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Thesis That I Is Bilateral

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Contents 1 Introduction 3 2 Hopf algebras 5 2.1 Algebras . 5 2.2 Coalgebras . 8 2.3 Lie algebras . 11 2.3.1 Universal enveloping algebra . 13 2.4 Bialgebras and Hopf algebras . 15 2.5 The Milnor-Moore theorem . 21 3 The Hopf algebra of rooted trees 29 3.1 Main de¯nitions . 29 3.2 Duality and Lie algebra of in¯nitesimal derivations . 38 3.3 The ladder tree Hopf algebra . 45 4 Insertion elimination Lie algebras 47 4.1 Derivations for the Hopf algebra of rooted trees . 47 4.2 Insertion Lie algebra . 49 4.3 Elimination Lie algebra . 52 4.4 The insertion-elimination Lie algebra . 54 5 The ladder Lie algebra 59 5.1 Motivations and generalities . 59 5.2 The structure of the Lie algebra LL . 65 5.3 Cohomology of LL ........................ 77 6 Extensions of Lie algebras 81 6.1 Extension of Lie algebras . 81 1 2 CONTENTS 6.1.1 Abelian extensions . 88 7 Appendix 1 93 7.1 Some elementary homological algebra . 93 7.2 Cohomology of Lie algebras . 96 7.2.1 Derivations . 100 8 Appendix 2 103 8.1 Cohomology of the lie algebra gl(n) . 103 Chapter 1 Introduction The goal of this dissertation is to begin the analysis of a class of combinatorial Lie algebras, which has been introduced by Alain Connes and Dirk Kreimer in [7], in their approach to the renormalization of perturbative quantum ¯eld theories [5, 6, 2], and [9] for the general framework. In their approach, a main role is played by the Hopf algebra structure de¯ned over the set of Feynman diagrams underlying the theory. The main features of such a Hopf algebra are captured by the Hopf algebra of rooted trees Hrt in its bare and dressed version. The properties of such a combinatorial Hopf algebra, becomes then crucial for the understanding of the combinatorics which is behind renormal- ization [12, 13, 14]. L Hrt is a commutative, Z¸0-graded and connected (ker² = i>0 Hi) Hopf al- ? gebra. By the Milnor-Moore theorem [21], its dual Hrt is isomorphic to the ? universal enveloping algebra of a Lie algebra P (Hrt), which can be faithfully represented into the Lie algebra of the in¯nitesimal characters of Hrt. From ? a more detailed analysis [7], it follows that the Lie algebra P (Hrt) has two other distinguished representations, D+ and D¡, where the former is the Lie algebra of the insertion operators and the latter the Lie algebra of the elim- ination operators. Since both D+ and D¡ are Lie algebras of derivations for + Hrt, it is natural to seek for a larger Lie algebra which contains both D and D¡ as sub Lie algebras. Such a Lie algebra is the insertion elimination Lie algebra LL introduced in [7]. Since the full insertion elimination Lie algebra is a quite complicated object, it is natural to seek for some distinguished sub Hopf algebra of Hrt, and then begin the analysis of the insertion elimination Lie algebra naturally attached to it. The choice of the ladder Hopf algebra HL is then quite a natural one. 3 4 CHAPTER 1. INTRODUCTION On the one hand, it is a fairly simple Hopf algebra, on the other hand it has a non trivial physical content, [3], [16]. The main achievement of the present work is the description of the structure of the insertion elimination Lie algebra, which is naturally associated to the ladder Hopf algebra of rooted trees. The outline of the present work is the following: In the ¯rst chapter, we introduce all the algebraic structures which are used in the following chapters. The second chapter is devoted to the detailed analysis of the Hopf algebra of rooted trees. In particular we give a summary of the results contained in [21], suitable to the present purposes. The third and the fourth chapters are the core of the present work. In the third chapter, we introduce and motivate the class of the insertion- elimination Lie algebras. The fourth chapter contains the analysis of the structure of the insertion-elimination Lie algebra which is naturally associ- ated to the Hopf algebra of the ladder rooted trees. There, we describe the structure of such a Lie algebra, its relations with some other well known in¯nite dimensional Lie algebra, and ¯nally, we describe its cohomology in some details . In the ¯fth chapter, we give a survey of the theory of the extensions of Lie algebras. There we carefully describe this theory, which is particularly rele- vant for what is discussed in chapter four. We conclude the exposition with two appendices, where the main results about the cohomology of Lie algebras and the cohomology of the general linear group are stated. Chapter 2 Hopf algebras In this chapter all the algebraic structures relevant for the present work will be introduced. A particular care will be taken of the class of connected graded Hopf algebras, for which a structure theorem will be proved. The last section contains a detailed account of the Milnor-Moore theorem, which is a key result for the topic of the present work. The references for the present chapter are: [11] and [21]. 2.1 Algebras We will introduce in this section the main notions from the theory of Hopf algebras. In what follows, we will assume that the base ¯eld k is the ¯eld of complex numbers C of the ¯eld of real numbers R and all the tensor products will be assumed over the ¯eld k. De¯nition 1 A k-algebra with unit is a k-vector space A together with two linear maps, multiplication m : A ­ A ¡! A and unit u : k ¡! A such that the following two diagrams are commutative: A ­ A ­ A ¡¡¡!m­id A ­ A ? ? ? ? id­my ym A ­ A ¡¡¡!m A 5 6 CHAPTER 2. HOPF ALGEBRAS (Associativity) u­id / o id­u k ­ AK A ­ A A ­ k KK ss KK ss KK m ss =» KK ss =» K% ² yss A (Unit) De¯nition 2 Let V and W be two k-vector spaces. De¯ne: ¿ : V ­ W ¡! W ­ V (2.1) saying that ¿(v ­ w) = w ­ v for each v ­ w 2 V ­ W . The map ¿ is called twist map. De¯nition 3 (A; u) is said to be commutative if ¿ ± m = m, i.e if the fol- lowing diagram is commutative: A ­ A m / A KK x; KK xx KK xx ¿ KK xxm K% xx A ­ A (Commutativity) Algebras over a given ¯eld k form a category, whose morphisms are de¯ned as follow: De¯nition 4 A morphism Á between two algebras (A1; m1; u1) and (A2; m2; u2) is a linear map Á : A1 ¡! A2 such that m2 ±(Á­Á) = Á±m1 and Á±u1 = u2. 2.1. ALGEBRAS 7 To show that algebras and their morphisms form a category we need only to check that: Proposition 1 Given Á1 : A1 ¡! A2 and Á2 : A2 ¡! A3, morphisms of algebras Á2 ± Á1 : A1 ¡! A3 is a morphism of algebras; Proof We need to check that: ¡ ¢ m3 ± (Á2 ± Á1) ­ (Á2 ± Á1) = (Á2 ± Á1) ± m1: ¡ ¢ ¡ ¢ m3 ± (Á2 ± Á1) ­ (Á2 ± Á1) = m3 ± Á2(Á1) ­ Á2(Á1) : Since Á2 is an al- gebra¡ morphism¢ the last term of the previous equality can be written as: Á2 m2(Á1 ­ Á1) . Since also Á1 is an algebra morphism we can rewrite the last formula as: Á2(Á1 ± m1), that is what we wanted to show. Ä From now on, by algebra will be meant associative algebra unless speci¯ed di®erently. Example 1 The ground ¯eld k with multiplication mk : k ­ k ¡! k (which corresponds to the natural multiplication) and unit uk : k ¡! k, de¯ned by uk(1) = 1, is a commutative k-algebra. For any given algebra (A; m; u), the unit u : k ¡! A is a morphism between the algebra (k; mk; uk) and the algebra (A; m; u). Example 2 Let (A; mA; uA) and (B; mB; uB) be two algebras over the ¯eld k. We can de¯ne¡ an algebra structure¢ on the tensor product A ­ B via the 0 0 0 0 0 following: m­ (a ­ b) ­ (a ­ b ) = mA(a ­ a ) ­ mB(b ­ b ), for a; a 2 A 0 and b; b 2 B. The unit is given by uA ­ uB. For any given algebra A, we can de¯ne the following notions: De¯nition 5 A given subvector space B ½ A is called an A sub-algebra if for each x; y 2 B, m(x; y) 2 B, i.e if and only if the restriction of the multiplication map to B takes values in B. Moreover we also have the following: De¯nition 6 A subalgebra I ½ A is called right (left) ideal if m(A; I) ½ I (m(I;A) ½ I). If I ½ A is called bilateral if it is left and right ideal. 8 CHAPTER 2. HOPF ALGEBRAS Proposition 2 If I is a bilateral ideal then the quotient space A = A=I has a natural structure of algebra: m : A ­ A ¡! A, m(x; y) = m(x; y). Proof The only thing we need to check is that the multiplication is well de¯ned. This follows from the hypothesis that I is bilateral. Ä Let us now introduce one more notion: De¯nition 7 An augmentation for A is an algebra morphism " : A ¡! k (where k is endowed with the algebra structure de¯ned in example 1). An algebra (A; m; u) with an augmentation map will be called augmented algebra. Proposition 3 Let A be an augmented algebra, with augmentation map ".
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