Non-associative systems and their applications

• Grant number: 16/06133-0
• Support Opportunities: Scholarships abroad - Research
• Start date: October 15, 2016
• End date: April 14, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Alexandre Grichkov
• Grantee: Alexandre Grichkov
• Host Investigator: Rolando Jimenez Benitez
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: Universidad Nacional Autónoma de México, Oaxaca (UNAM), Mexico
• Associated research grant: 14/09310-5 - Algebraic structures and their representations, AP.TEM

Abstract

We expect to obtaint significant results in the following directions: 1. Lie algebras:a) to classify simple Lie algebras of dimension < 8 and 2-algebras of di-mension < 16:b) classify simple finite dimensional Lie algebras of toral rank two in char-acteristic 2.c) construct new simple finite dimensional Lie algebras and 2-algebras,d) describe the simple Lie algebras in characteristic 2 with 1-dimensionalCartan subalgebra.2. Groups with triality:a) prove analog of Zelmanov`s Theorem for groups with trialityfor exponent pn; p > 3 -prime number. Note this analog is the generalizationof Zelmanov`s Theorem, becouse we suppose that the group with triality G hasnot expoent pn; but only the elements from M(G) have this expoent;b) construct the groups with triaity corresponding to the code Moufang loopsand free loop in variety generated by those loops, 3. Loops de Moufang: a) Study of cyclic extentions of groups in the variety of Moufang loops, b) Prove the analog of Weak Burnside problem (Zelmanov`s Theorem) for Moufang loops for expoent pn; p > 3,c) Construct bases of theory of representations of Moufang loops,d) Find the variety of Moufang loops cooresponding to Malcev algebras isomorphic to central extension of associative algebras in the variety of Malcev algebras and prove the speciality of thos Malcev algebras. 4. Finite geometries and related quasi-groups and loops. a) Classify free nilpotent Steiner loops of degree two, b) Describe extensions of Steiner loops, c) Construct an innite Steiner loop of minimal growth in the variety of metabian loops,d) Prove that group of half-automorphisms of Generalized Steiner loop is isomorphic to group of automorphisms of the corresponding 4-geometry, e) Construct non-trivial examples of metabilian 4-geometries, using theory of central extension of loops, corresponding to Generalized Steiner quasi-groups. During this period I plan ro participate in two International Congress (Mex-ico and Oxaca) and give two mini-couses (Oxaca and Cuernavaca) about groups with triality and related topics. (AU)