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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]

Given a cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K -algebra A , we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A # H , involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial "Hopf" algebra H p a r is involved in the definition and we construct a projective resolution of B. [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]

In this paper, we study if, for a given simple module over a Hopf algebra, there exists a virtual module such that their tensor product is the regular module. This is related to a conjecture by Donald Knutson, later disproved and refined by Savitskii, stating that for every irreducible character of a finite group, there exists a virtual character such that their tensor product is the regular character. We also introduce the Knutson Index as a measure of Knutson's Conjecture failure, discuss its algebraic properties and present counter-examples to Savitskii's Conjecture. [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, 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 show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSL n (q) (n ≥ 4) , PSL 3 (q) (q > 2) , or PSp 2 n (q) (n ≥ 3) , is isomorphic to a group algebra. [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]

In the paper, we discuss the partial actions of Sweedler Hopf algebra on the generalized quaternion algebra, and give the sufficient and necessary conditions to make the actions be the partial actions. As an application, we give a complete description of all partial actions of Sweedler Hopf algebra on the generalized quaternion algebra. [ABSTRACT FROM AUTHOR]

Let B be a right coideal subalgebra of a Hopf algebra H. We construct a left H*-module structure on the Hochschild cohomology of B with coefficients in B x M where M is a B-bimodule as well as a right H-comodule satisfying a compatible condition. We prove that the structure is a canonical invariant which coincides with one given by Krähmer (2012) for quantum homogeneous spaces. In particular, Krähmer's group-like element turns out to be determined by B uniquely, which plays an important role in describing the dualizing complex of B. [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]

This paper constructs a novel Hopf algebra cf (UT •) on the class functions of the unipotent upper triangular groups UT n (F q) over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of GL n (F q) class functions. Furthermore, cf (UT •) contains a Hopf subalgebra which is isomorphic to a known combinatorial Hopf algebra, previously used to prove a conjecture about chromatic quasisymmetric functions. Some additional Hopf algebraic properties are also established. [ABSTRACT FROM AUTHOR]

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map on the tensor algebra by considering a deformed quasi-shuffle product of words. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties. [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]

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]

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]

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]

In this paper, we study cyclic cohomology theory on a regular multiplier Hopf algebra. Given a modular pair consisting of a group-like element, a character and in involution for a regular multiplier Hopf algebra, we associate a cyclic module. The approach is based on the Connes-Moscovici cyclic module of Hopf algebras and we obtain the same cyclic modules when our algebra has an identity. [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]

The Suzuki algebra AµλNn, introduced by Suzuki Satoshi in 1998, is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra AµλN2n and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type A1, A1 × A1, A2, A2 × A2, Super type A2(q; I2) and the Nichols algebra ufo(8). There are 64, 4m and m² -dimensional Nichols algebras of non-diagonal type over AµλN2n . The 64-dimensional Nichols algebras are of dihedral rack type D4. The 4m and m2 -dimensional Nichols algebras B(Vabe) discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over AµλNn. Using a result of Masuoka, we prove that dim B(Vabe) = ∞ under the condition b² = (ae)−1, b ∈ Gm for m ≥ 5. [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]

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension 4p for a prime p > 5. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property. [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]

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]

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]

The theory of groups has a twofold symmetry, sending a group to its opposite. Groups invariant under the symmetry are abelian. The theory of quasigroups has a richer, sixfold symmetry, obtained by permuting the multiplication with its two divisions. The Sixfold Way identifies the various classes of quasigroups which are invariant under the respective subgroups of the symmetry group of the theory. Quantum quasigroups provide a self-dual framework to unify the study of quasigroups and Hopf algebras. The goal of this paper is to classify the symmetry classes of quantum quasigroups. Corresponding to the Sixfold Way for classical quasigroups, we are able to identify a Sevenfold Way for general classes exhibiting a symmetry, and initiate a study of a fuller symmetry which holds for linear quantum quasigroups. [ABSTRACT FROM AUTHOR]

We introduce two new bases of Q S y m , the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in NSym , the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions ρ and ω , which generalize the classical involution ω on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and Q S y m respectively. We then use the involutions ρ and ω to translate these and other known results to our reverse and row-strict reverse bases. [ABSTRACT FROM AUTHOR]

We classify pointed Hopf algebras of discrete corepresentation type over an algebraically closed field K with characteristic zero. For such algebras H , we explicitly determine the algebra structure up to isomorphism for the link indecomposable component B containing the unit. It turns out that H is a crossed product of B and a certain group algebra. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the semiring of quotients of a H− semimodule semialgebra A with respect to the filter F of H−stable ideals I of A with right and left annihilators of I are zero. We establish a connection between semiring of quotients of a semialgebra A and the ring of quotients of the algebra of differences A 4. We also introduce the Hopf algebra action of the semiring of quotients and study the connection between the smash product of semialgebra of quotients and the smash product of algebra of quotients. [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]

In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A , we show that the finite generation condition on A -modules can be replaced by a condition on any affine commutative A -module algebra R under the assumption that R is integral over its invariant subring R A. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A ′ over a finite field, where A can be viewed as a deformation of A ′ , and prove that if the finite generation condition holds for A ′ , then the same condition holds for A. [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]

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R (H) , which we call its remnant and a co-quasitriangular structure on H induces a YBE solution on R (H) , which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, H , this secondary group structure appears as the projection of the transmutation of H. Finally, for any YBE solution, we obtain a SupLat-FRT Hopf algebra in SupLat, whose remnant recovers the universal skew brace of the solution. [ABSTRACT FROM AUTHOR]

Let (M , Q) be a dg manifold. The space of vector fields with shifted degrees (X (M) [ − 1 ] , L Q) is a Lie algebra object in the homology category H ((C M ∞ , Q) − mod) of dg modules over (M , Q) , the Atiyah class α M being its Lie bracket. The triple (X (M) [ − 1 ] , L Q ; α M) is also a Lie algebra object in the Gabriel-Zisman homotopy category Π ((C M ∞ , Q) − mod). In this paper, we describe the universal enveloping algebra of (X (M) [ − 1 ] , L Q ; α M) and prove that it is a Hopf algebra object in Π ((C M ∞ , Q) − mod). As an application, we study Fedosov dg Lie algebroids and recover a result of Stiénon, Xu, and the second author on the Hopf algebra arising from a Lie pair. [ABSTRACT FROM AUTHOR]

By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial. [ABSTRACT FROM AUTHOR]

The objects of study in this paper are Hopf algebras H which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra A by a finite dimensional Hopf algebra H ‾ : = H / A + H , where A + is the augmentation ideal of A. Basic structural and homological properties are recalled and classes of examples are listed. A structure theorem is proved when (⁎) H ‾ is semisimple and cosemisimple, showing that in this case the noncommutativity of H arises from the action of a finite group. For example, when (⁎) holds and H is prime and pointed, it is a crossed product of a smooth affine commutative domain by a finite group, and the simple H -modules are described by a type of Clifford's theorem. [ABSTRACT FROM AUTHOR]

A Hopf Galois structure on a finite field extension L / K is a pair (H , μ) , where H is a finite cocommutative K -Hopf algebra and μ a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree 2 p n , with p an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree 2 p 2 , we determine exactly the possible sets of Hopf Galois structure types. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial representations of groups and Hopf algebras, our globalization coincides with those described earlier in literature. Finally, we introduce Hopf partial comodules over a bialgebra as geometric partial comodules in the monoidal category of (global) modules. By applying our globalization theorem we obtain an analogue of the fundamental theorem for Hopf modules in this partial setting. [ABSTRACT FROM AUTHOR]