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Introduction to Lie Algebras

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Introduction to Lie Algebras Introduction to Lie algebras N. Perrin November 9, 2015 2 Contents I Lie algebras 7 1 Algebras 9 1.1 Basics .......................................... ... 9 1.2 Derivations ..................................... ..... 10 1.3 Productofalgebras............................... ....... 10 1.4 Restriction and extension of the scalars . ............. 10 1.5 Exercices ....................................... 11 2 Lie algebras 13 2.1 Definition ....................................... 13 2.2 Adjointrepresentation . ......... 14 2.3 Ideals .......................................... 14 2.4 AbelianLieagebras ............................... ...... 15 2.5 Derived, central ascending and central descending series ................. 15 2.6 Extensions...................................... ..... 17 2.7 Semidirectproducts .............................. ....... 18 2.8 Exercices ....................................... 19 3 Envelopping algebra I 21 3.1 Definition ....................................... 21 3.2 Enveloppingalgebraofaproduct . .......... 22 3.3 Envelopping algebra of the opposite Lie algebra . ............... 23 3.4 Exercices ....................................... 24 4 Representations 25 4.1 Definition ....................................... 25 4.2 Tensorproductofrepresentations . ............ 26 4.3 Representations in the space of morphisms . ............. 26 4.4 Invariants ...................................... ..... 27 4.5 Invariantbilinearforms . ......... 28 4.6 Casimirelement .................................. ..... 30 4.7 Exercices ....................................... 32 5 Nilpotent Lie algebras 35 5.1 Definition ....................................... 35 5.2 Engel’sTheorem .................................. ..... 36 5.3 Maximalnilpotentideal . ........ 37 5.4 Exercices ....................................... 39 3 4 CONTENTS 6 Semisimple and nilpotent elements 41 6.1 Semisimpleendomorphisms . ........ 41 6.2 Semisimple and nilpotent decomposition . ............. 41 6.3 Exercice........................................ 44 7 Solvable Lie algebras 45 7.1 Definition ....................................... 45 7.2 Radical ......................................... 46 7.3 Lie’sTheorem.................................... ..... 46 7.4 Cartan’scriterion............................... ........ 47 7.5 Exercices ....................................... 49 8 Semisimple Lie algebras 51 8.1 Definition ....................................... 51 8.2 First caracterisation of semisimple Lie algebras . ................. 51 8.3 Semisimplicity of representations . ............. 53 8.4 SimpleLiealgebras............................... ....... 54 8.5 Jordan-Chevalley decomposition . ........... 55 8.6 ExamplesofsemisimpleLiealgebras . ........... 56 8.7 Exercices ....................................... 60 II Classification of complex semisimple Lie algebras 63 9 Cartan subalgebras 65 9.1 definition ....................................... 65 9.2 Regularelements ................................. ...... 65 9.3 Cartan subalgebras associated with regular elements . ................. 66 9.4 ConjugacyofCartansubalgebras . .......... 67 9.5 Thesemisimplecase ............................... ...... 69 9.6 Exercices ....................................... 71 10 The Lie algebra sl2 73 10.1 Definition, standard basis and simplicity . ............... 73 10.2 Representations, weights and primitive elements . ................. 74 10.3 The subrepresentation generated by a primitive element ................. 74 10.4 Structure of finite dimensional representations . .................. 75 10.5Exercices ...................................... ..... 77 11 Root systems 79 11.1Definition ...................................... ..... 79 11.2Weylgroup ...................................... 81 11.3 Invariantbilinearform . .......... 81 11.4Dualrootsystem ................................. ...... 82 11.5 Relative position of two roots . ........... 82 11.6 Systemofsimpleroots . ........ 83 11.7 TheCartanmatrix ................................ ...... 88 11.8 TheCoxetergraph ................................ ...... 89 11.9 Irreduciblerootsystems . .......... 89 CONTENTS 5 11.10Exercices ..................................... ...... 91 12 Classification of Connected Coxeter graphs 93 12.1 Contractionofanedge. ........ 93 12.2Classification .................................. ....... 94 12.3 Dynkin diagrams and classification of root systems . ................ 97 12.4Exercices ...................................... 101 13 Classification of complex semisimple Lie algebras 103 13.1 Decomposition of the Lie algebra . ........... 103 13.2 Structure theorem for complex semisimple Lie algebras .................. 104 13.3 Free Lie algebras, Lie algebras defined by generators and relations . 106 13.4 Serre’spresentation. .......... 107 13.5Exercices ...................................... 115 14 Representations of semisimple Lie algebras 117 14.1Weights ........................................ 117 14.2 Primitiveelements . .. .. .. .. .. .. .. .. .. ........ 118 14.3 Highest weight representations . ............ 119 14.4 Finite dimensional representations . .............. 120 14.5 Application to the Weyl group . ......... 122 14.6Characters..................................... 122 14.7 Weyl’scharacterformula. .......... 124 15 Envelopping algebra II 125 16 Groups 127 6 CONTENTS Part I Lie algebras 7 Chapter 1 Algebras In this chapter we recall very basic facts on general algebra. We state the results without proofs (with the exception of basic properties of derivations) the proof are identical to the similar statement for commutative algebras. 1.1 Basics Definition 1.1.1 An algebra A over k is a vector space over k together with a bilinear map A A A × → denoted (x,y) xy. In symbols we have: 7→ x(y + z)= xy + xz and (x + y)z = xz + yz for all (x,y,z) A3, • ∈ (ax)(by) = (ab)(xy) for all (a, b) K2 and (x,y) A2. • ∈ ∈ Remark 1.1.2 Remark that we assume neither the commutativity of the product nor its associativity. Remark also that A has, a priori, no unit. Proposition & Definition 1.1.3 Let A be an algebra, the vector space A together with the multi- plication defined by (x,y) yx is again an algebra called the opposite algebra and denoted Aop. 7→ Proposition & Definition 1.1.4 A subvector space B of A which is stable for the multiplication has a natural structure of algebra inherited from the algebra stucture of A. Such a subvector space B is called a subalgebra of A. Definition 1.1.5 A subvector space I of an algebra A is called an left ideal (resp. right ideal ) if xy I for all x A and y I (resp. for all x I and y A). ∈ ∈ ∈ ∈ ∈ A left and right ideal is called a two-sided ideal. Proposition 1.1.6 For I a two-sided ideal of an algebra A, the quotient has a natural structure of algebra defined by x y = xy where x is the class of an element x of A in the quotient A/I. · Definition 1.1.7 For I a two-sided ideal of an algebra A, the algebra A/I is called the quotient algebra of A by I. Definition 1.1.8 Let f be a linear map between two algebras A and B. The map f is called a morphism of algebra if we have the equality f(xy)= f(x)f(y) for all (x,y) A2. ∈ Example 1.1.9 The morphism A A/I from an algebra A to the quotient algebra is a morphism → of algebras. 9 10 CHAPTER 1. ALGEBRAS 1.2 Derivations As derivations are less classical objects, we shall give proofs of their basic properties. Definition 1.2.1 An endomorphism D of an algebra A is called a derivation of A if the equality D(xy)= xD(y)+ D(x)y holds for all (x,y) A2. ∈ Proposition 1.2.2 The kernel of a derivation of A is a subalgebra of A. Proof. Let x and y be in the kernel, we need to prove that xy is again in the kernel. This follows from the definition of a derivation. Proposition 1.2.3 Let D1 and D2 be two derivation of an algebra A, then the commutator [D1, D2]= D D D D is again a derivation of A. 1 2 − 2 1 Proof. Let us compute [D1, D2](xy) using the fact that D1 and D2 are derivations: [D , D ](xy)= D (xD (y)+ D (x)y) D (xD (y)+ D (x)y) 1 2 1 2 2 − 2 1 1 = xD1D2(y)+ D1(x)D2(y)+ D1D2(x)y + D2(x)D1(y)) (xD D (y)+ D (x)D (y)+ D D (x)y + D (x)D (y)) − 2 1 2 1 2 1 1 2 = [D1, D2](x)y + x[D1, D2](y) and the result follows. 1.3 Product of algebras Proposition & Definition 1.3.1 Let A and B be two algebras over k and consider the map A B × → A B defined by ((x,y), (x ,y )) (xx ,yy ). This defines a structure of algebra over k on A B × ′ ′ 7→ ′ ′ × called the product algebra of A and B. Proposition 1.3.2 The algebras A and B are supplementary two-sided ideals in A B. × Proposition 1.3.3 Let C be an algebra such that A and B are supplementary two sided ideals in C, then C is isomorphic to the product algebra A B. × 1.4 Restriction and extension of the scalars Let k0 be a subfield of k and k1 be an extension of k. By restriction of the scalar, we may consider any algebra A over k as an algebra over k0. Proposition & Definition 1.4.1 Let A be an algebra over k, the vector A seen as a vector space over k0 and with its multiplication over k is an algebra over k0 called the algebra obtained by restriction of the scalars from k to k0. Proposition & Definition 1.4.2 Let A be an algebra over k, the algebra structure over k on A k 1 ⊗k 1 defined by (x a)(y b)= xy ab is called the algebra obtained by extension of the scalar from k to ⊗ ⊗ ⊗ k1. 1.5. EXERCICES 11 1.5 Exercices Exercice 1.5.1 Prove the assertions in Propositions & Definition 1.1.3, 1.1.4, 1.3.1, 1.4.1, 1.4.2 in Propositions 1.1.6, 1.3.2, 1.3.3 and in Example 1.1.9. Exercice 1.5.2 Prove that all the Definitions and statements in this
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