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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]

Let |$\mathcal{B}_{\mathfrak{q}}$| be a finite-dimensional Nichols algebra of diagonal type with braiding matrix |$\mathfrak{q}$|⁠ , |$\mathcal{L}_{\mathfrak{q}}$| be the corresponding Lusztig algebra as in [ 4 ], and |$\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$| be the corresponding quantum Frobenius map as in [ 5 ]. We prove that the finite-dimensional Lie algebra |$\mathfrak{n}^{\mathfrak{q}}$| is either 0 or the positive part of a semisimple Lie algebra |$\mathfrak{g}^{\mathfrak{q}}$|⁠ , which is determined for each |$\mathfrak{q}$| in the list of [ 25 ]. [ABSTRACT FROM AUTHOR]

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]

We study pre-Nichols algebras of quantum linear spaces and of Cartan type with finite GK-dimension. We prove that except for a short list of exceptions involving only roots of order 2, 3, 4, 6, any such pre-Nichols algebra is a quotient of the distinguished pre-Nichols algebra introduced by Angiono generalizing the De Concini-Kac-Procesi quantum groups. There are two new examples, one of which can be thought of as G2 at a third root of one. [ABSTRACT FROM AUTHOR]

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given. [ABSTRACT FROM AUTHOR]

We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras [9] belonging to a one-parameter family; we call these algebras large quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of [13]. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras U q ⊃ U q ⩾ ⊃ U q + are also described. Besides all (multiparameter) big quantum groups of De Concini–Kac–Procesi and big quantum super groups at roots of unity, our framework also contains the quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeiler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. The previous approaches to the above problems relied on reductions to rank two cases and direct calculations of Poisson brackets, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on perfect pairings between restricted and non-restricted integral forms. [ABSTRACT FROM AUTHOR]

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra ℬ (V) of a Yetter–Drinfeld module V on which a Lie algebra 𝔤 acts by biderivations. As a special case, we find unrolled versions of the small quantum group. [ABSTRACT FROM AUTHOR]

We compute all Nichols algebras of rigid braided vector spaces of dimension 2 that admit a non-trivial quadratic relation. [ABSTRACT FROM AUTHOR]

The finite-dimensional simple modules over the Drinfeld double of the bosonization of the Nichols algebra 픬(7) are classified. [ABSTRACT FROM AUTHOR]

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs (n, q), any finite-dimensional pointed Hopf algebra H with G(H) ≃ PSLn(q) or SLn(q) is isomorphic to a group algebra. [ABSTRACT FROM AUTHOR]

After the classification of the finite-dimensional Nichols algebras of diagonal type [17, 18], the determination of its defining relations [6, 7], and the verification of the generation in degree-s1 conjecture [6], there is still one step missing in the classification of complex finite-dimensional Hopf algebras with abelian group, without restrictions on the order of the latter: the computation of all deformations or liftings. A technique towards solving this question was developed in [5], built on cocycle deformations. In this paper, we elaborate further and present an explicit algorithm to compute liftings. In our main result we classify all liftings of finite-dimensional Nichols algebras of Cartan type A, over a cosemisimple Hopf algebra H. This extends [2], where it was assumed that the parameter is a root of unity of order >3 and that H is a commmutative group algebra. When the parameter is a root of unity of order 2 or 3, new phenomena appear: the quantum Serre relations can be deformed; this allows in turn the power root vectors to be deformed to elements in lower terms of the coradical filtration, but not necessarily in the group algebra. These phenomena are already present in the calculation of the liftings in type A2 at a parameter of order 2 or 3 over an abelian group [11, 19], done by a different method using a computer program. As a byproduct of our calculations, we present new infinite families of finite-dimensional pointed Hopf algebras. [ABSTRACT FROM AUTHOR]

Let Bq be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix q ∈ kθ×θ. Let Lq be the Lusztig algebra associated to Bq [AAR]. We present Lq as an extension (as braided Hopf algebras) of Bq by Zq where Zq is isomorphic to the universal enveloping algebra of a Lie algebra nq. We compute the Lie algebra nq when θ = 2. [ABSTRACT FROM AUTHOR]

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups. [ABSTRACT FROM AUTHOR]

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a rack collapses. [ABSTRACT FROM AUTHOR]

Abstract: We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting. [Copyright &y& Elsevier]

We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras. [ABSTRACT FROM AUTHOR]

We investigate a family of Hopf algebras of dimension 72whose coradical is isomorphic to the algebra of functions on S3. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other. [ABSTRACT FROM AUTHOR]

We classify finite-dimensional Hopf algebras whose coradical is isomorphic to the algebra of functions on 𝕊3. We describe a new infinite family of Hopf algebras of dimension 72. [ABSTRACT FROM PUBLISHER]

Abstract: We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a complete reducibility theorem in this category. [Copyright &y& Elsevier]

Abstract: We study Poisson symmetric spaces of group type with Cartan subalgebra “adapted” to the Lie cobracket. [Copyright &y& Elsevier]

We consider issues related to the origins, sources and initial motivations of the theory of Hopf algebras. We consider the two main sources of primeval development: algebraic topology and algebraic group theory. Hopf algebras are named from the work of Heinz Hopf in the 1940’s. In this note we trace the infancy of the subject back to papers from the 40’s, 50’s and 60’s in the two areas mentioned above. Many times we just describe—and/or transcribe parts of—some of the relevant original papers on the subject. [ABSTRACT FROM AUTHOR]

Abstract: We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goes as follows: to any double groupoid, we associate an abelian group bundle and a second double groupoid, its frame. The frame satisfies that every box is determined by its edges, and thus is called a ‘slim’ double groupoid. In the first step, we prove that every double groupoid is obtained as an extension of its associated abelian group bundle by its frame. In the second, independent, step we prove that every slim double groupoid with a filling condition is completely determined by a factorization of a certain canonically defined ‘diagonal’ groupoid. [Copyright &y& Elsevier]

Abstract: We show that indecomposable exact module categories over the category RepH of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to equivariant Morita equivalence. Specifically, any indecomposable exact module category is equivalent to the category of finite-dimensional modules over a left comodule algebra. This is an alternative approach to the results of Etingof and Ostrik. For this, we study the stabilizer introduced by Yan and Zhu and show that it coincides with the internal Hom. We also describe the correspondence of module categories between RepH and . [Copyright &y& Elsevier]

Let π∈Sm be a product of disjoint cycles of the same length, C the conjugacy class of π, and ρ an irreducible representation of the centralizer of π. We prove that either the Nichols algebra B(C,ρ) is infinite dimensional or the braiding of the Yetter-Drinfeld module is negative. [ABSTRACT FROM AUTHOR]

Abstract: The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal group-like element and quantum and Frobenius–Perron dimensions of simple objects are computed. [Copyright &y& Elsevier]

The explicit list of all almost factorizable Lie bialgebra structures on real absolutely simple Lie algebras is given. By “absolutely simple” we mean a real Lie algebra whose complexification is simple. [ABSTRACT FROM AUTHOR]

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX,cq), where X is a rack and q is a 2-cocycle on X with values in C×. Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras. [Copyright &y& Elsevier]

Let k be a field. Let also (F,G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (σ,τ) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra kGτ#σkF combine into a braided Hopf algebra. We also discuss diagonal realizations of such braided Hopf algebras in the category of Yetter–Drinfeld modules over a finite group. [Copyright &y& Elsevier]

Studies compact matrix quantum groups arising from Herminitan Yang-Baxter coalgebras. Construction of a coquasitriangular bialgebra; Consideration of the problem of passing from the bialgebra to a Hopf algebra; Instances of appearance of quantum general linear groups of orthogonal type.

We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius–Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements. [Copyright &y& Elsevier]