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If A is an associative algebra, then we can define the adjoint Lie algebra A(−) and Jordan algebra A(+). It is easy to see that any associative Rota–Baxter (RB) operator on A induces a Lie and Jordan RB operator on A(−) and A(+), respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra H4, which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra H4(−). [ABSTRACT FROM AUTHOR]

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]

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]

A class of algebras $ D(m, d, \xi) $ introduced by [22] were not pointed and generated by the coradical of $ D(m, d, \xi) $. Let $ D $ be the quotient of $ D(m, d, \xi) $ module the principle ideal $ (g^m-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.

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]

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.

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.

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 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]

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]

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]

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]

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]

Let D(Rm,n(q)) be the Drinfeld double of Radford Hopf algebra Rm,n(q) and D(Hs,t) be the Drinfeld double of generalized Taft algebra Hs,t. Both D(Rm,n(q)) and D(Hs,t) have very symmetric structures. We calculate all Hopf automorphisms of D(Rm,n(q)) and D(Hs,t), respectively. Furthermore, we prove that the Hopf automorphism group AutHopf(D(Rm,n(q))) is isomorphic to the direct sum Zn⨁Zm of cyclic groups Zm and Zn, the Hopf automorphism group AutHopf(D(Hs,t)) is isomorphic to the semi-direct products k*⋊Zd of multiplicative group k* and cyclic group Zd, where s=td,k*=k\{0}, and k is an algebraically closed field with char (k)∤t.

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]

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.

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]

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]

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]

This paper is an attempt to construct a special kind of Hopf pairing $ \langle-, -\rangle:H^\bullet\otimes H\rightarrow {𝕜} $. Specifically, $ H^\bullet $ 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^\circ $ of them, which are given by generators and relations; 2) the Hopf pairings desired are determined by choosing certain Hopf subalgebras $ H^\bullet $ of $ H^\circ $, where $ \langle-, -\rangle $ becomes the evaluation.

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]

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]

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]

In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.) We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

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]

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]

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]

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]

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]