3 tóm tắt luận án tiếng anh

The first is the result of a thorough classification of the Lien + 1, n class of real Liealgebras that can be solved for n + 1 in the given dimension n by the first cohomologyspace of it

INTRODUCTION 1Chapter 0 Some basic knowledge and results 5Chapter1 Real solvable Lie algebra classes having small dimensional or

Chapter2 Some classes of solvable quadratic Lie algebras and

INTRODUCTION

Lie groups and Lie algebras (collectively called Lie theory) was initiated by SophusLie, a Norwegian mathematician since the 70s of the 18th century and developed bymany mathematicians around the world during nineteenth and early twentieth centuriessuch as Felix Klein, Friedrich Engel, Wilhelm Killing, Elie Cartan, Hermann Weyl, etc

As a result, fundamental Lie Theory problems such as classification of Lie groups andLie algebras, quadratic Lie algebras, Lie superalgebras, quadratic Lie superalgebras,etc and cohomology computations often receive the attention of the mathematicalcommunity

A Lie group is a group that is also a differentiable manifold, in which group erations are compatible with differentiable structures In the process of studying theleft-invariant vector field of Lie groups, Lie algebra was born In Lie theory, quiteinterestingly, there is a one-to-one correspondence between the set of singly connectedLie groups and the set of Lie algebras In this thesis, we approach the problem ofclassification on the class of Lie algebras

op-According to the Levi-Malshev Theorem, every finite-dimensional Lie algebra over

a field of characteristic 0 can be decomposed as a semi-direct product of a semi-simplesubalgebra and a solvable ideal (see Levi [21] in 1905 and Malshev [12] in 1945) Fromthat, the problem of classification of general Lie algebras is reduced to the classification

of semi-simple Lie algebras and solvable Lie algebras in which the problem of classifyingsemi-simple Lie algebras was thoroughly solved by Cartan [20] in 1894 (on C) and byGantmacher [10] (on R)

As for the class classification problem of solvable Lie algebras, although there aresome classifications in the particular case The classification in the general case, so far,

is still an open problem

In general, the solvable Lie algebra classification problem is quite complex, peopleoften find ways to narrow the class of objects to be classified to make it easier tocontrol Specifically, there are at least three approaches The first is the way to classify

in terms of dimensionality (ie, to fix the number of dimensions of the Lie algebras to

be classified) The second is the way to classify according to the structure (that is,

to add one or more special properties to the class of Lie algebras to be classified) Thethird is combining both dimensional and structural classifications in a logical way

On the direction of classification by dimensionality, Mubarakzyanov [13, 14] hasclassified solvable Lie algebras of dimension 4 and 5 over fields of characteristic zero.Results for the six-dimensional case were also obtained by Mubarakzyanov [15] Fordecades, all classification problems of dimension 7 and more have remained unsolved,even with the help of specialized computational software Obviously, it is more feasible

to approach the solvable Lie algebra classification problem in terms of structure or acombination of both directions: fixed dimensionality and structural addition In thisthesis, we go in the direction of supplementing the structure and at the same timecoordinating with the direction of fixed number of dimensions

The problem we are interested in the thesis is studying and classifying real solvableLie algebras having small codimensional derived algebras We have achieved somepositive results as follows

The first is the result of a thorough classification of the Lie(n + 1, n) class of real Liealgebras that can be solved for n + 1 in the given dimension n by the first cohomologyspace of its derived algebra, corresponding to the coadjoint representation (see Theorem

1.1) The second is the assertion that classifying real solvable Lie algebras of dimension

n + 2 with a given n-dimensional derived algebra (denoted by Lie(n + 2, n)) is a wildproblem (see Theorem 1.2) Finally, there is a special subclass of class Lie(n + 2, n)

in which the wild property is broken (see Theorem 1.3)

On the other hand, also in the direction of structure, recently appeared a solvableLie algebraic object with the additional structure of a non-degenerate and invariantbilinear form with respect to the Lie bracket These are called quadratic Lie alge-bras Since the last few decades, the problem of classifying quadratic Lie algebras andquadratic Liesuper algebras has attracted the attention of a number of mathematicians.Therefore, we have a basis to continue to study this classification problem

The cohomology description has also been fully solved on the class of semi-simpleLie algebras However, for the class of solvable Lie algebras, the number of results

of Lie Heisenberg algebra h2n+1, [18] by Pouselee on the cohomology of an extension of

an one-dimensional Lie algebra hZi by Lie Heisenberg algebra h2n+1

Since then, we have set the task to study the problem of classifying some classes ofsolvable Lie algebras, solvable quadratic Lie algebras, solvable quadratic Lie superal-gebras, and compute cohomology on several classes The thesis is titled: “ComputingCohomology and classification problems of Lie algebras, quadratic Lie superalgebras.”The successful implementation of the thesis topic has scientific significance and cer-tain contributions to Lie theory in particular, to Algebra, Geometry and Topology ingeneral Specifically, the topic has the following specific contributions:

(1) We present an effective method to classify (n + 1)-dimensional real solvable Lie gebras having one-codimensional derived algebras provided that a full classification

al-of n-dimensional nilpotent Lie algebras is given

(2) In addition, the problem of classifying (n+2)-dimensional real solvable Lie algebrashaving two-codimensional derived algebras is proved to be wild In this case, weclassify a subclass of the considered Lie algebras which are extended from theirderived algebras by a pair of derivations containing at least one inner derivation.(3) The cohomology of all solvable Lie algebras with one-dimensional derived algebrahas been fully described

(4) By applying Pouseele’s method concerning the extension of one-dimensional Liealgebra by the Heisenberg Lie algebra we obtain all Betti numbers bkfor the generalDiamond Lie algebra

(5) Describe the cohomology of all low-dimensional quadratic Lie algebras, the ond group of the Jordan-type quadratic Lie algebras, of elememtary quadratic Liesuperalgebras has been classify

(6) Explicitly describe the space of skew-symmetric derivations of low-dimensionalsolvable quadratic Lie algebras From there, compute their second cohomologygroup

(7) Applying double extension and general double extension combined with some sults of classification of adjoint orbits of symplectic Lie algebras, we classify allsolvable quadratic Lie algebras and solvable quadratic Lie superalgebras of dimen-sion in low

re-The results of the thesis have been reported at a number of national and tional Mathematical Conferences:

interna-ˆ Scientific research conference for Master’s and PhD students of Ho Chi Minh CityUniversity of Education, October 2016

ˆ National Conference on Algebra - Geometry - Topology in Buon Ma Thuot City,Dak Lak, December 2016

ˆ International Mathematical Conference on Algebra - Geometry at Mahidol versity, Bangkok, Thailand (ICMA-MU 2017), May 2017

Uni-ˆ Mathematical and Applied Mathematics Conference at University of Science andTechnology, Vietnam National University, Ho Chi Minh City, August 2017

ˆ Scientific Conference of the University of Natural Sciences, Vietnam NationalUniversity, Ho Chi Minh City, 11th time, November 2018

ˆ National Conference on Algebra - Geometry - Topology in Ba Ria - Vung Tau,December 2019

Despite many efforts, the thesis is hard to avoid shortcomings We look forward

to receiving comments from reviewers and readers so that we have the opportunity torevise, correct and improve our work We sincerely thank you

Chapter 0

Some basic knowledge and results

This chapter is devoted to the general introduction of the concepts of Lie groups, Liealgebras, quadratic Lie algebras, quadratic Lie super algebras and their cohomologies,and briefly outlines some results and known properties

0.1 Lie groups, Lie algebras and cohomology

In this section, we introduce the concept of Lie groups, Lie algebras and somerelated concepts such as ideal of a Lie algebra, Lie algebraic homomorphism, semi-directproduct, solvable Lie algebras, nilpotent Lie algebras, cohomology of a Lie algebra.Moreover, we recall the Levi-Malcev Theorem, the concept of the wild problem, andthe weak similarity

0.2 Quadratic Lie algebras and cohomology

In this section, we introduce the concept of quadratic Lie algebras, homomorphismsand some related concepts such as ideals, non-degenerate ideals, isometric isomor-phisms, orthogonal direct sum Besides, we present the concept of the super-Poissonproduct and apply it to the congruence calculation of the quadratic Lie algebra

0.3 Quadratic Lie superalgebras

In this section, we introduce the concept of quadratic Lie superalgebras, cohomology

of a Lie superalgebra, superderivation, anti-symmetric super derivation, the graddedsuper-Poisson bracket and apply it to the computation of cohomology of a quadraticLie superalgebra

5

coho-we introduce the concept of the wild problem and show that the problem of classifyingreal solvable Lie algebras having two codimensional derived algebras is proved to bewild also gives a classification of a subclass of this class when wildness is broken Inaddition, we will also present new results on the description of all cohomology of theclass of real Lie algebras solvable having one dimensional derived algebras with coef-ficients on the base field and cohomology of a a special subclass of real solvable Liealgebras having one codimensional derived algebras, that is, the general Diamond Liealgebra class The main results of this chapter have been published in 3 internationalpapers: The first is “Classification of real solvable Lie algebras whose simply connectedLie groups have only zero or maximal dimensional coadjoint orbits ” in the Journal ofRevista de la UM in 2016 (a separate case of Theorem1.1), the second is “On the prob-lem of classifying solvable Lie algebras having small codimensional derived algebras”

in the Journal of Communications in Algebras in 2022 (Theorem 1.1, 1.2, 1.3) and thethird is “Cohomology of some families of Lie algebras and quadratic Lie algebras” inEast-West Journal of Mathematics in 2018

6

1.1 Classification of real solvable Lie algebras with

one-codimensional derived algebras

Firstly, Let us consider a n-dimensional nilpotent Lie algebra a, for each Lie algebra

g in Lie(n + 1, n) whose derived algebra a is an extension of a by a derivation of a.Theorem 1.1 For an arbitrary n-dimensional nilpotent Lie algebra a, the problem

of classifying all Lie algebras in Lie(n + 1, n) with derived algebra isomorphic to a isequivalent to the problem of classifying outer derivations in the first cohomology space

H1(a, a) satisfying the equivalent conditions in Proposition 1.1.1, up to proportionalsimilarity by the automorphism group of a

Proposition 1.1.1 Let a be a n-dimensional nilpotent Lie algebra and g = RY ⊕Dawith D ∈ Der(a) as above By renumbering (if necessary), we assume that a1 =span{X1, , Xm} with 0 ≤ m < n Then the following conditions are equivalent:

1.2 Classification of real solvable Lie algebras with

two-codimensional derived algebras

Each extension of h by a pair of derivations is not necessarily of class Lie(n + 2, n).The following are necessary and sufficient conditions of the derivaton pair so that aextensive Lie algebra of class Lie(n + 2, n) Using this condition, we prove that the

classification problem Lie(n + 2, n) contains the problem of classifying pairs of squarematrices, up to weakly similarity

Theorem 1.2 The problem of classifying Lie Lie(n + 2, n) is wild

From the wildness of the Lie(n + 2, n) classification problem, we consider the specialcases (see Belitskii et al [3], Section 3) Because of the wildnes in Theorem 1.2, Wenaturally consider the subclass Liead(n + 2, n) of Lie(n + 2, n) such that the derivationpair of extension has at least one inner derivation For this subclass, we have:

Theorem 1.3 Given a n-dimensional nilpotent Lie algebra h, the problem of ing Liead(n + 2, n) with Lie derived algebra h is equivalent to the problem of classifyingthe equivalence class of outer derivatives of the first cohomology space of R⊕h satisfyingthe equivalence conditions in Proposition 1.2.1, up to proportional similarity

classify-Proposition 1.2.1 Let h be a n-dimensional nilpotent Lie algebra and g = RZ ⊕D

k = RZ ⊕D (RY ⊕D 0 h) has D ∈ Der(k), D(k) ⊂ h and D0 ∈ Der(h) as above Byrenumbering (if necessary), we assume h1 = span{X1, , Xm} with 0 ≤ m < n Thenthe following conditions are equivalent:

1.3 Computing cohomology of Lie algebras having

small dimensional or codimensional derived gebras

al-Theorem 1.4 The Bettti numbers of Lie algebras of class Lie(n, 1) is described asfollows:

−



2m

i − i m+1

and α(p, 0) = α(0, p) is the number of elements of the set

k 2

k 2

k−2 2

k−2 2





 if 2 ≤ k ≤ n,2



n

n+1 2

n+1 2

n−1 2

n−1 2

k−1 2

k−1 2

k+1 2

k+1 2

to 7 have just been classified in [8] by Duong Minh Thanh and Ushirobira in 2014.Then, the second Betti number of nilpotent Jordan-type Lie algebra are also computedexplicitly The main results of the chapter have been published in 02 papers The firstpaper is “Betti numbers and the space of anti-symmetric derivations of quadratic Liealgebras of dimension less than or equal to 7” in 2015 and the second paper is “SecondBetti numbers of the nilpotent Jordan-type Lie algebras” in 2019 Both papers werepublished in Journal of Natural Sciences, Ho Chi Minh City University of Education.

2.1 Classification of the skew-symmetric derivations

of solvable Lie algebras of dimension ≤ 7

Theorem 2.1 The space of skew-symmetric derivations and the second Betti numbers

of seven-dimensional solvable quadratic Lie algebras is described in the following table:

11

On each space of skew-symmetric derivations of a quadratic Lie algebra, after moving the inner derivations, we get the second cohomology group From this wededuce the following corollary.

re-Corollary 2.1.1 The second cohomology of all indecomposable solvable quadratic Liealgebras of dimension less than or equal to 7 is given by the following table:

Đại số Lie Nhóm đối đồng điều thứ hai

2.2 Describing cohomology of solvable quadratic Lie

algebras of dimension in low

In this section, we compute the cohomology of solvable quadratic Lie algebras ofdimension in low by the method of calculating super-Poisson brackets

Theorem 2.2 The cohomology of indecoposable solvable Lie algebras of dimensionsless than or equal to 7 are described as follows: