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 skew braces and their ideals

The first-named author is partially supported by CCP CoDiMa (EP/M022641/1) and the OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541). The second-named author is supported by the ERC Advanced grant 320974. The third-named author is supported by PICT-201-0147, MATH-AmSud 17MATH-01 and ERC Advanced grant 320974.We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions.PostprintPeer reviewe

Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang-Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.Comment: 18 pages. Postprint versio

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

RICOSTITUZIONE IMMUNITARIA IN PAZIENTI SOTTOPOSTI A TRAPIANTO ALLOGENICO DI CELLULE STAMINALI EMOPOIETICHE: VALUTAZIONI A LUNGO TERMINE

Allogenic stem cell transplantation represents an important therapeutic option for the treatment of a number of haematological diseases. Particularly in the context of onco-haematological diseases this treatment has been shown to improve outcome thanks to the graft versus tumour (GVT) effect. Although recent improvements in transplantation procedures, still some limitations to the applicability of these treatment persist. First, not all patients may have available an HLA (human leukocyte antigens) identical donor and secondly, conditioning regimens before transplantation may be too toxic for elderly patients or for patients with other co-morbidities. To overcome this limitations in last few years many researchers have been exploiting new approaches. The introduction of not fully matched transplantations (mismatched or haploidentical) and reduced intensity regiments have partially overcome the limitations. In the onco-haematological context the most common complications of allogenic stem cell transplantation are relapse, opportunistic infections and activation of transplanted immune system against the normal tissues of the host (graft versus host disease \u2013 GVHD). All of these represent alterations of the normal functions of the immune system and demonstrate the importance of monitoring immunity in allo-transplanted patients. Studies in these field are mostly limited to the evaluation of the first year after transplantation and data from longer follow up are limited. In this study we monitored patients undergoing haematopoietic stem cell transplantation from an alternative donor after reduced intensity regimen over one year after transplantation (up to 4-5 years after transplantation). Particularly we evaluated 6 patients (age at transplantation 34; range 15-49) undergoing haploidentical stem cell transplantataion associated with T cell depletion in vitro (immunomagnetic selection of CD34 stem cells) and in vivo (anti CD52 antibody \u2013 Alemtuzumab) followed by infusion of CD8 depleted donor lymphocytes and 18 patients (age at transplantation 40; range 22-60) undergoing transplantation from a match unrelated donor (MUD) followed by in vivo T cell depletion (anti thymocyte globulins \u2013 ATG). All the patients have been analysed at the times of clinical remission and not under pharmacological treatment. In order to compare data obtained by the single cohorts to a normal situation we also evaluated 10 healthy donors (age 37; range 24-55). The evaluation of the immune recovery has been carried out through 4 different techniques: - analysis of chimerism through amplification of 9 different short tandem repeats (STR); - evaluation of the lymphoid sub population B, T and NK (and their maturation stages) through flow cytometry immunophenotype; - analysis of the thymic productivity trough quantification of the episomal DNA sjTREC (signal joint T-cell receptor excision circle); - evaluation of the receptorial complexity of the T and B cell compartment trough analysis of the CDR3 (complementary determinating region 3) of the \u3b2 chain of the T cell receptor and of the heavy chain of the immunoglobulins. Our results show no significant differences between the two groups of patients analysed in the long term immune reconstitution, neither respectively the counts of the lymphoid sub population nor regarding the complexity values for the B and T cell receptors comparing the data to the ones from healthy subjects. We show a reduction in the thymic productivity that persist over 3 years post transplantation although there are no difference comparing the two cohorts of patients. Even though the counts of the main populations normalize between 1 and 2 years after transplantation the analysis of the maturation of the B and T cell compartments highlights the persistence of alterations in the normal maturation process in all the follow up points. Particularly both patients undergoing haploidentical or MUD transplantation present an increase in the B na\uefve subpopulation and a decrease in the B memory subsets demonstrating an alteration in the normal B cell development probably due to an alteration in the normal function of the germinal center. Concerning the T cell compartment it has been seen a decrease in the production of na\uefve T cells, that reflects the low thymic production, and an increase in the effector and terminal memory compartment. These might be a mechanism involved in the control of the oncological disease . In conclusion, patients that do not relapse and do not experience other clinical problems are able to recover immunity in the long term although thymic production remains low. No significant difference are found between the two types of transplantation highlighting that haploidentical transplantation is a good alternative to HLA identical transplantation implying less problems for donor recruitment. Analysis of the maturation steps of the B and T cell compartment demonstrated the persistence of alterations long term after transplantation indicating that this evaluation should be further studied in order to elucidate the mechanism at the basis of the immune recovery. Moreover this could be a good marker for monitoring the clinical course of the patients