MINISTRY OF EDUCATION AND TRAINING HCMC University of Education SOCIALIST REPUBLIC OF VIETNAM Independence - Freedom - Happiness INFORMATION PAGE OF NEW CONTRIBUTIONS TO ACADEMY AND THEORY OF THE THESIS (In English) - Author's name: Cao Tran Tu Hai - Supervisors : Assoc.Prof.Dr Le Anh Vu Dr Duong Minh Thanh - Thesis title: Computing cohomology and classification problems of Lie algebras, quadratic Lie superalgebras - The science of the thesis: Mathematics - Major: Geometry and Topology - Course: 2015-2019 - Code: 62 46 01 05 - Name of training institution: Ho Chi Minh City University of Pedagogy CONTENTS OF NEW CONTRIBUTIONS OF THE THESIS (Academic, theoretical, new points drawn from research results and thesis survey) The content of the thesis's new contributions has scientific significance and has made certain contributions to Lie theory in particular, to the Field of Algebra, Geometry and Topology in general New results are listed below We study the classification of finite-dimensional real solvable Lie algebras having derived algebras of low-codimension Specifically, we give a complete classification of (n+1)dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of n-dimensional nilpotent Lie algebras is given In addition, the problem of classifying (n+2)-dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild In this case, we classify a subclass of the considered Lie algebras which are extended from their derived algebras by a pair of derivations containing at least one inner derivation The results of the thesis have actively contributed to the explicit calculation of the cohomology of real solvable Lie algebras In particular, the cohomology of the class of all solvable Lie algebras with 1-dimensional derived algebra has been fully described By applying Pouseele's method related to the extension of one-dimensional Lie algebra by the Heisenberg Lie algebra, the thesis has explicitly calculated all Betti numbers for a subclass of real solvable Lie algebras with 1-codimensional derivate algebra, which is the class of general Diamond Lie algebras By applying the method of computing super-Poisson brackets, the thesis has described the cohomology of all solvable quadratic Lie algebras in low dimension, the second cohomology group of Jordan type nilpotent quadratic Lie algebras, the first and second cohomology groups of all the elementary quadratic Lie superalgebras that have been classified Thanks to the explicit description of the space of antisymmetric derivations of solvable quadratic Lie algebras in low dimension, the thesis has explicitly calculated their second cohomology group Using double extension and general double extension combined with some results of classification of adjoint orbits of symplectic Lie algebras, the thesis has classified solvable quadratic Lie algebras and solvable quadratic Lie superalgebras in low dimension SUPERVISORS Assoc.Prof.Dr Le Anh Vu Dr Duong Minh Thanh PhD STUDENT Cao Tran Tu Hai