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Lie Algebras Lie algebras Alexei Skorobogatov March 20, 2007 Introduction For this course you need a very good understanding of linear algebra; a good knowl- edge of group theory and the representation theory of finite groups will also help. The main sources for these notes are the books [6] and [8]. We give complete proofs of all statements with the exception of the conjugacy of Cartan subgroups, the uniqueness theorem for semisimple Lie algebras, and the existence theorem for exceptional semisimple Lie algebras. These results can be found in [4]. Contents 1 Basic definitions and examples 2 2 Theorems of Engel and Lie 4 3 The Killing form and Cartan’s criteria 8 4 Cartan subalgebras 12 5 Semisimple Lie algebras 15 6 Root systems 19 7 Classification and examples of semisimple Lie algebras 27 1 1 Basic definitions and examples Let k be a field of characteristic zero. Most of the time k will be R or C. A not necessarily associative algebra is a vector space over k with a bilinear product, i.e. a product satisfying left and right distributivity laws. Two classes of such algebras have particularly good properties and so are very useful: associative algebras, where multiplication satisfies the associativity axiom (ab)c = a(bc), and Lie algebras, where the product is traditionally written as the bracket [ab], and satisfies skew-symmetry and the Jacobi identity : [aa] = 0, [a[bc]] + [b[ca]] + [c[ab]] = 0. We have 0 = [a + b, a + b] = [aa] + [bb] + [ab] + [ba] = [ab] + [ba], so that the bracket is indeed skew-symmetric in the usual sense. Examples of Lie algebras 1. Abelian Lie algebras. Any vector space with the zero product [ab] = 0. 2. R3 with vector product a b. It can be defined by bilinearity and skew- × symmetry once we postulate e1 e2 = e3, e2 e3 = e1, e3 e1 = e2. × × × The Jacobi identity is a standard exercise in vector algebra. 3. From any associative algebra A we construct a Lie algebra on the same vector space by setting [ab] = ab ba. The Lie algebra of n n-matrices is called gl(n). − × 4. Let sl(n) be the subspace of gl(n) consisting of matrices with zero trace. Since Tr(AB) = Tr(BA), the set sl(n) is closed under [ab] = ab ba, and hence is a Lie − algebra. 5. Let o(n) be the subspace of gl(n) consisting of skew-symmetric matrices, that is, AT = A. Then − (AB BA)T = BT AT AT BT = ( B)( A) ( A)( B) = (AB BA), (1) − − − − − − − − − so that o(n) is closed under [ab] = ab ba, and hence is a Lie algebra. − If we want to emphasize the dependence of the ground field k we write gl(n, k), sl(n, k), o(n, k). To define a Lie bracket on a vector space with basis e1, . , en we need to specify r the structure constants clm, that is, elements of k such that n r [el, em] = clmer. r=1 X 2 For example, 1 0 0 1 0 0 H = ,X+ = ,X = 0 1 0 0 − 1 0 − − is a basis of the vector space sl(2). One easily checks that [H, X+] = 2X+, [H, X ] = 2X , [X+,X ] = H. − − − − Similarly, 0 0 0 0 0 1 0 1 0 R = 0 0 1 ,R = 0 0 0 ,R = 1− 0 0 x y z 0 1− 0 1 0 0 0 0 0 − is a basis of the vector space o(3). We have [Rx,Ry] = Rz, [Ry,Rz] = Rx, [Rz,Rx] = Ry. Definition 1.1 A homomorphism of Lie algebras is a linear transformation which preserves the bracket. An isomorphism is a bijective homomorphism. A Lie subal- gebra is a vector subspace closed under the bracket. An ideal of a Lie algebra g is a Lie subalgebra a g such that [ag] a. By skew-symmetry of the bracket any ⊂ ⊂ ideal is two-sided. The quotient algebra g/a is then defined in the obvious way, asa quotient vector space with the inherited bracket operation. More examples of Lie algebras 6. Upper triangular n n-matrices A = (aij), × aij = 0 if i > j, form a subalgebra of the full associative algebra of n n-matrices. × The attached Lie algebra will be denoted by t(n). 7. Upper triangular n n-matrices A = (aij) such that aii = 0 for all i also form × a subalgebra of the algebra of n n-matrices. The attached Lie algebra will be × denoted by n(n). Exercises. 1. Prove that o(2) and n(2) are abelian 1-dimensional Lie algebras, hence they are isomorphic to k with zero bracket. 2. Prove that the Lie algebra from Example 2 is isomorphic to o(3) by comparing the structure constants. 3. Let k = R or C. The Lie algebras sl(2), o(3), t(2), n(3) all have dimension 3. Are any of these isomorphic? (The answer will depend on k.) Definition 1.2 The derived algebra g0 of a Lie algebra g is the ideal [gg] gener- ated by [ab], for all a, b g. ∈ 3 It is clear that g/[gg] is the maximal abelian quotient Lie algebra of g. This construction can be iterated as follows to define the derived series of g: g(1) = [gg], g(r+1) = [g(r)g(r)]. By induction we show that the g(r) are ideals of g (the first inclusion is due to the Jacobi identity): [g(r+1)g] = [[g(r)g(r)]g] [[g(r)g]g(r)] [g(r)g(r)] = g(r+1). ⊂ ⊂ If g(n) = 0 for some n, then g is called solvable. We note that any automorphism of g preserves g(r). The last non-zero ideal g(r) is visibly abelian. An example of a solvable Lie algebra is t(n), or any abelian Lie algebra. We can also iterate the construction of the derived algebra in another way: g1 = g(1) = [gg], gr+1 = [grg]. This means that gr is generated by iterated brackets of r+1 elements [a1[a2[a3 ... [arar+1] ...]]]. This series of ideals is called the lower central series, and if it goes down to zero, then g is called nilpotent. The ideals gr are also preserved by the automorphisms of g. An example of a nilpotent Lie algebra is n(n). Any abelian Lie algebra is also nilpotent. More exercises. 4. Compute the derived series of t(n), n(n), sl(n). Hence deter- mine which of these Lie algebras are solvable. 5. Compute the lower central series of t(n), n(n), sl(n). Hence determine which of these Lie algebras are nilpotent. 6. Prove that g(r) gr for any g and r. In particular, this implies that every ⊂ nilpotent algebra is solvable. Show by an example that the converse is false. (Hint: consider the Lie algebra attached to the algebra of affine transformations of the line x ax + b, a, b k. This algebra can be realized as the algebra of 2 2-matrices 7→ ∈ × with the zero bottom row.) 2 Theorems of Engel and Lie A (matrix) representation of a Lie algebra g is a homomorphism g gl(n); the → number n is called the dimension of the representation. A representation is called faithful if its kernel is 0. A representation g gl(V ), where V is a vector space, → is called irreducible if the only g-invariant subspace W V , W = V , is W = 0. ⊂ 6 (Recall that a subspace W V is g-invariant if gW W .) ⊂ ⊂ Any subalgebra of gl(n) comes equipped with a natural representation of dimen- sion n. To an arbitrary Lie algebra g we can attach a representation in the following way. Definition 2.1 Elements a g act as linear transformations of the vector space g ∈ by the rule x [ax]. The linear transformation of g attached to a g is denoted 7→ ∈ by ad(a). 4 Lemma 2.2 ad is a homomorphism g gl(g), called the adjoint representation. → Proof To check that ad is a homomorphism we need to prove that ad([ab]) = [ad(a)ad(b)] = ad(a)ad(b) ad(b)ad(a). − The left hand side sends x to [[ab]x], whereas the right hand side sends x to [a[bx]] − [b[ax]]. By the Jacobi identity this is the same as [a[bx]] + [a[xb]] + [x[ba]], and by skew-symmetry this equals [x[ba]] = [[ba]x] = [[ab]x]. QED − Recall that a linear transformation is nilpotent if its n-th power is zero for some n. It easily follows from the definition of a nilpotent Lie algebra thatif g is nilpotent, then ad(x) is nilpotent for any x g. The converse is also true. Engel’s theorem ∈ says that if g has a faithful representation ρ : g gl(n) such that ρ(x) is nilpotent → for every x g, then g is nilpotent. A key result in the direction of Engel’s theorem ∈ is the following proposition. Proposition 2.3 If ρ : g gl(V ) is a representation on a (non-zero) vector space → V such that ρ(a) is a nilpotent linear transformation for any a g, then V has a ∈ non-zero vector x such that gx = 0. Proof We prove this for all finite-dimensional Lie algebras inductively by dim g, the case of dim g = 1 being clear. If ρ is not faithful, then dim (g/Ker ρ) < dim g, so that the statement for g follows from that for dim (g/Ker ρ). Hence we assume g gl(V ), so that the elements of g are nilpotent linear transformations of V given ⊂ by nilpotent matrices.
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