THÔNG TIN TÀI LIỆU
CONTENTS INTRODUCTION Chapter Some basic knowledge and results Chapter Real solvable Lie algebra classes having small dimensional or codimensional derived algebras and cohomology computation Chapter Some classes of solvable quadratic Lie algebras and cohomology computation 11 Chapter Some classes of solvable quadratic Lie superalgebras and cohomology computation 17 LIST OF AUTHOR’S PAPERS 21 BIBLIOGRAPHY 22 1 INTRODUCTION Lie groups and Lie algebras (collectively called Lie theory) was initiated by Sophus Lie, a Norwegian mathematician since the 70s of the 18th century and developed by many mathematicians around the world during nineteenth and early twentieth centuries such 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 and Lie algebras, quadratic Lie algebras, Lie superalgebras, quadratic Lie superalgebras, etc and cohomology computations often receive the attention of the mathematical community A Lie group is a group that is also a differentiable manifold, in which group operations are compatible with differentiable structures In the process of studying the left-invariant vector field of Lie groups, Lie algebra was born In Lie theory, quite interestingly, there is a one-to-one correspondence between the set of singly connected Lie groups and the set of Lie algebras In this thesis, we approach the problem of classification on the class of Lie algebras According to the Levi-Malshev Theorem, every finite-dimensional Lie algebra over a field of characteristic can be decomposed as a semi-direct product of a semi-simple subalgebra and a solvable ideal (see Levi [21] in 1905 and Malshev [12] in 1945) From that, 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 classifying semi-simple Lie algebras was thoroughly solved by Cartan [20] in 1894 (on C) and by Gantmacher [10] (on R) As for the class classification problem of solvable Lie algebras, although there are some 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, people often find ways to narrow the class of objects to be classified to make it easier to control 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) The third is combining both dimensional and structural classifications in a logical way On the direction of classification by dimensionality, Mubarakzyanov [13, 14] has classified solvable Lie algebras of dimension and over fields of characteristic zero Results for the six-dimensional case were also obtained by Mubarakzyanov [15] For decades, all classification problems of dimension 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 a combination of both directions: fixed dimensionality and structural addition In this thesis, we go in the direction of supplementing the structure and at the same time coordinating with the direction of fixed number of dimensions The problem we are interested in the thesis is studying and classifying real solvable Lie algebras having small codimensional derived algebras We have achieved some positive results as follows The first is the result of a thorough classification of the Lie(n+1, n) class of real Lie algebras that can be solved for n + in the given dimension n by the first cohomology space 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 + with a given n-dimensional derived algebra (denoted by Lie(n + 2, n)) is a wild problem (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 solvable Lie algebraic object with the additional structure of a non-degenerate and invariant bilinear form with respect to the Lie bracket These are called quadratic Lie algebras Since the last few decades, the problem of classifying quadratic Lie algebras and quadratic 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-simple Lie algebras However, for the class of solvable Lie algebras, the number of results on cohomology computations is still very limited and in the general case is still an open problem The cohomology description and computation of the Betti numbers of a solvable Lie algebra are of interest to the Mathematical community Some typical works can be mentioned here, which is Santharoubane’s [19] work on the cohomology 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 of solvable Lie algebras, solvable quadratic Lie algebras, solvable quadratic Lie superalgebras, and compute cohomology on several classes The thesis is titled: “Computing Cohomology and classification problems of Lie algebras, quadratic Lie superalgebras.” The successful implementation of the thesis topic has scientific significance and certain contributions to Lie theory in particular, to Algebra, Geometry and Topology in general Specifically, the topic has the following specific contributions: (1) We present an effective method to classify (n + 1)-dimensional real solvable Lie algebras having one-codimensional derived algebras provided that a full classification of n-dimensional nilpotent Lie algebras is given (2) In addition, the problem of classifying (n+2)-dimensional real solvable Lie algebras having two-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 (3) The cohomology of all solvable Lie algebras with one-dimensional derived algebra has been fully described (4) By applying Pouseele’s method concerning the extension of one-dimensional Lie algebra by the Heisenberg Lie algebra we obtain all Betti numbers bk for the general Diamond Lie algebra (5) Describe the cohomology of all low-dimensional quadratic Lie algebras, the second group of the Jordan-type quadratic Lie algebras, of elememtary quadratic Lie superalgebras has been classify 4 (6) Explicitly describe the space of skew-symmetric derivations of low-dimensional solvable quadratic Lie algebras From there, compute their second cohomology group (7) Applying double extension and general double extension combined with some results of classification of adjoint orbits of symplectic Lie algebras, we classify all solvable quadratic Lie algebras and solvable quadratic Lie superalgebras of dimension in low The results of the thesis have been reported at a number of national and international Mathematical Conferences: Scientific research conference for Master’s and PhD students of Ho Chi Minh City University of Education, October 2016 National Conference on Algebra - Geometry - Topology in Buon Ma Thuot City, Dak Lak, December 2016 International Mathematical Conference on Algebra - Geometry at Mahidol Uni- versity, Bangkok, Thailand (ICMA-MU 2017), May 2017 Mathematical and Applied Mathematics Conference at University of Science and Technology, Vietnam National University, Ho Chi Minh City, August 2017 Scientific Conference of the University of Natural Sciences, Vietnam National University, Ho Chi Minh City, 11th time, November 2018 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 to revise, correct and improve our work We sincerely thank you Chapter Some basic knowledge and results This chapter is devoted to the general introduction of the concepts of Lie groups, Lie algebras, 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 some related concepts such as ideal of a Lie algebra, Lie algebraic homomorphism, semi-direct product, solvable Lie algebras, nilpotent Lie algebras, cohomology of a Lie algebra Moreover, we recall the Levi-Malcev Theorem, the concept of the wild problem, and the weak similarity 0.2 Quadratic Lie algebras and cohomology In this section, we introduce the concept of quadratic Lie algebras, homomorphisms and some related concepts such as ideals, non-degenerate ideals, isometric isomorphisms, orthogonal direct sum Besides, we present the concept of the super-Poisson product 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 gradded super-Poisson bracket and apply it to the computation of cohomology of a quadratic Lie superalgebra Chapter Real solvable Lie algebra classes having small dimensional or codimensional derived algebras and cohomology computation In this chapter, we will study the classification problem and compute the cohomology of real Lie algebra classes solvable having small dimensional or codimensional derived algebras First, we will introduce a new result on the full classification of the class of real Lie algebras solvable whose derived algebras are of codimension Next, we introduce the concept of the wild problem and show that the problem of classifying real solvable Lie algebras having two codimensional derived algebras is proved to be wild also gives a classification of a subclass of this class when wildness is broken In addition, we will also present new results on the description of all cohomology of the class of real Lie algebras solvable having one dimensional derived algebras with coefficients on the base field and cohomology of a a special subclass of real solvable Lie algebras having one codimensional derived algebras, that is, the general Diamond Lie algebra class The main results of this chapter have been published in international papers: The first is “Classification of real solvable Lie algebras whose simply connected Lie groups have only zero or maximal dimensional coadjoint orbits ” in the Journal of Revista de la UM in 2016 (a separate case of Theorem 1.1), the second is “On the problem 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 the third is “Cohomology of some families of Lie algebras and quadratic Lie algebras” in East-West Journal of Mathematics in 2018 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 is equivalent to the problem of classifying outer derivations in the first cohomology space H (a, a) satisfying the equivalent conditions in Proposition 1.1.1, up to proportional similarity by the automorphism group of a Proposition 1.1.1 Let a be a n-dimensional nilpotent Lie algebra and g = RY ⊕D a with D ∈ Der(a) as above By renumbering (if necessary), we assume that a1 = span{X1 , , Xm } with ≤ m < n Then the following conditions are equivalent: g ∈ Lie(n + 1, n); g1 = a; span{Xm+1 , , Xn } ⊂ D(a) + a1 ; rank (dij )i>m = n − m, where (dij )n ∈ Matn (R) is the matrix representation of D corresponding to the base {X1 , , Xn }; ˜ : a/a1 → a/a1 is a linear isomorphic, where D ˜ is induced from D on the D quotient space a/a1 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 a extensive 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 square matrices, 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 special cases (see Belitskii et al [3], Section 3) Because of the wildnes in Theorem 1.2, We naturally consider the subclass Liead (n + 2, n) of Lie(n + 2, n) such that the derivation pair 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 classifying Liead (n + 2, n) with Lie derived algebra h is equivalent to the problem of classifying the equivalence class of outer derivatives of the first cohomology space of R⊕h satisfying the equivalence conditions in Proposition 1.2.1, up to proportional similarity Proposition 1.2.1 Let h be a n-dimensional nilpotent Lie algebra and g = RZ ⊕D k = RZ ⊕D (RY ⊕D0 h) has D ∈ Der(k), D(k) ⊂ h and D0 ∈ Der(h) as above By renumbering (if necessary), we assume h1 = span{X1 , , Xm } with ≤ m < n Then the following conditions are equivalent: g ∈ Lie(n + 2, n); g1 = h; span{Xm+1 , , Xn } ⊂ D(k) + D0 (h) + h1 ; ˜ + Im D ˜ , where D ˜ : k/h1 → k/h1 and D ˜ : h/h1 → h/h1 are reduced h/h1 = Im D homomorphisms of D and D0 respectively on quotient Lie algebra k/h1 and h/h1 1.3 Computing cohomology of Lie algebras having small dimensional or codimensional derived algebras Theorem 1.4 The Bettti numbers of Lie algebras of class Lie(n, 1) is described as follows: (i) b1 (aff(R)) = 1, b2 (aff(R)) = n − ; ≤ k ≤ n (ii) bk (aff(R) ⊕ Rn−2 ) = k 2m 2m ; (iii) bk (h2m+1 ) = bn−k (h2m+1 ) = − k−2 k m > 1, ≤ k ≤ m n − n − + , (iv) bk (h3 ⊕ Rn−3 ) = bn−k (h3 ⊕ Rn−3 ) = k k−2 n > 3, ≤ k ≤ n (v) n−2m−1 n−2m−1 bk (h )= )= 2m+1 ⊕ R bn−k (h2m+1 ⊕ R n−1 n−1 − ; n > 2m + > 3, ≤ k ≤ m = k k−2 n−2m−1 bk (h2m+1 ⊕ Rn−2m−1 )= ) = bn−k (h 2m+1 ⊕ R n − 2m − i=0 k−i 2m 2m ; where Ci = i − i i + m+1 − i − m+1 n > 2m + > 3, m + ≤ k ≤ n2 = min{k,2m+1} P Ci Consider the class of solvable Lie algebras g2n+2 (Λ) over C, whose the Lie bracket satisfies: [Y0 , Xi ] = λi Xi , [Y0 , Yi ] = −λi Yi , [Xi , Yi ] = λi X0 , i = 1, 2, , n Given two positive integers p and q, let α(p, q) be the number of elements of the set ( ) p q X X λi1 , , λip , λj1 , , λjq : λil = λjl l=1 l=1 and α(p, 0) = α(0, p) is the number of elements of the set ( ) p X λi1 , , λ ip : λil = l=1 When p < or q < 0, α(p, q) = is a convention Then we have the result of Betti number as follows: 10 Theorem 1.5 With the above notations, the Betti numbers of the Diamond Lie algebra are described as follows: (i) If k = 0, 1, 2n + or 2n + then bk (g2n+2 ) = (ii) If ≤ k ≤ n then X bk (g2n+2 ) = α(p, q) + p+q=k (iii) Nếu k = n + X bk (g2n+2 ) = X p+q=k−1 α(p, q)+2 p+q=n+1 (iv) If n + ≤ k ≤ 2n then X bk (g2n+2 ) = α(p, q) + p+q=k−1 X X α(p, q) − p+q=k−2 p+q=n X p+q=k−2 α(p, q)−2 p+q=n−1 α(p, q) − X p+q=k α(p, q) p+q=k−3 X α(p, q)−2 X α(p, q) − α(p, q) − X α(p, q) p+q=n−2 X α(p, q) p+q=k+1 Here we consider a few special cases to illustrate the above results as follows: (i) λ1 = λ2 = = λn : if k = 0, k = 1, n n n n − if ≤ k ≤ n, k k−2 k−2 k 2 2 n n n n − bk (g2n+2 ) = if k = n + 1, n+1 n+1 n−1 n−1 2 n n n n − if n + ≤ k ≤ 2n, k−1 k−1 k+1 k+1 2 2 k = 2n + 1, k = 2n + (ii) λ1 = λ2 = = λn−1 = λ, λn = 2λ: n−1 n−1 n−1 n−1 bk (f) = + k k k k −1 −1 2 2 n−1 n−1 + k−1 (−1) +1 (−1)k−1 −3 k k + + 2 2 Thus, we obtain all the Betti numbers of g2n+2 through the Betti number of f Chapter Some classes of solvable quadratic Lie algebras and cohomology computation In this chapter, we discuss solvable quadratic Lie algebras, i.e solvable Lie algebras equipped with a non-degenerate invariant symmetric bilinear form First, we will introduce the concept of double extension and T ∗ extension as important tools for solving classification problem of solvable quadratic Lie algebras Next, we will introduce the spatial description of the anti-symmetric derivations and explicitly compute the Betti number of the class of solvable Lie algebras of dimension less than or equal to 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 computed explicitly The main results of the chapter have been published in 02 papers The first paper is “Betti numbers and the space of anti-symmetric derivations of quadratic Lie algebras of dimension less than or equal to 7” in 2015 and the second paper is “Second Betti numbers of the nilpotent Jordan-type Lie algebras” in 2019 Both papers were published 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 ≤ 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 12 Quadratic Lie algebras Skew-symmetric derivations The second Betti numbers ⊥ g4 ⊕ C3 ⊥ g5 ⊕ C2 y z −x1 −x2 −x3 x y −b t −x1 −x2 ⊥ g6,1 ⊕ C ⊥ g6,2 ⊕ C 0 x 0 0 0 0 −z −y x1 x2 0 −x 0 0 0 −u 0 u 0 0 w v z 0 0 −x 0 0 −c 0 0 −t b −x −y x1 c −z −y1 0 x y1 0 −y2 0 u A 0 t t D = B −A −C C 0 a c y 0 x3 −w −v x2 y2 −u 11 0 b d z 0 0 0 0 0 0 D= 0 t −a −b x1 0 h −c −d x2 −t −h −y −z x3 −x1 −x2 −x3 0 0 13 Quadratic Lie algebras Skew-symmetric derivations The second Betti numbers a a x 0 0 b y 0 0 0 0 0 0 ⊥ g6,3 ⊕ C D= z −a 0 x1 0 t −b −a x2 −z −t −x −y x3 −x1 −x2 −x3 0 0 2x b −a 0 0 x −c b 0 0 x 0 0 g7,1 D= e 0 −b 0 y −2x 0 0 d −b −x −y −d −e a c −x b −a 0 0 0 −c y 0 0 0 0 0 0 g7,2 D= e 0 −y 0 0 d −b 0 0 x 0 0 −d −x −e a c c −a 0 0 −c b 0 0 0 0 0 0 g7,3 D = −b a 0 0 0 −d e b −c 0 d −f −a c 0 −e f 0 a −b In the above table, the basis on the vector spaces C, C2 and C3 is the orthonormal basis; 14 x, y, z, t, h, a, b, c, d, e, f, u, v, w, x1 , x2 , x3 , y1 , y2 are real numbers; A is one of the square matrices × with zero trace, B is the skew-symmetric × matrix, C = y1 y2 y3 with y1 , y2 , y3 ∈ C On each space of skew-symmetric derivations of a quadratic Lie algebra, after removing the inner derivations, we get the second cohomology group From this we deduce the following corollary Corollary 2.1.1 The second cohomology of all indecomposable solvable quadratic Lie algebras of dimension less than or equal to is given by the following table: Đại số Lie Nhóm đối đồng điều thứ hai tồn phương g4 {0} g5 span {[X1∗ ∧ Z1∗ − X2∗ ∧ Z2∗ ] , [X1∗ ∧ Z2∗ ] , [X2∗ ∧ Z1∗ ]} [X ∗ ∧ Z ∗ ] , [X ∗ ∧ Z ∗ ] , [X ∗ ∧ Z ∗ ] , [X ∗ ∧ Z ∗ ] , [X ∗ ∧ Z ∗ ] , 3 2 span [X ∗ ∧ Z ∗ ] , [X ∗ ∧ Z ∗ − X ∗ ∧ Z ∗ ] , [X ∗ ∧ Z ∗ − X ∗ ∧ Z ∗ ] 1 3 2 3 g6,1 g6,2(±1) span {[X1∗ ∧ Z1∗ ] , [X1∗ ∧ Z2∗ ] , [X2∗ ∧ Z1∗ ]} g6,2(λ) , λ 6= ±1 span {[X1∗ ∧ Z1∗ ]} g6,3 span {[X2∗ ∧ Z1∗ ]} g7,1 span {[2X1∗ ∧ Z1∗ + X2∗ ∧ Z2∗ + X3∗ ∧ Z3∗ ] , [X1∗ ∧ X3∗ ]} g7,2 span {[X2∗ ∧ X3∗ ] , [T ∗ ∧ Z2∗ ]} g7,3 {0} 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 of dimension in low by the method of calculating super-Poisson brackets Theorem 2.2 The cohomology of indecoposable solvable Lie algebras of dimensions less than or equal to are described as follows: 15 H1 H2 H3 bk g4 X1∗ {0} X2∗ ∧ Z1∗ ∧ Z2∗ (1,0,1) g5 X1∗ , Z1∗ ∧ X2∗ , Z2∗ ∧ X1∗ , T ∗ ∧ X1∗ ∧ Z1∗ , T ∗ ∧ X1∗ ∧ Z2∗ , (2,3,3,2) X2∗ g6,1 T ∗ ∧ X2∗ ∧ Z1∗ X1∗ , X1∗ ∧ Z2∗ , X1∗ ∧ Z3∗ , X1∗ ∧ X2∗ ∧ Z1∗ , X1∗ ∧ X2∗ ∧ Z2∗ , X2∗ , X2∗ ∧ Z1∗ , X2∗ ∧ Z3∗ , X1∗ ∧ X2∗ ∧ Z3∗ , X1∗ ∧ X3∗ ∧ Z1∗ , X3∗ X3∗ ∧ Z1∗ , X3∗ ∧ Z2∗ , X1∗ ∧ X3∗ ∧ Z2∗ , X2∗ ∧ X3∗ ∧ Z1∗ , X1∗ ∧ Z1∗ − X3∗ ∧ Z3∗ , X1∗ ∧ Z2∗ ∧ Z3∗ , X2∗ ∧ Z1∗ ∧ Z3∗ , X2∗ ∧ Z2∗ − X3∗ ∧ Z3∗ X3∗ ∧ Z1∗ ∧ Z2∗ , (3,8,12,8,3) X1∗ ∧ Z1∗ ∧ Z2∗ + X3∗ ∧ Z2∗ ∧ Z3∗ , X1∗ ∧ Z1∗ ∧ Z3∗ − X2∗ ∧ Z2∗ ∧ Z3∗ , X2∗ ∧ Z1∗ ∧ Z2∗ − X3∗ ∧ Z1∗ ∧ Z3∗ g6,2 (λ) X3∗ X1∗ ∧ Z1∗ (1,1,2,1,1) X1∗ ∧ Z1∗ ∧ Z3∗ − λX2∗ ∧ Z2∗ ∧ Z3∗ (λ 6= ±1, 0) g6,2 (1) X1∗ ∧ X3∗ ∧ Z1∗ , X3∗ X1∗ ∧ Z1∗ , X1∗ ∧ Z2∗ , X1∗ ∧ X3∗ ∧ Z1∗ , X1∗ ∧ Z2∗ ∧ Z3∗ , X2∗ ∧ Z1∗ X2∗ ∧ Z1∗ ∧ Z3∗ , (1,3,4,3,1) X1∗ ∧ Z1∗ ∧ Z3∗ − X2∗ ∧ Z2∗ ∧ Z3∗ g6,2 (−1) X3∗ X1∗ ∧ Z1∗ , X1∗ ∧ X2∗ , X1∗ ∧ X3∗ ∧ Z1∗ , X1∗ ∧ X2∗ ∧ Z3∗ , Z1∗ ∧ Z2∗ Z1∗ ∧ Z2∗ ∧ Z3∗ , (1,3,4,3,1) X1∗ ∧ Z1∗ ∧ Z3∗ + X2∗ ∧ Z2∗ ∧ Z3∗ g6,3 X3∗ X1∗ ∧ Z2∗ X1∗ ∧ X3∗ ∧ Z1∗ , X1∗ ∧ Z2∗ ∧ Z3∗ , (1,1,3,1,1) X1∗ ∧ Z1∗ ∧ Z3∗ + X2∗ ∧ Z2∗ ∧ Z3∗ g7,1 X3∗ , X1∗ ∧ X3∗ , Z1∗ ∧ Z2∗ Z1∗ g7,2 X3∗ , X3∗ (2,2,3,3,2,2) X1∗ ∧ Z1∗ ∧ Z2∗ − X3∗ ∧ Z2∗ ∧ Z3∗ X2∗ ∧ X3∗ , X1∗ ∧ Z1∗ Z2∗ g7,3 Z1∗ ∧ Z2∗ ∧ Z3∗ , X1∗ ∧ X3∗ ∧ Z2∗ , T ∗ ∧ X2∗ ∧ X3∗ , T ∗ ∧ X2∗ ∧ Z2∗ , (2,2,3,3,2,2) X1∗ ∧ Z1∗ ∧ Z3∗ − T ∗ ∧ Z2∗ ∧ Z3∗ {0} X1∗ ∧ X2∗ ∧ Z3∗ , X1∗ ∧ Z1∗ ∧ Z3∗ − X2∗ ∧ Z2∗ ∧ Z3∗ − 2T ∗ ∧ Z1∗ ∧ Z2∗ (1,0,2,2,0,1) 16 2.3 The second Betti number of the Jordan-type quadratic Lie algebras Given a nilpotent Jordan block Jn , vector space q = C2n with canonical basis {X1 , , Xn , Y1 , , Yn } Given a linear mapping C : q → q with a matrix in canonical basis Jn C= t −Jn Then C ∈ o(2n) Call j2n = q ⊕ CX0 ⊕ CY0 the one-step double extension of q by C Consider the vector space q = C2n+1 with canonical basis {X1 , , Xn , T, Y1 , , Yn } Given a linear mapping C : q → q with a matrix in canonical basis Jn+1 M , C= −Jnt where M = (mij ) is the matrix (n + 1) × n with all zero terms except mn+1,n = −1 Then C ∈ o(2n + 1) Call j2n+1 = q ⊕ CX0 ⊕ CY0 the one-step double extension of q by C Theorem 2.3 With the above notation, the second Betti number of all nilpotent Jordan-type Lie algebras is given by the following formula: hni + với n ≥ hni n + 1 (ii) b2 (j3 ) = b2 (g5 ) = 3, b2 (j2n+1 ) = + + với n ≥ 2 (i) b2 (j4 ) = b2 (g6,1 ) = 8, b2 (j2n ) = n + Chapter Some classes of solvable quadratic Lie superalgebras and cohomology computation Chapter is also the last chapter of the thesis We present the results of classifications of solvable seven-dimensional quadratic Lie superalgebras and solvable eightdimensional quadratic Lie superalgebras with the six-dimensional even part (Theorem 3.1 and 3.2) Part of the new results of these two items have been published in the paper “Classification of eight-dimensional solvable Lie super algebras with six-dimensional indivisible even parts” published in Journal of Natural Sciences, Ho Chi Minh City University of Education in 2016 In addition, the results of the first and second cohomologies of the class of elementary quadratic Lie superalgebras are also introduced (Theorem 3.3) These new results were published in the paper “The Second Cohomology Group of Elementary Quadratic Lie Superalgebras” published in the East-West Journal of Mathematics in 2017 3.1 Classification of solvable seven-dimensional incomposable solvable quadratic Lie superalgebras Theorem 3.1 Let g be a seven-dimensional incomposable solvable Lie superalgebra (i) If g has a trivial odd part, then g is isometrically isomorphic to one of the following quadratic Lie algebras: 17 18 g7,1 = span {X1 , X2 , X3 , T, Z1 , Z2 , Z3 } : [X3 , X2 ] = X1 , [X3 , T ] = X2 , [X3 , Z1 ] = −Z2 , [X3 , Z2 ] = −T, [X2 , Z1 ] = [T, Z2 ] = Z3 g7,2 = span {X1 , X2 , X3 , T, Z1 , Z2 , Z3 } : [X3 , X1 ] = X1 , [X3 , T ] = X2 , [X3 , Z1 ] = −Z1 , [X3 , Z2 ] = −T, [X1 , Z1 ] = [T, Z2 ] = Z3 g7,3 = span {X1 , X2 , X3 , T, Z1 , Z2 , Z3 }: [X3 , X1 ] = X1 , [X3 , X2 ] = −X2 , [X3 , Z1 ] = −Z1 , [X3 , Z2 ] = Z2 , [X1 , Z1 ] = [Z2 , X2 ] = Z3 , [X1 , X2 ] = T, [X1 , T ] = −Z2 , [X2 , T ] = Z1 (ii) If g has a non-trivial odd part, then g is isometrically isomorphic to one of the following quadratic Lie superalgebras: gs7,1 (λ, µ) = g0 ⊕ g1 = span {X1 , X2 , T, Z1 , Z2 } ⊕ span {Y1 , T1 }: [X1 , X2 ] = T, [X1 , T ] = −Z2 , [X2 , T ] = Z1 , [X1 , T1 ] = λY1 , [X2 , T1 ] = µY1 , [T1 , T1 ] = λZ1 + µZ2 gs7,2 (λ, µ) = g0 ⊕ g1 = span {X1 , X2 , T, Z1 , Z2 } ⊕ span {Y1 , T1 }: [X1 , X2 ] = T, [X1 , T ] = −Z2 , [X2 , T ] = Z1 , [X1 , Y1 ] = λY1 , [X1 , T1 ] = −λT1 , [X2 , Y1 ] = µY1 , [X2 , T1 ] = −µT1 , [Y1 , T1 ] = λZ1 + µZ2 3.2 Classification of solvable eight-dimensional incomposable solvable quadratic Lie superalgebras with six-dimensional even parts The main result in this section is classification of eight-dimensional incomposable solvable quadratic Lie superalgebras with six-dimensional even parts: Theorem 3.2 Let g be an eight-dimensional solvable quadratic Lie superalgebra with six-dimensional even part If g is indecomposable, then g is isometrically isomorphic to one of the following quadratic Lie superalgebras: 19 gs8,6,1 (λ, µ, ν) = g0 ⊕ g1 = g6,1 ⊕ span{Y1 , T1 }: [X1 , T1 ] = −λT1 , [X2 , Y1 ] = µY1 , [X2 , T1 ] = −µT1 , [X3 , Y1 ] = νY1 , [X3 , T1 ] = −νT1 , [Y1 , T1 ] = λZ1 +µZ2 +νZ3 , where λ, µ, ν are non-zero complex numbers gs8,6,2 (λ, µ) = g0 ⊕g1 = g6,1 ⊕span{Y1 , T1 }: [X3 , T1 ] = Y1 , [X1 , T1 ] = λY1 , [X2 , T1 ] = µY1 , [T1 , T1 ] = λZ1 + µZ2 + Z3 , where λ, µ are complex numbers gs8,6,3 (λ) = g0 ⊕ g1 = g6,2 (λ) ⊕ span{Y1 , T1 }: [X3 , T1 ] = Y1 , [T1 , T1 ] = Z3 , where λ is a non-zero complex number gs8,6,4 (λ, µ) = g0 ⊕ g1 = g6,2 (λ) ⊕ span{Y1 , T1 }: [X3 , Y1 ] = µY1 , [X3 , T1 ] = −µT1 , [Y1 , T1 ] = µZ3 , where λ, µ are non-zero complex numbers 1 gs8,6,5 (λ) = g0 ⊕ g1 = g6,2 (λ) ⊕ span{Y1 , T1 }: [X3 , Y1 ] = Y1 , [X3 , T1 ] = − T1 , 2 [Z1 , T1 ] = Y1 , [Y1 , T1 ] = Z3 , [T1 , T1 ] = X1 , where λ is a non-zero complex number 1 gs8,6,6 (µ) = g0 ⊕ g1 = g6,2 ⊕ span{Y1 , T1 }: [X3 , Y1 ] = Y1 , [X3 , T1 ] = − T1 , 2 [Z1 , T1 ] = Y1 , [Z2 , T1 ] = Y1 , [Y1 , T1 ] = Z3 , [T1 , T1 ] = X1 + µX2 , where µ is a non-zero complex number gs8,6,7 = g0 ⊕ g1 = g6,3 ⊕ span{Y1 , T1 }: [X3 , T1 ] = Y1 , [T1 , T1 ] = Z3 gs8,6,8 (λ) = g0 ⊕g1 = g6,3 ⊕span{Y1 , T1 }: [X3 , Y1 ] = λY1 , [X3 , T1 ] = −λT1 , [Y1 , T1 ] = λZ3 , where λ is a non-zero complex number 1 gs8,6,9 = g0 ⊕ g1 = g6,3 ⊕ span{Y1 , T1 } : [X3 , Y1 ] = Y1 , [X3 , T1 ] = − T1 , [Z2 , T1 ] = 2 Y1 , [Y1 , T1 ] = Z3 , [T1 , T1 ] = X2 10 gs8,6,10 (λ) = g0 ⊕g1 = (g4 ⊕ C2 )⊕span{Y1 , T1 }: [X, T1 ] = λY1 , [U1 , T1 ] = Y1 , [U2 , T1 ] = iY1 , [T1 , T1 ] = λZ + U1 + iU2 , where {U1 , U2 } is the basis of C2 , λ is a non-zero complex number 11 gs8,6,11 (λ, µ) = g0 ⊕ g1 = (g4 ⊕ C2 ) ⊕ span{Y1 , T1 }: [X, Y1 ] = λY1 , [X, T1 ] = −λT1 , [U1 , Y1 ] = µY1 , [U1 , T1 ] = −µT1 , [U2 , Y1 ] = iµY1 , [U2 , T1 ] = −iµT1 , [Y1 , T1 ] = λZ+µU1 +iµU2 , where {U1 , U2 } is the basis of C2 , λ, µ non-zero complex numbers 20 12 gs8,6,12 (λ, µ) = g0 ⊕ g1 = (g5 ⊕ C) ⊕ span{Y1 , T1 }: [X1 , T1 ] = λY1 , [X2 , T1 ] = µY1 , [U, T1 ] = Y1 , [T1 , T1 ] = λZ1 + µZ2 + U, where {U } is the basic of C, λ, µ are complex numbers not all zero 13 gs8,6,13 (λ, µ, ν) = g0 ⊕ g1 = (g5 ⊕ C) ⊕ span{Y1 , T1 }: [X1 , Y1 ] = λY1 , [X1 , T1 ] = −λT1 , [X2 , Y1 ] = µY1 , [X2 , T1 ] = −µT1 ,[U, Y1 ] = νY1 , [U, T1 ] = −νT1 , [Y1 , T1 ] = λZ1 + µZ2 + νU, where {U } is the basic of C, λ and µ are complex numbers not all zero, ν 6= 3.3 The first and second cohomologies of elementary quadratic Lie superalgebras In this section, by applying the results on the relationship between the coboundary operator and the graded super-Poisson bracket, we will present the description of the first and second cohomologies of elementary quadratic Lie superalgebras havings been classified in [7] Recall that there are exactly three elementary quadratic Lie superalgebras: gs4,1 , gs4,2 , gs6 Theorem 3.3 With the above notations, the first and second cohomologies of elementary quadratic Lie superalgebras are defined as follows (i) H (gs4,1 , C) = span Y0∗ , Y1∗ , H (gs4,1 , C) = span Y0∗ ⊗ X1∗ , X1∗ Y1∗ − 2X0∗ ∧ Y0∗ where {X0∗ , Y0∗ , X1∗ , Y1∗ } is the dual basis of {X0 , Y0 , X1 , Y1 } (ii) H (gs4,2 , C) = span Y0∗ , H (gs4,2 , C) = {0} where {X0∗ , Y0∗ , X1∗ , Y1∗ } is the dual basis of {X0 , Y0 , X1 , Y1 } (iii) H (gs6 , C) = span Y0∗ , Z1∗ , T1∗ , n h 2 i h ∗ 2 i H (gs6 , C) = span Y0∗ ⊗ X1∗ , Y0∗ ⊗ Y1∗ , Z1∗ , T1 , X1∗ Z1∗ − X0∗ ∧ Y0∗ , Y1∗ T1∗ − X0∗ ∧ Y0∗ where {X0∗ , Y0∗ , X1∗ , Y1∗ , Z1∗ , T1∗ } is the dual basis of {X0 , Y0 , X1 , Y1 , Z1 , T1 } 21 LIST OF AUTHOR’S PAPERS Cao Trần Tứ Hải Dương Minh Thành (2015), “Số Betti không gian đạo hàm phản xứng đại số Lie toàn phương giải số chiều ≤ 7”, Tạp chí Khoa học Tự nhiên, Trường ĐHSP TP.HCM, số 5(70), tr 86-96 Cao Trần Tứ Hải Dương Minh Thành (2016), “Phân loại siêu đại số Lie toàn phương giải chiều với phần chẵn bất khả phân chiều”, Tạp chí Khoa học Tự nhiên, Trường ĐHSP TP.HCM, số 12 (90), tr 162-174 Cao Trần Tứ Hải Dương Minh Thành (2019), “Số Betti thứ hai đại số Lie lũy linh kiểu Jordan”, Tạp chí Khoa học Tự nhiên, Trường ĐHSP TP.HCM số 16 (16), tr 877-890 Le Anh Vu, Ha Van Hieu, Nguyen Anh Tuan, Cao Tran Tu Hai, Nguyen Thi Mong Tuyen (2016), “Classification of real solvable Lie algebras whose simply connected Lie groups have only zero or maximal dimensional coadjoint orbits”, Revista de la UMA, Vol 57, no 2, 119-143 Cao Tran Tu Hai, Duong Minh Thanh and Le Anh Vu (2017), “The Second Cohomology Group of Elementary Quadratic Lie Superalgebras”, East-West Journal of Mathematics, Vol 19 , no 1, 32-42 Cao Tran Tu Hai, Duong Minh Thanh and Le Anh Vu (2018), “Cohomology of some families of Lie algebras and quadratic Lie algebras”, East-West Journal of Mathematics, Vol 20, no 2, 188-201 Le Anh Vu, Cao Tran Tu Hai, Duong Quang Hoa, Nguyen Anh Tuan, Vo Ngoc Thieu (2022), “On the problem of classifying solvable Lie algebras having small codimensional derived algebras”, Communications in Algebras, Vol 50, no 9, 3775–3793 BIBLIOGRAPHY [1] W Bai, W Liu (2017), “Cohomology of Heisenberg Lie Superalgebras”, J of Math Physics 58 021701 [2] I Bajo, S Benayadi, and M Bordemann (2007), “Generalized double extension and descriptions of quadratic Lie superalgebras”, arXiv:0712.0228v1 [3] G R Belitskii, R Lipyanski, V V Sergeichuk, “Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild”, Linear Algebra Appl 407 (2005) 249–262 [4] M Bordemann (1997), “Nondegenerate invariant bilinear forms on nonassociative algebras”, Acta Math Univ Comenianac LXVI, no 2, 151 – 201 [5] P Donovan, M R Freislich (1973), The representation theory of finite graphs and associated algebras,Carleton Lecture Notes 5, Ottawa [6] M T Duong (2014), “The Betti number for a family of solvable Lie algebras”, Bulletin of the Malaysian Mathematical Sciences Society, 40, Issue 2, 735–746 [7] M T Duong and R.Ushirobira (2014), “Singular quadratic Lie superalgebras”, J of Algebra 407, 372–412 [8] M T Duong and R.Ushirobira (2014), “Solvable quadratic Lie algebras of dimension ≤ 8”, arXiv:1407.6775v1 [9] V Futorny, T Klymchuk, A P Petravchuk, V V Sergeichuk (2018), “Wildness of the problems of classifying two-dimensional spaces of commuting linear operators and certain Lie algebras”, Linear Algebra Appl 536 201–209 [10] F R Gantmacher (1939), “On the classification of real simple Lie groups”, Sb Math., (2), 217–250 22 23 [11] N Jacobson (1944), “Schur’s Theorem on commutative matrices”,Bull Amer Math Soc 50 (6) 431–436 [12] Malcev A I (1945), “On solvable Lie algebras”, Izv Ross Akad Nauk Ser Mat., (5), 329–356 [13] G M Mubarakzyanov (1963), “On solvable Lie algebras”, Izv Vyssh Uchebn Zaved Mat 114–123 [14] G M Mubarakzyanov (1963), “Classification of real structures of Lie algebras of fifth order”, Izv Vyssh Uchebn Zaved Mat 99–106 [15] Mubarakzyanov, G M (1963), “Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element”, Izv Vyssh Uchebn Zaved Mat 4:104– 116 [16] T D Pham, M T Duong, A V Le (2012), “Solvable quadratic Lie algebras in low dimensions”, East-West J of Math 14 (2) 208-218 [17] G Pinczon and R Ushirobira (2007), "New applications of graded Lie algebras to Lie algebras, generalized Lie algebras and Cohomology", J Lie Theory 17 (3), 633 – 668 [18] H Pouseele (2005), “On the cohomology of extensions by a Heisenberg Lie algebra”, Bull Austral.Math Soc 71, 459-470 [19] L J Santharoubane (1983), “Cohomology of Heisenberg Lie algebras”, Proc Amer Math Soc 87, 23-28 [20] E J Cartan (1894), Sur la structure des groupes de transformations finis et continus, Ph.D Thesis, Nony, Paris [21] E E Levi (1905), “Sulla struttura dei gruppi finiti e continui”, Atti Accad Sci Torino Cl Sci Fis Mat Natur., 40, 551–565
Ngày đăng: 25/06/2023, 13:28
Xem thêm: