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This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of Mq(n) and the observation that K [Mq(n)] is a quadratic algebra, and can be equipped with a multiplier Hopf ∗-algebra structure in the sense of quantum permutation groups developed byWang and an observation by Rollier–Vaes. In our next paper, we will propose the study of multiplier Hopf graph algebras. The current paper can be viewed as a precursor to this upcoming work, serving as a crucial intermediary bridging the gap between the abstract concept of multiplier Hopf algebras and the well-developed field of graph theory, thereby establishing connections between them! This survey review paper is dedicated to the 78th birthday anniversary of Professor Alfons Van Daele. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product \text {Rep}(H)\rtimes \text {Aut}(H). These are quantum invariants of knots endowed with a homomorphism of the knot group to \text {Aut}(H). We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the SL(n,\mathbb {C})-twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant. [ABSTRACT FROM AUTHOR]

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel'd twists is presented, Lie bialgebra structures of which have been investigated by the authors recently. [ABSTRACT FROM AUTHOR]

In 1998, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In 2010, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra H in a braided category B, they associate a paracocyclic object in B. In this paper, we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for H and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with H to that associated with an H-module coalgebra via a categorical version of the Connes–Moscovici trace. [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]

Given a Hopf algebra H , Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld H -module algebra A gives rise to an associative A -bialgebroid structure on the smash product algebra A ♯ H. They also exhibited an antipode map making A ♯ H the total algebra of a Lu's Hopf algebroid over A. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair A and A op of braided commutative Yetter–Drinfeld H -module algebras, and output is a symmetric Hopf algebroid A ♯ H ≅ H ♯ A op over  A. This construction does not require that the antipode of H is invertible. [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]

Davydov–Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category C are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center Z (C) relative to C . From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of Z (C) . Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras Λ C k ⋊ C [ Z 2 ] , the Taft algebras and the small quantum group of sl 2 at a root of unity. [ABSTRACT FROM AUTHOR]

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a λ -TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative λ -TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the λ -TD shuffle product. We then show that the free commutative λ -TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative λ -TD algebra is connected and filtered, and thus is a left counital Hopf algebra. [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]

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]

A class of left bialgebroids whose underlying algebra A ♯ H is a smash product of a bialgebra H with a braided commutative Yetter–Drinfeld H -algebra A has recently been studied in relation to models of field theories on noncommutative spaces. In Borowiec and Pachoł (2017 J. Phys. A: Math. Theor. 50 055205) a proof has been presented that the bialgebroid A F ♯ H F where HF and AF are the twists of H and A by a Drinfeld 2-cocycle F = ∑ F 1 ⊗ F 2 is isomorphic to the twist of bialgebroid A ♯ H by the bialgebroid 2-cocycle ∑ 1 ♯ F 1 ⊗ 1 ♯ F 2 induced by F. They assume H is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter–Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with a general coaction and the correct prebraiding, and even without the assumption of quasitriangularity. [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]

Algebraic quantum groupoids have been developed by two of the authors of this note (AVD and SHW) in a series of papers [A. Van Daele and S. Wang, Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples, Banach Center Publ. 98 (2012) 367–415; A. Van Daele and S. Wang, Weak multiplier Hopf algebras I. The main theory, J. Reine Angew. Math. 705 (2015) 155–209; A. Van Daele and S. Wang, Weak multiplier Hopf algebras II. The source and target algebras, preprint (2014), arXiv: 1403.7906v2 [math.RA]; A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], see also [A. Van Daele, Algebraic quantum groupoids. An example, preprint (2017), arXiv: 1702.04903 [math.RA]]. By an algebraic quantum groupoid, we understand a regular weak multiplier Hopf algebra with enough integrals. Regular multiplier Hopf algebroids are obtained also by two authors of this note (TT and AVD) in [T. Timmermann and A. Van Daele, Regular multiplier Hopf algebroids. Basic theory and examples, Commun. Algebra 46 (2017) 1926–1958]. Integral theory and duality for those have been studied by one author here (TT) in [T. Timmermann, Integration on algebraic quantum groupoids, Int. J. Math. 27 (2016) 1650014, arXiv: 1507.00660 [QA]; T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]. In these papers, the term algebraic quantum groupoid is used for a regular multiplier Hopf algebroid with a single faithful integral. Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. In the paper, Weak multiplier Hopf algebras III. Integrals and duality [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], one of the main results is that the dual of an algebraic quantum groupoid, admits a dual of the same type. In the paper, On duality of algebraic quantum groupoids [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]], a result of the same nature is obtained for regular multiplier Hopf algebroids with a single faithful integral. The duality of regular weak multiplier Hopf algebras with a single integral can be obtained from the duality of regular multiplier Hopf algebroids (see [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]). That is however not the obvious way to obtain this result. It is more difficult and less natural than the direct way followed in [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]]. We will discuss this statement further in the paper. Nevertheless, it is interesting to investigate the relation between the two approaches to duality in greater detail. This is what we do in this paper. We build further on the intimate relation between weak multiplier Hopf algebras and multiplier Hopf algebroids as studied in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. We now add the presence of integrals. That seems to be done best in a framework of dual pairs. It is in fact more general than the duality of these objects coming with integrals. We are convinced that the material we present in this paper will provide a deeper understanding of the duality of algebraic quantum groupoids, both within the framework of weak multiplier Hopf algebras, as well as more generally for multiplier Hopf algebroids. Finally, we feel it is also appropriate to include some historical comments on the development of these duality theories. [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]

Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A. Balan firstly introduced the notion of right-right Yetter-Drinfeld modules over H and studied its Galois extension. As a continuation, the aim of this paper is to introduce more properties of Yetter-Drinfeld modules. First, we will describe all the other three kinds of Yetter-Drinfeld modules over H, and the monoidal and braided structure of the categories of Yetter-Drinfeld modules explicitly. Furthermore, we will prove that the category H H YD f d of finite dimensional left-left Yetter-Drinfeld modules is rigid. Then we will compute explicitly the canonical isomorphisms in H H YD f d . Finally, as an application, we will rewrite the isomorphisms in the case of coquasitriangular dual quasi-Hopf algebra. [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]

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]

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]

Let U q (g) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra g. Let λ be a nonzero dominant integral weight of g , and let V be the corresponding type 1 finite-dimensional irreducible representation of U q (g). Starting from this data, the R -matrix formalism for quantum groups outputs a Hopf algebra U R λ (g) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley–Serre type presentation which can be recovered from that of U q (g) by adding a single invertible quantum Cartan element. We simultaneously establish that U R λ (g) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining U R λ (g) , with respect to a natural grading by the root lattice of g compatible with the weight space decomposition of End (V). [ABSTRACT FROM AUTHOR]

In this paper we study the classifying spaces of graph products of simplicial groups and connected Hopf algebras over a field, and show that they can be uniformly treated under the framework of polyhedral products. It turns out that these graph products are models of the loop spaces of polyhedral products over a flag complex and their homology, respectively. Certain morphisms between graph products are also considered. In the end we prove the structure theorems of such graph products in the form we need. [ABSTRACT FROM AUTHOR]

In this paper we mainly consider the notion of Drinfeld double for two weak multiplier Hopf (⁎-)algebras which are paired with each other. Then we show that the Drinfeld double is again a weak multiplier Hopf (⁎-)algebra. Furthermore, we study integrals on the Drinfeld double. Finally, we establish the correspondence between modules over a Drinfeld double D (A) and Yetter-Drinfeld modules over a weak algebraic quantum group A. [ABSTRACT FROM AUTHOR]

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over the 2 -rank Taft algebra A ¯ are construted and classified by Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class ring of the category of Yetter-Drinfeld modules over A ¯ is described explicitly by generators and relations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of natural transformation between morphisms of quasigroupoids and between morphisms of weak Hopf quasigroups. Also, we prove that natural transformations between morphisms of finite quasigroupoids come from natural transformations between morphisms of weak Hopf quasigroups and, on the other hand, we obtain that every natural transformation for morphisms defined between pointed cosemisimple weak Hopf quasigroups comes from a natural transformation between finite quasigroupoids. [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]

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]

Let A / B be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. Then the problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension will be studied. Finally, more relations for the crossed products and smash products will be obtained as applications. [ABSTRACT FROM AUTHOR]

In this paper, we define a new type restricted quantum group U ̄ q (s l 2 * ) and determine its Hopf Poincaré-Birkhoff-Witt-deformations U ̄ q (s l 2 * , κ) in which U ̄ q (s l 2 * , 0) = U ̄ q (s l 2 * ) and the classical restricted Drinfeld–Jimbo quantum group U ̄ q (s l 2 ) is included. We show that U ̄ q (s l 2 * ) is a basic Hopf algebra, then uniformly realize U ̄ q (s l 2 * ) and U ̄ q (s l 2 ) via some quotients of (deformed) preprojective algebras corresponding to the Gabriel quiver of U ̄ q (s l 2 * ). Moreover, we obtain a uniform tensor-categorical realization of U ̄ q (s l 2 * ) and U ̄ q (s l 2 ) , which is consistent with the above-mentioned Hopf-algebraic realization. [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]

For a commutative ring R and a commutative cocommutative Hopf algebra H finitely generated projective as an R-module, Tilborghs in (Math J Okayama Univ 32:43–52, 1990), established an anti-isomorphism of groups between the Brauer group BD(R, H) of H-dimodule and the Brauer B D (R , H ∗) of H ∗ -dimodule algebras, where H ∗ is the linear dual of H. In this paper, we generalize this result by constructing an anti-isomorphism of groups between BD(S, H), the Brauer group of dyslectic (S, H)-dimodule algebras and B D (S op , H ∗) , the Brauer group of dyslectic (S op , H ∗) -dimodule algebras, where S is an H-commutative H-dimodule algebra and S op is the opposite algebra of S. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a λ -Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a λ -Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras. [ABSTRACT FROM AUTHOR]

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In this paper we consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in [AAH1], and its restricted version in odd characteristic introduced in [AAH2]. We show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in loc. cit. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of osp(1|2). [ABSTRACT FROM AUTHOR]

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX , where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q -cycle sets, q -braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting. [ABSTRACT FROM AUTHOR]

Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor U T : C T → C 'slackly' preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k -linear exact slack Hopf monads on Vect k , that is comagma algebras such that a modified fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness. [ABSTRACT FROM AUTHOR]

In this paper, we define a G -graded Hopf module associative pseudoalgebra A for any grading group G and Hopf algebra H , and construct a smash product A # H of A with H , which is also a G -graded associative pseudoalgebra. Then we establish a Mortia context between A # H and A H , the fixed sub-pseudoalgebra under the action of H. In addition, we prove a similarity of the Schur-Weyl type duality between the group algebra k [ S e (n) ] of Sergeev group S e (n) and a super pseudoalgebra. [ABSTRACT FROM AUTHOR]

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed 8m-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations. [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]

The Lie superalgebra p s l (2 | 2) is recognized as a quite special algebra. In mathematics, it has the vanishing Killing form and allows for the three-dimensional central extension. In physics, it shows up as the symmetry of the one-dimensional Hubbard model and the asymptotic S-matrix of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In this paper, we present the Drinfeld realization of the Yangian algebra associated with the centrally extended Lie superalgebra p s l (2 | 2). Furthermore, we show that it possesses Hopf algebra structures, particularly the coproducts. The idea to prove the existence of manifest coproducts is as follows. First, we shall introduce them to Levendorskii's realization, a system of a finite truncation of Drinfeld generators. Second, we show that Levendorskii's realization is isomorphic to the Drinfeld realization by induction. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study two smash products A ★ H for a left H-quasimodule algebra A over a Hopf quasigroup H over a field K and B # U for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. Then, we prove our duality theorem (A ★ H) # H * ≅ A ⊗ (H # H *) ≅ A ⊗ M n (K) ≅ M n (A) in the setting of a Hopf quasigroup H of dimension n. As an application of our result, we consider a special case of a finite quasigroup. [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 characterize in bicategorical terms actions of monoidal categories on the categories of representations of algebras and of relative Hopf modules. For this purpose we introduce 2-cocycles in any 2-category . We observe that under certain conditions the structures of pseudofunctors between bicategories are in one-to-one correspondence with (twisted) 2-cocycles in the image bicategory. In particular, for certain pseudofunctors to Cat , the 2-category of categories, one gets 2-cocycles in the free completion 2-category under Eilenberg–Moore objects, constructed by Lack and Street. We introduce (co)quasi-bimonads in and a suitable bicategory of Tambara (co)modules over (co)quasi-bimonads in fitting the setting of the latter pseudofuntors. We describe explicitly the involved 2-cocycles in this context and show how they are related to Sweedler's and Hausser–Nill 2-cocycles in , which we define. This allows us to recover some results of Schauenburg, Balan, Hausser and Nill for modules over commutative rings. We fit a version of the 2-category of bimonads in , which we introduced in a previous paper, in a similar setting as above and recover a result of Laugwitz. We observe that pseudofunctors to Cat in general determine what we call pseudo-actions of hom-categories, which correspond to the whole range of a 2-cocycle, so that the described actions of categories appear as restrictions of these 2-cocycles to endo-hom categories. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras k G # σ , τ k Z 2 constructed through abelian extensions of k Z 2 by k G for an abelian group G. We prove that there are only two forms of them and we get a complete list of all universal R -matrices of the generalized Kac-Paljutkin algebra H 2 n 2 (see Section 2 for the definition). [ABSTRACT FROM AUTHOR]

We survey a vast array of known results and techniques in the area of polynomial identities in pointed Hopf algebras. Some new results are proven in the setting of Hopf algebras that appeared in papers of D. Radford and N. Andruskiewitsch - H.-J. Schneider. [ABSTRACT FROM AUTHOR]

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In this paper, we generalize the Nguyen–Spradlin–Volovich–Wen (NSVW) tree formula from the MHV sector to any helicity sector. We find a close connection between the Permutohedron and the KLT relation, and construct a non-trivial mapping between them, linking the amplitudes in the gauge and gravity theories. The gravity amplitude can also be mapped from a determinant followed from the matrix-tree theorem. Besides, we use the binary tree graphs to manifest its Lie structure. In our tree formula, there is an evident Hopf algebra of the permutation group behind the gravity amplitudes. Using the tree formula, we can directly re-derive the soft/collinear limit of the amplitudes. [ABSTRACT FROM AUTHOR]