Pointed Hopf Algebras and Quasi-isomorphisms

Abstract

We show that a large class of finite dimensional pointed Hopf algebras is quasi-isomorphic to their associated graded version coming from the coradical filtration, i.e. they are 2-cocycle deformations of the latter. This supports a slightly specialized form of a conjecture in [M]. IntroductionRecently there has been a lot of progress in determining the structure of pointed Hopf algebras. This has led to a discovery of whole new classes of such Hopf algebras and to some important classification results. See the survey article [AS4] for an introduction and references. A lot of these classes contain infinitely many non-isomorphic Hopf algebras of the same dimension, thus disproving an old conjecture of Kaplansky. Masuoka showed in [M], and in a private note, that for certain of these new families the Hopf algebras are all 2-cocycle deformations of each other. This led him to weaken Kaplansky's conjecture, stating that up to quasi-isomorphisms there should only be a finite number of Hopf algebras of a given dimension. This was disproved in [EG] for a family of Hopf algebras of dimension 32. However, our results support Masuoka's conjecture in a really big class of * This work will be part of the author's Ph.D. thesis written under the supervision of Professor H.-J. Schneider. The author is a member of the Graduiertenkolleg "Mathematik im Bereich ihrer Wechselwirkung mit der Physik" at Munich University.

Construction of quantized enveloping algebras by cocycle deformation

“…Emphasizing a Hopf-Galois theoretic treatment, we discussed in [6] cocycle deformation of a certain class of pointed Hopf algebras, containing U q , and proved that each of those Hopf algebras is a cocycle deformation of the naturally associated, simpler graded Hopf algebra. This generalizes the preceding results by Didt [7], and by Kassel and Schneider [8]. In this paper we treat with a larger class of Hopf algebras including non-pointed ones, working in the context of pre-Nichols algebras; see below.…”

“…If µ = 0, u(D, λ, 0) is presented as H λ , in the same way as in the preceding examples; see [6,Appendix]. By Theorem 5.2, u(D, λ, 0) is a cocycle deformation of u(D, 0, 0); this was proved by Didt [7]. By some additional argument, it is proved by [6, Theorem A.1] (cf.…”

Abelian and non-abelian second cohomologies of quantized enveloping algebras

“…They also proved the same result for the Frobenius-Lusztig kernels u q , which is a finite-dimensional quotient Hopf algebra of U q defined when q is a root of 1; this result also follows from our Theorem 4.3, in the same way as above. Didt [8] obtained results on cocycle deformations of finite-dimensional Nichols algebras and their liftings. Among others, he obtained the result [8, Theorem 1] for those algebras defined with zero root vector parameters [2,3], which generalizes the above-cited result for u q .…”

Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

“…Since its introduction in 1998 by Andruskiewitsch and Schneider, the Lifting Method [AS98] grew to one of the most powerful and most fruitful methods to study Hopf algebras [AS00], [BDR02], [Did05], [KR09], [ABM10], [Mom10], [ARS10], [AS10], [MPSW], [GG], [Mas]. Although it originates from a purely Hopf algebraic problem, the method quickly showed a strong relationship with other areas of mathematics such as • quantum groups [Ros98], [AS10],…”