arXiv:2001.06621v1 [math.RA] 18 Jan 2020 ON PRO-SOLVABLE LIE ALGEBRAS WITH MAXIMAL PRO-NILPOTENT IDEALS m 0 AND m 2 . K.K. ABDURASULOV 1 , B.A. OMIROV 1,2 , G.O. SOLIJANOVA 2 Abstract. The paper is devoted to the study of pro-solvable Lie algebras whose maximal pro- nilpotent ideal is either m 0 or m 2 . Namely, we describe such Lie algebras and establish their com- pleteness. Triviality of the second cohomology group for one of the obtained algebra is established. 1. Introduction Pro-solvable and pro-nilpotent Lie algebras are an important and interesting class of Lie algebras, which generalize the class of solvable and nilpotent Lie algebras, respectively. The definition of pro-solvable (respectively, pro-nilpotent) Lie algebra L is divided into two parts; the first part of the definition is the condition of potentially solvability (respectively, nilpotency) and the second part of the definition is dim L [i] /L [i+1] < ∞ (respectively, dim L i /L i+1 < ∞) for any i ≥ 1. Since the study of infinite-dimensional solvable and pro-nilpotent Lie algebras is a complex problem, they should be studied by adding additional restrictions. One of such important restrictions for the study of solvable Lie algebras is fixing its nilradical, while for nilpotent Lie algebras one of successful restrictions is condition on dimensions of L i /L i+1 . Here we apply these approaches for the study of pro-solvable Lie algebras by fixing their maximal pro-nilpotent ideals. Due to works [5] and [6] we have some examples of pro-nilpotent Lie algebras. Among pro-nilpotent Lie algebras we consider those which have the most simple structure, they are m 0 , m 1 and m 2 . It is known that unique pro-solvable Lie algebra with maximal pro-solvable Lie algebra m 1 = {e i | i ∈ N} is algebra W ≥0 = {e i | i ∈ N ∪{0}}. Note that for the algebras m 0 , m 1 and m 2 the conditions dim(L 1 /L 2 )=2, dim(L i /L i+1 )=1 i ≥ 1 hold true. Similar to finite-dimensional solvable Lie algebras considered in [1] we focus our study for pro- solvable Lie algebras with maximal pro-nilpotent ideals and maximal dimension of complementary subspace to the ideals. In this work we describe pro-solvable Lie algebras generated by m 0 (respectively, m 2 ) and its special kinds of derivations under the condition that complementary subspace to m 0 has maximal dimension. We also prove that such algebras are complete. Furthermore, the triviality of the second cohomology group of one of them is proved. Throughout the paper we consider complex Lie algebras with countable basis such that any element of the algebra can be represented as a finite linear combination of basis elements. Moreover, by maximal ideal we shall assume maximal by including ideal. 2. Preliminaries In this section we give necessary definitions and preliminary results. Definition 2.1. An algebra (L, [·, ·]) is called a Lie algebra if it satisfies the properties [x,x]=0, [x, [y,z ]] + [y, [z,x]] + [z, [x,y]] = 0 for all x,y,z ∈ L. The second condition is called Jacobi identity. Definition 2.2. A linear map d : L → L of an algebra (L, [−, −]) is said to be a derivation if for all x,y ∈ L, the following condition holds: d([x,y]) = [d(x),y]+[x,d(y)] . For a given x ∈ L, ad x denotes the map ad x : L → L such that ad x (y)=[x,y], ∀y ∈ L. One can check that a map ad x is a derivation. We call this kind of derivations inner derivations. 2010 Mathematics Subject Classification. 17B40, 17B56, 17B65. Key words and phrases. Lie algebra, potentially nilpotent Lie algebra, pro-nilpotent Lie algebra, cohomology group. 1