsically as the left/right symmetric part of the coassociativity diagram. [ABSTRACT FROM AUTHOR]

A partial Hopf–Galois theory is developed for partial H-module algebras, and we recover analogs of classical results for Hopf algebras. [ABSTRACT FROM AUTHOR]

Let H be a Hopf algebra of finite representation type. For the grouplike elements and skew-primitive elements in H, we prove some general results about the eigenvalues and eigenvectors of the McKay matrix in the framework of Green ring. As an example, we investigate the McKay matrix WV of Taft algebra H n (q) for tensoring with an indecomposable H n (q) -module V : = M (2 , 0). Furthermore, we compute the characteristic polynomial of WVin terms of a kind of generalized Fibonacci polynomial, and give some eigenvector(s) of each eigenvalue for WVby relating to the generalized Fibonacci polynomial. At last, we determine the eigenspaces of all eigenvalues. [ABSTRACT FROM AUTHOR]

We study the G -extensions C of a near-group fusion category of type (Z 2 , 1) . If C is braided we prove that C can be reconstructed from pointed fusion categories by Z 2 -extensions or Z 2 -equivariantizations. Furthermore, if C is also integral, or C is equivalent as a tensor category to the category of finite dimensional representations of a semisimple Hopf algebra, we prove that C is group-theoretical, which completes the classification of these categories in the sense of Morita equivalence. [ABSTRACT FROM AUTHOR]

With the main motivation to present some alternatives to the cosemisimplicity of twisted Smash coproduct, this paper first gives some basic constructions which make sense to consider the derived functors of some left exact functors from the category of comodules to vector spaces. Then, the Hochschild—Serre spectral sequence for twisted Smash coproduct coalgebras is obtained, which extends some results on homological properties given in the category of comodules over Smash coproduct in the literature. [ABSTRACT FROM AUTHOR]

Hopf-Galois objects are natural generalizations of Hopf algebras with a Galois-theoretic flavor. In this paper, we prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, introduce a concept of Poisson Hopf-Galois algebra and establish a relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. More importantly, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Noncommutative Geometry is the property of European Mathematical Society - EMS - Publishing House GmbH and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Motivated by q-shuffle products determined by Singer from qanalogues of multiple zeta values, we build in this article a generalisation of the shuffle and stuffle products in terms of weak shuffle and stuffle products. Then, we characterise weak shuffle products and give as examples the case of an alphabet of cardinality two or three. We focus on a comparison between algebraic structures respected in the classical case and in the weak case. As in the classical case, each weak shuffle product can be equipped with a dendriform structure. However, they have another behaviour towards the quadri-algebra and the Hopf algebra structure. We give some relations satisfied by weak stuffle products. [ABSTRACT FROM AUTHOR]

We apply quantum group methods to quantum computing, starting with the notion of interacting Frobenius Hopf algebras for ZX calculus with noncommutative algebra and noncocommutative coalgebra. We introduce the notion of *-structures in ZX calculus at this algebraic level and construct examples based on the quantum group u q (sl 2) at a root of unity. We provide an abstract formulation of the Hadamard gate related to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX calculus to take place in a braided tensor category. In the ribbon case, the Hadamard gate coming from braided self-duality obeys a modular identity. We give the example of b q (sl 2), the self-dual braided version of u q (sl 2). [ABSTRACT FROM AUTHOR]

By means of a new notion of subforests of an angularly decorated rooted forest, we give a combinatorial construction of a coproduct on the free Rota–Baxter algebra on angularly decorated rooted forests. We show that this coproduct equips the Rota–Baxter algebra with a bialgebra structure and further a Hopf algebra structure. [ABSTRACT FROM AUTHOR]

Copyright of Universitas Scientiarum is the property of Pontificia Universidad Javeriana and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We show that an L ∞ -algebra can be extended to a graded Hopf algebra with a codifferential. Then, we twist this extended L ∞ -algebra with a Drinfel'd twist, simultaneously twisting its modules. Taking the L ∞ -algebra as its own (Hopf) module, we obtain the recently proposed braided L ∞ -algebra. The Hopf algebra morphisms are identified with the strict L ∞ -morphisms, whereas the braided L ∞ -morphisms define a more general L ∞ -action of twisted L ∞ -algebras. [ABSTRACT FROM AUTHOR]

There are two notions of exponent of finite-dimensional Hopf algebras introduced and studied in the literature. In this note, we discuss and compare their properties including invariance and finiteness. Specifically, one notion is invariant under twisting and taking the Drinfeld double, exactly as the other one. We also find that if the non-cosemisimplicity and dual Chevalley property hold, both exponents are infinite in characteristic 0 but finite in positive characteristic. [ABSTRACT FROM AUTHOR]

For a finite-index II 1 subfactor N ⊂ M , we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on M in a trace-preserving fashion and fixing N pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse. [ABSTRACT FROM AUTHOR]

A class of finite-dimensional Hopf algebras which generalize the notion of Taft algebras is studied. We give necessary and sufficient conditions for these Hopf algebras to omit a pair in involution. That is, to not have a group-like and a character implementing the square of the antipode. As a consequence, we prove the existence of an infinite set of examples of finite-dimensional Hopf algebras without such pairs. Implications for the theory of anti-Yetter–Drinfeld modules as well as biduality of representations of Hopf algebras are discussed. [ABSTRACT FROM AUTHOR]

In this study we introduce q-deformed split quaternions, that is, this deformation reduces to classical split quaternions as q-1 where q is a real parameter. It is also shown that there is a quantum group associated with q-deformed split quaternions, which is isomorphic to SUq (1,1). [ABSTRACT FROM AUTHOR]

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functorZWB ! yA, where B is the category of bottom tangles in handlebodies and yA is the degree-completion of the category A of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, Ais the linear PROP governing "Casimir Hopf algebras", which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2-tensor. The functor Z induces a canonical isomorphism grB Š A, where grB is the associated graded of the Vassiliev-Goussarov filtration on B. To each Drinfeld associator ' we associate a ribbon quasi-Hopf algebra H' in yA, and we prove that the braided Hopf algebra resulting from H' by "transmutation" is precisely the image by Z of a canonical Hopf algebra in the braided category B. Finally, we explain how Z refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant. [ABSTRACT FROM AUTHOR]

This work presents a classification of Uq(sl2)-symmetries on the quantum disc. The principal invariant of such classification, the grading jump, is introduced. It turns out that, under the present subjects, the grading jump can take only 3 values: 0, 1, -1. The subcollection of the complete collection of symmetries is extracted in such a way that the selected symmetries satisfy certain compatibility condition for involutions. [ABSTRACT FROM AUTHOR]

We give a series of infinite dimensional noncommutative and noncocommutative pointed Hopf algebras, which are Artin-Schelter Gorenstein Hopf algebras with injective dimensions 3. Radford's Hopf algebra and Gelaki's Hopf algebra are homomorphic images of these Hopf algebras. We determine the irreducible representations of them. We describe the Grothendieck rings of them. We obtain that two non-isomorphic Hopf algebras could have isomorphic Grothendieck rings. [ABSTRACT FROM AUTHOR]

Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity. [ABSTRACT FROM AUTHOR]