Solvable 3-Lie algebras with a maximal hypo-nilpotent ideal N

This paper obtains all solvable 3-Lie algebras with the m-dimensional filiform 3-Lie algebra N (m >= 5) as a maximal hypo-nilpotent ideal, and proves that the m-dimensional filiform 3-Lie algebra N can't be as the nilradical of solvable non-nilpotent 3-Lie algebras. By means of one dimensional extension of Lie algebras to the 3-Lie algebras, we get some classes of solvable Lie algebras directly

3-Lie algebras with an ideal N

AbstractWe define the hypo-nilpotent ideal in n-Lie algebras and obtain all solvable 3-Lie algebras with an m-dimensional simplest filiform 3-Lie algebra as a maximal hypo-nilpotent ideal. We prove that the dimension of such solvable 3-Lie algebras is at most m+2, and there is no solvable 3-Lie algebra with the simplest filiform 3-Lie algebra as the nilradical

PRL2/PTP4A2 phosphatase is important for hematopoietic stem cell self-renewal

Hematopoietic stem cell (HSC) self-renewal is tightly controlled by cytokines and other signals in the microenvironment. While stem cell factor (SCF) is an early acting cytokine that activates the receptor tyrosine kinase KIT and promotes HSC maintenance, how SCF/KIT signaling is regulated in HSCs is poorly understood. The protein tyrosine phosphatase 4A (PTP4A) family (aka PRL [phosphatase of regenerating liver] phosphatases), consisting of PTP4A1/PRL1, PTP4A2/PRL2, and PTP4A3/PRL3, represents an intriguing group of phosphatases implicated in cell proliferation and tumorigenesis. However, the role of PTP4A in hematopoiesis remains elusive. To define the role of PTP4A in hematopoiesis, we analyzed HSC behavior in Ptp4a2 (Prl2) deficient mice. We found that Ptp4a2 deficiency impairs HSC self-renewal as revealed by serial bone marrow transplantation assays. Moreover, we observed that Ptp4a2 null hematopoietic stem and progenitor cells (HSPCs) are more quiescent and show reduced activation of the AKT and ERK signaling. Importantly, we discovered that the ability of PTP4A2 to enhance HSPC proliferation and activation of AKT and ERK signaling depends on its phosphatase activity. Furthermore, we found that PTP4A2 is important for SCF-mediated HSPC proliferation and loss of Ptp4a2 decreased the ability of oncogenic KIT/D814V mutant in promoting hematopoietic progenitor cell proliferation. Thus, PTP4A2 plays critical roles in regulating HSC self-renewal and mediating SCF/KIT signaling