Conjugacy classes of p-cycles of type D in alternating groups

We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in arXiv:0812.4628. We also determine all the subracks of those conjugacy classes which are not of type D.Comment: Second paragraph of subsection 2.2 rewritten. 4-th sentence of subsection 2.4 rewritten. More explanations added in Remark 2.4. Lemma 2.5 and Corollary 2.7 added. Appendix removed and put it as Remark 3.1. Remark 3.2 (former 3.1) reorganized. References: [Da], [EGSS], [H], [IS] added, [GPPS] removed. Communications in Algebra (2014

On Nichols algebras associated to simple racks

This is a report on the present state of the problem of determining the dimension of the Nichols algebra associated to a rack and a cocycle. This is relevant for the classification of finite-dimensional complex pointed Hopf algebras whose group of group-likes is non-abelian. We deal mainly with simple racks. We recall the notion of rack of type D, collect the known lists of simple racks of type D and include preliminary results for the open cases. This notion is important because the Nichols algebra associated to a rack of type D and any cocycle has infinite dimension. For those racks not of type D, the computation of the cohomology groups is needed. We discuss some techniques for this problem and compute explicitly the cohomology groups corresponding to some conjugacy classes in symmetric or alternating groups of low order.Comment: 26 pages, minor change

On Nichols algebras over PGL(2,q) and PSL(2,q)

We compute necessary conditions on Yetter-Drinfeld modules over the groups \mathbf{PGL}(2,q)=\mathbf{PGL}(2,\FF_q) and \mathbf{PSL}(2,q)=\mathbf{PSL}(2,\FF_q) to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that there is no non-trivial finite-dimensional pointed Hopf algebra over the Mathieu groups $M_{20}$ and $M_{21}=\mathbf{PSL}(3,4)$.Comment: Minor change

On Nichols algebras over SL(2,Fq) and GL(2,Fq)

We compute necessary conditions on Yetter-Drinfeld modules over the groups SL(2,Fq) and GL(2,Fq) to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with a group of group-likes isomorphic to one of these groups.Comment: Major exposition revision, including referees remarks. To appear in J. Math. Phys. 13 page

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

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.Comment: v2: 35 pages, 6 tables, 14 figure

The logbook of Pointed Hopf algebras over the sporadic groups

In this notes we give details of the proofs performed with GAP of the theorems of our paper "Pointed Hopf Algebras over the Sporadic Simple Groups".Comment: 22 pages, final versio

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

It is shown that Nichols algebras over alternating groups A_m, m>4, are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to A_m is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups S_m are all infinite-dimensional, except maybe those related to the transpositions considered in [FK], and the class of type (2,3) in S_5. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra of (X,q) is infinite dimensional, for q an arbitrary cocycle. arXiv:0904.3978 is included here.Comment: Changes in version 7: We eliminate the Subsection 3.3 and references to type C throughout the paper. We remove Lemma 3.24, Proposition 3.28 and Example 3.29 (old numbering), since they are not needed in the present paper. Several minor mistakes are corrected. The proof of Step 2 in Theorem 4.1 is adjuste

Heart Transplantation in High-Risk Recipients Employing Donor Marginal Grafts Preserved With Ex-Vivo Perfusion

Extending selection criteria to face donor organ shortage in heart transplantation (HTx) may increase the risk of mortality. Ex-vivo normothermic perfusion (EVP) limits ischemic time allowing assessment of graft function. We investigated the outcome of HTx in 80 high-risk recipients transplanted with marginal donor and EVP-preserved grafts, from 2016 to 2021. The recipients median age was 57 years (range, 13–75), with chronic renal failure in 61%, impaired liver function in 11% and previous cardiac surgery in 90%; 80% were mechanically supported. Median RADIAL score was 3. Mean graft ischemic time was 118 ± 25 min, “out-of-body” time 420 ± 66 min and median cardiopulmonary bypass (CPB) time 228 min (126–416). In-hospital mortality was 11% and ≥moderate primary graft dysfunction 16%. At univariable analysis, CPB time and high central venous pressure were risk factors for mortality. Actuarial survival at 1 and 3 years was 83% ± 4%, and 72% ± 7%, with a median follow-up of 16 months (range 2–43). Recipient and donor ages, pre-HTx extracorporeal life support and intra-aortic balloon pump were risk factors for late mortality. In conclusion, the use of EVP allows extension of the graft pool by recruitment of marginal donors to successfully perform HTx even in high-risk recipients