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Lie Algebras

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  • 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 conjugacyof Cartan subgroups, the uniqueness theorem for semisimple Lie algebras, and theexistence theorem for exceptional semisimple Lie algebras. These results can befound in [4].

    Contents

    1 Basic definitions and examples 2

    2 Theorems of Engel and Lie 4

    3 The Killing form and Cartans 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 ofsuch algebras have particularly good properties and so are very useful: associativealgebras, where multiplication satisfies the associativity axiom

    (ab)c = a(bc),

    and Lie algebras, where the product is traditionally written as the bracket [ab], andsatisfies 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 bracketis indeed skew-symmetric in the usual sense.

    Examples of Lie algebras 1. Abelian Lie algebras. Any vector space with thezero 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 vectorspace 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 Liealgebra.

    5. Let o(n) be the subspace of gl(n) consisting of skew-symmetric matrices, thatis, AT = A. Then(AB BA)T = BTAT ATBT = (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 specifythe structure constants crlm, that is, elements of k such that

    [el, em] =nr=1

    crlmer.

    2

  • For example,

    H =

    (1 00 1

    ), X+ =

    (0 10 0

    ), X =

    (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,

    Rx =

    0 0 00 0 10 1 0

    , Ry = 0 0 10 0 01 0 0

    , Rz = 0 1 01 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 whichpreserves 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 isa Lie subalgebra a g such that [ag] a. By skew-symmetry of the bracket anyideal is two-sided. The quotient algebra g/a is then defined in the obvious way, as aquotient 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 forma subalgebra of the algebra of n n-matrices. The attached Lie algebra will bedenoted 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 comparingthe 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 g 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. Thisconstruction 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 theJacobi 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 ofg 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+1elements [a1[a2[a3 . . . [arar+1] . . .]]]. This series of ideals is called the lower centralseries, and if it goes down to zero, then g is called nilpotent. The ideals gr are alsopreserved 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 whichof these Lie algebras are nilpotent.

    6. Prove that g(r) gr for any g and r. In particular, this implies that everynilpotent 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 linex 7 ax + b, a, b k. This algebra can be realized as the algebra of 2 2-matriceswith the zero bottom row.)

    2 Theorems of Engel and Lie

    A (matrix) representation of a Lie algebra g is a homomorphism g gl(n); thenumber n is called the dimension of the representation. A representation is calledfaithful 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 6= V , is W = 0.(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 followingway.

    Definition 2.1 Elements a g act as linear transformations of the vector space gby the rule x 7 [ax]. The linear transformation of g attached to a g is denotedby 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 byskew-symmetry this equals [x[ba]] = [[ba]x] = [[ab]x]. QEDRecall 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 that if g is nilpotent,then ad(x) is nilpotent for any x g. The converse is also true. Engels theoremsays that if g has a faithful representation : g gl(n) such that (x) is nilpotentfor every x g, then g is nilpotent. A key result in the direction of Engels theoremis the following proposition.

    Proposition 2.3 If : g gl(V ) is a representation on a (non-zero) vector spaceV such that (a) is a nilpotent linear transformation for any a g, then V has anon-zero vector x such that gx = 0.

    Proof We prove this for all finite-dimensional Lie algebras inductively by dim g, thecase of dim g = 1 being clear. If is not faithful, then dim (g/Ker ) < dim g,so that the statement for g follows from that for dim (g/Ker ). Hence we assumeg gl(V ), so that the elements of g are nilpotent linear transformations of V givenby nilpotent matrices. Consider the restriction of the adjoint representation of gl(V )to g. The Lie algebra g acts on gl(V ) sending the elements of g to g. Thus ad(a),a g, is a linear transformation of g given by x 7 ax xa. We havead(a)x = axxa, ad(a)2x = a2x2axa+xa2, ad(a)3x = a3x3a2xa+3axa2xa3, . . .

    (2)so that ad(a)r consists of the terms like amxarm. Thus an = 0 implies ad(a)2n1 =0.

    Let m g, m 6= g, be a maximal subalgebra (possibly zero). The restriction ofad to m leaves m invariant, hence we have the quotient representation of m on g/m.By the inductive assumption it has a non-zero vector on which m acts by 0. Chooseits representative v g, then [m, v] m, v / m. The vector space spanned by mand v is a subalgebra of g; by the maximality of m it equals g.

    Let U 6= 0 be the subspace of V consisting of all the vectors killed by m; notethat U 6= 0 by the inductive assumption. Let us show that U is g-invariant. SincemU = 0 we need to show that vU U , that is, mvu = 0 for any m m, u U .Write mv = vm + [mv], thus mvu = vmu + [mv]u = 0 since [mv] m, and everyelement of m kills u. Now consider the nilpotent linear transformation of U defined

    5

  • by v. Clearly, U has a non-zero vector killed by v, but this vector is killed by thewhole g. QED

    Let V be the space of linear functions V k. A linear transformation :

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