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On the Classification of Complex Semisimple Lie Algebras of Finite Dimension

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On the Classification of Complex Semisimple Lie Algebras of Finite Dimension On the classification of complex semisimple Lie algebras of finite dimension Christoph Bartsch∗ and Thomas B. Mielingy Faculty of Physics, University of Vienna Boltzmanngasse 5, 1090 Vienna, Austria (Dated: January 5, 2018) We discuss the mathematical methods which lead to the full classification of semisimple Lie algebras. Following the book Groups, Representations and Physics by H. F. Jones in outline, we discuss various criteria for the semisimplicity of Lie algebras, introduce the root space decomposition and the notion of simple roots. The Cartan matrix leads to the pictorial representation of root systems by means of Dynkin diagrams, which we derive explicitly for the group SU(n). I. GENERAL DEFINITIONS Definition 1 (Lie algebra). A Lie algebra is a vector space g over a field K endowed with a binary operation [·; ·]: g × g ! g, called Lie bracket, which is linear, alternating and satisfies the Jacobi identity. Remark 1. Since the Lie bracket is bilinear and alternating, it is also anticommutative. Example. Well known examples are square matrices with the commutator, (in classical mechanics) functions on phase space equipped with the Poisson bracket, the Lie bracket of vector fields as well 3 as R with the cross product. Definition 2 (Lie subalgebra, Ideal). Let g be a Lie algebra. A subspace h ⊆ g which is closed under the Lie bracket is called a Lie subalgebra. If a subalgebra i ⊆ g satisfies [g; i] ⊆ i, i is called an ideal in g. Definition 3 (Abelian Lie algebra). A Lie algebra g is said to be abelian if its Lie bracket is zero. In [1], the requirement of a simple Lie algebra to be non-abelian is not stated. We use this standard definition, taken from [2]. Definition 4 (Simple Lie algebra). A non-abelian Lie algebra g is called simple if it has no non- trivial ideals, i.e. the only ideals in g are f0g and g itself. Definition 5 (Semisimple Lie algebra). A Lie algebra g is called semisimple if f0g is the only abelian ideal in g. Remark 2. A semisimple Lie algebra is not abelian. Definition 6 (Generators of a Lie algebra). A subset of a Lie algebra g is said to generate g if the smallest subalgebra of g containing the set is g itself. The elements of a generating subset are called generators. Definition 7 (General linear Lie algebra). Let V be a vector space. The general linear Lie algebra over V , denoted by gl(V ), is the vector space of endomorphisms of V (linear maps from V into V ) with the Lie bracket defined as the commutator. ∗ [email protected] y [email protected] 2 Definition 8 (Structure constants). Let g be a complex Lie algebra of dimension n with generators n fTigi=1. For α; β 2 f1; :::; ng we can decompose the Lie bracket [Tα;Tβ] in terms of the generators. γ The coefficients fαβ in the decomposition γ [Tα;Tβ] = ifαβ Tγ n are called structure constants (of g in the basis fTigi=1). II. THE ADJOINT REPRESENTATION AND THE KILLING FORM Definition 9 (Adjoint representation). The adjoint representation of a Lie algebra g is defined as the linear map ad : g ! gl(g) ad(x)(y) := [x; y]: Remark 3. This is indeed a representation of the Lie algebra g, meaning that ad is a Lie algebra homomorphism, which is to say ad([x; y]) = ad x ad y − ad y ad x for all x; y 2 g. Proof. Let x; y; z 2 g. By the anticommutativity of the Lie bracket and the Jacobi identity we have ad([x; y])(z) = [[x; y]; z] = −[z; [x; y]] = [x; [y; z]] + [y; [z; x]] = [x; [y; z]] − [y; [x; z]] = ad(x) ad(y)z − ad(y) ad(x)z: Remark 4. The transformation matrix of the adjoint representation is given by β β (ad Tα) γ = ifαγ ∗α which we can easily verify by denoting by T the dual vector of Tα and writing β ∗β ∗β ∗β δ β (ad Tα) γ = T (ad Tα(Tγ)) = T ([Tα;Tγ]) = T (ifαγ Tδ) = ifαγ Definition 10 (Killing form). Let g be a finite dimensional Lie algebra over a field K. The Killing form is the symmetric bilinear form K : g × g ! K K(x; y) := tr(ad(x) ad(y)): Remark 5. As usual for a two-form, the coefficients of the Killing form K with respect to a basis n of generators fTigi=1 are defined as Kij = K(Ti;Tj) i i such that for two vectors x = ξ Ti and y = η Ti in g, the Killing form can be computed as i j K(x; y) = Kijξ η due to the following identity. i j i j i j K(x; y) = K(ξ Ti; η Tj) = ξ η K(Ti;Tj) = ξ η Kij 3 Remark 6. Due to the trace being invariant under cyclic permutations of its arguments, it holds that K(X; [Y; Z]) = K([X; Y ];Z) This property is sometimes called ’associativity’. Remark 7. We can use the Killing form to lower indices, eg. the ones of the structure constants δ by fαβγ = fαβ Kδγ. We can rewrite this in the following way δ δ δ ifαβγ = ifαβ Kδγ = ifαβ K(Tδ;Tγ) = K(ifαβ Tδ;Tγ) = K([Tα;Tβ];Tγ) which shows that fαβγ is completely antisymmetric. Due to the antisymmetry of the Lie bracket it is antisymmetric in the first indices, but due to the ’associativity’ property of the Killing form it is also antisymmetric in the last two indices and consequently antisymmetric as regards the first and the last index1. Theorem 1 (Cartan). Let g be a finite dimensional Lie algebra over R or C. Then g is semisimple if and only if the Killing form is non-degenerate. Proposition 1. Let g be a semisimple Lie algebra over R or C. Then g is a direct sum of simple Lie algebras. Proof. See the supplementary material online. We can summarise our findings in the following way. Corollary 1. Let g be a finite dimensional real or complex Lie algebra. Then the following three statements are equivalent. 1. g is semi-simple 2. The Killing form on g is non-degenerate 3. g is a direct sum of simple Lie algebras III. THE CARTAN BASIS OF A LIE ALGEBRA Jones does not define the notion of Cartan subalgebras in [1]. We use the definition given in [2]. (The more general definition for Lie algebras which are not necessarily complex or semisimple is irrelevant for our purposes.) Definition 11 (Cartan subalgebra). Let g be a complex semisimple Lie algebra. A subalgebra h ⊂ g is called a Cartan subalgebra, if h is a maximal abelian subalgebra (i.e. there exists no abelian subalgebra of which h is a proper subalgebra) and if for each h 2 h the adjoint representation ad(h) is diagnonalizable. Remark 8. One can show that every semisimple complex Lie algebra contains at least one ad- diagnonalizable element, so it has at least one Cartan subalgebra. (See eg. [3, p. 39].) Furthermore, all Cartan subalgebras of a given Lie algebra are isomorphic and thus have the same dimension [4]. 1 There is barely anything to do: fαβγ = −fβαγ = fβγα = −fγβα 4 Definition 12 (Rank of a Lie algebra). Let g be a complex semisimple Lie algebra. The rank of g, denoted by rk g, is defined as the dimension of any Cartan subalgebra of g. The following definition is taken from [5]. In [1], Jones does not work with roots in such an algebraic manner but only considers their components with respect to some basis of a Cartan subalgebra h. The underlying concept is that, due to h being abelian, the adjoint actions of all elements in h commute and can thus be simultaneously diagonalized. Then every common eigenvector x 2 g of all adjoint actions of h induces a linear map λx : h ! C by ad h(x) = λx(h)x for all h 2 h. Definition 13 (Roots of a semisimple Lie Algebra). Let g be a complex semisimple Lie algebra and let h be a Cartan subalgebra of g. Denote by h∗ the dual space of h. For λ 2 h∗ define the subspace gλ := fx 2 g j 8h 2 h :[h; x] = λ(h)xg If λ 6= 0 and gλ 6= f0g, λ is said to be a root (with respect to h) and gλ is called the root space of λ. A non-zero vector in gλ is called a root vector. Remark 9. It holds that g0 = h. Remark 10. Let g be a complex semisimple Lie algebra of finite dimension and h a Cartan r ∗ subalgebra generated by fHigi=1 where r = rk g. Let λ 2 Φ ⊂ h be a root. As usual for a dual r vector, the components of λ (with respect to the basis fHigi=1) are denoted by λi := λ(Hi). Proposition 2. It can be shown that for each root λ, the root space gλ is one-dimensional. Remark 11 (Root space decomposition). Let g be a complex semisimple Lie algebra and let h be a Cartan subalgebra of g. Denote by Φ the set of all roots of g with respect to h. As a vector space, g can be decomposed as M g = h ⊕ gλ λ2Φ This decomposition immediately suggests the use of adapted bases. Definition 14 (Cartan Basis). Let g be a complex semisimple Lie algebra of rank r. A Cartan r r basis is a basis B = fHigi=1 [ fEλgλ2Φ where the fHigi=1 generate a Cartan subalgebra h and the fEλgλ2Φ span the root spaces with respect to h (with Φ being the set of all roots wrt.
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