OMIROV BAKHROM科研成果

Homepage 新建主栏目 Research 名称 In recent years the non-commutative and non-associative analogies of classical constructions become of interest in connection with their applications in many branches of mathematics, biology (population, genetics, etc.) and physics (string theory, quantum field theory, etc.). The non-commutative analogues of Lie (super)algebras are Leibniz (super)algebras. Although some deep results on Leibniz (super)algebras were obtained, the crucial problems about structure of Leibniz (super)algebras, which correspond to known structures of quotient Lie (super)algebras, are still open. In mathematical physics important subjects are degenerations, contractions and deformations of Lie algebras. The structure theory of Lie algebras plays an important role in Mathematics and Physics, especially in quantum field theory. It has been deeply investigated for many years by mathematicians, namely algebraists and specialists in representation theory, and by theoretical physicists. Leibniz algebras present a non-commutative analogue of Lie algebras and they were introduced by Loday in 1993 as algebras that satisfy the Leibniz identity: [x,[y, z]]=[[x, y], z] ？ [[x, z], y]. Note that if a Leibniz algebra has the additional property [x, x]=0 for any element of algebra, then the Leibniz identity coincides with Jacobi identity. Therefore, Lie algebras are particular cases of Leibniz algebras. Another generalization of Lie algebras is so-called Mal’cev algebras and the intersection of the varieties of Leibniz and Mal’cev algebras coincides exactly with the variety of Lie algebras. During the last 30 years the theory of Leibniz algebras has been actively investigated. Some (co)gomology and deformations properties; relations with R-matrices and Yang-Baxter equations; result on various types of decompositions; structure of semisimple, solvable and nilpotent Leibniz algebras; classifications of some classes of graded nilpotent Leibniz algebras were obtained in numerous papers devoted to Leibniz algebras. In fact, many results of theory of Lie algebras have been extended to Leibniz algebras case. For instance, the classical results on Cartan subalgebras, Levi’s decomposition, properties of solvable algebras with given nilradical and others from the theory of Lie algebras are also true for Leibniz algebras. From classical theory of finite-dimensional Lie algebras it is known that an arbitrary Lie algebra is decomposed into a semidirect sum of the solvable radical and its semisimple subalgebra (Levi’s theorem). According to the Cartan-Killing theory a semisimple Lie algebra can be represented as a direct sum of simple ideals, which are completely classiffied. Thanks to Mal’cev’s and Mubarakzyanov′s results the study of solvable Lie algebras is reduced to the study of nilpotent ones. Therefore, the study of finite-dimensional Lie algebras is focused to nilpotent algebras. In fact, Barnes proved an analogue of Levi’s theorem for the case of Leibniz algebras. Namely, Leibniz algebra is decomposed into a semidirect sum of its solvable radical and a semisimple Lie algebra. Therefore, the main problem of the description of finite-dimensional Leibniz algebras consists of the study of solvable Leibniz algebras. The inherent properties of non-Lie Leibniz algebras imply that the subspace spanned by squares of elements of the algebra is a non-trivial ideal (further denoted by I). Moreover, ideal I is abelian and hence, it belongs into solvable radical and it is the minimal with respect to the property that quotient algebra by this ideal is a Lie algebra. For a given Leibniz algebra we can correspond several Lie algebras, such as quotient algebra by ideal I; the set of all right multiplications with commutator operation; quotient algebra by right annihilator (which is ideal, as well) etc. Actually the solvability of each mentioned Lie algebra is equivalent to the solvability of a given Leibniz algebra. Although semisimplicity of a Leibniz algebra is equivalent to semisimplicity of a quotient algebra by ideal I, nevertheless the result about decomposition of semisimple Leibniz algebra into a direct sum of simple ones is not true. Thus, the crucial problems about structures of Leibniz algebras, which correspond to semisimple, solvable quotient algebras, by subspace spanned by squares, are still open. The main idea of the fundamental Mal’cev’s work consists of that he introduced the so-called splittable type of solvable Lie algebras and for such type of algebras he proved their decomposition into a semidirect sum of nilradical and abelian subalgebra, whose elements in regular representation are diagonalizable. Note that the nilradical of Leibniz algebra is not radical in the sense of Kurosh, because the quotient algebra by nilradical may contain a nilpotent ideal. Nevertheless, similarly to the case of Lie algebras, it plays a crucial role in the description of solvable Leibniz algebras. The fact about that the square of a solvable Leibniz algebra is contained in the nilradical also excites to study of nilpotent algebras. The investigation of solvable Lie algebras with some special types of nilradical comes from different problems in Physics. For a solvable finite-dimensional Leibniz algebra with a given nilradical the reduction of operator of right multiplication on element of a complemented space to a nilradical is non nilpotent outer derivation of the nilradical. Therefore, the dimension of a complemented space to the nilradical is not greater than maximal number of nil-independent outer derivations of the nilradical of solvable Leibniz algebra. This is one of the important reasons to investigate in the project the derivations of nilpotent Leibniz algebras (especially, outer derivations or the first group of cohomology). Actually, in order to describe the solvable Leibniz algebras with a given nilradical it is sufficient to describe the properties of nil-independent outer derivation of the nilradical and this was the subject of various papers. Actually, the first group of cohomology of nilpotent Lie algebra in coefficient itself is non-trivial, i.e. an nilpotent Lie algebra has outer derivation. We note that the same problem for Leibniz algebras is still open. Moreover, Dixmier established that Hi(G, G)10 (1 ≤ i ≤ n) for n-dimensional Lie algebra G. Taking into account the importance Hi(L, L), i ≥ 1for any variety of algebras it is crucial to get answer to analogous result of Dixmier for Leibniz algebras case. Nowadays the geometric approach are developed to study of solvable and nilpotent algebras. In fact, any n-dimensional algebra over a field F with a fixed basis can be considered as a point in an n3-dimensional F-space of the structure constants with the Zariski topology. The changing of basis corresponds to the action of the linear reductive group GLn(F) as follows: (g*l)(x,y) = g(l(g-1(x), g-1(y))), where l is a bilinear mapping, which define the algebra structure. The orbits under this action are the isomorphic classes of algebras. Therefore, the description of n-dimensional algebras can be reduced to a geometric problem of classification of orbits under the action of the group GLn(F). Obviously, set of the structure constants of all n-dimensional algebras, which defined by identities, is an affine variety (for example associative, Jordan, alternative, Lie, Leibniz and other well-known types of (super)algebras). Note that solvable (respectively, nilpotent) Leibniz (super)algebras of the same dimension form also an invariant subvariety of the variety of Leibniz algebras under mentioned action. From algebraic geometry it is known that an algebraic variety is a union of irreducible components. The algebras with open orbits, under the mentioned action, in the variety of Leibniz algebras are called rigid. The closures of these open orbits give irreducible components of the variety. Hence the finding of rigid algebras is crucial problem from the geometrical point of view. To the description of properties of rigid Leibniz (super)algebras we shall focus one of the purpose of the project. In fact, rigid algebra is characterized by absence of any degeneration to rigid algebra, that is, orbit of rigid algebra do not belong to closure of orbit of any other algebra. Existence or absence of degeneration in a given variety of algebras is revealed by construction or by using invariant arguments. This approach is very effective in case of solvable and nilpotent algebras. There are various works devoted to geometrical description of nilpotent Lie algebras. Namely, Grunewald and O’Halloran proposed hypothesis that any n-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension, that is, is not rigid algebra in the variety of Lie algebras, although it can be rigid in subvariety of nilpotent Lie algebras. One of the objective of the planed tasks is clarify this hypothesis not only for Lie algebras but also for Leibniz algebras. The hypothesis based on fact that second group of cohomologies of rigid algebras are trivial, while for nilpotent Lie algebras, as it was mentioned before, they are always non trivial. Similarly to the case of Lie algebras Balavoine proved the general principles for deformations and rigidity of Leibniz algebras. Professional Occupation 名称 2023 –till now 2020 – 2023 2019 – 2020 2016 – 2019 Chair Professor, Institute for Advanced Study of Mathematics, Harbin Institute of Technology, China; Head of the Department "Algebra and Functional analysis", National University of Uzbekistan Fulbright Scholar, University of California San Diego, USA;Head of the Department "Algebra and Functional analysis", National University of Uzbekistan 2012 – 2016 Leading Scientific Fellow, Institute of Mathematics of Uzbek Academy of Sciences 2010 – 2012 Leading Scientific Fellow, Institute of Mathematics Uzbek Academy of Sciences 2010 – 2010 Visiting Professor of University of Seville, Spain (6 months) 2008 – 2009 Visiting Professor of University of Seville, Spain (6 months)2007 – 2008 Leading Scientific Fellow, Institute of Mathematics of Uzbek Academy of Sciences, 2005 – 2007 Senior Scientific Fellow, Institute of Mathematics of Uzbek Academy of Sciences 2003 – 2008 Associate Professor (part time), Faculty of Mathematics, Department of Algebra and Analysis, National University of Uzbekistan 2002 – 2005 Post Doctoral Fellow, Institute of Mathematics of Uzbek Academy of Sciences 2001 – 2002 Scientific Fellow, Institute of Mathematics of Uzbek Academy of Sciences 1999 – 2001 Junior Scientific Fellow, Institute of Mathematics of Uzbek Academy of Sciences 1996 – 1999 Institute of Mathematics of Uzbek Academy of Sciences, Ph.D student Awards 名称 Member of The World Academy of Sciences (TWAS) for the advancement of science in developing countries, since 1st January of 2022; "Top Researcher in Natural Sciences" Scopus Award 2018 by Elsevier, Uzbekistan; Web of Science Awards 2017 in the category of "Highly cited author", Clarivate Analytics, Uzbekistan; Awarded with the highest state prize of the Republic of Uzbekistan in the field of science and technology, 23.08.2017; The Supreme Attestation Committee of the Republic of Uzbekistan award with Diploma of the scientific title Professor of Algebra, 2013; Grant of the Academy of Sciences for the Developing World (TWAS) for purchase of equipment, No.:11-018RG/MATHS/AS_I-UNESCO FR:3240262715, 2012; Awarded with TWAS prize for Young Scientists in Developing Countries for 2010 year; TWAS Young Affiliate, Trieste, Italy for 2008-2013 years; The Supreme Attestation Committee of the Republic of Uzbekistan award with Diploma of the scientific title “Senior Scientific Researcher in Mathematical logic, Algebra and Theory of Numbers”, 2007; INTAS Young Scientist Fellowship - Postdoctoral Fellowship, INTAS (Belgium) Ref. Nr. 04-83-3035 for 2005-2006 years; The award of Uzbek Academy of Sciences Prize for young scientists, 2001.