Note on the Decomposition of Semisimple Lie Algebras Thomas B. Mieling (Dated: January 5, 2018) Presupposing two criteria by Cartan, it is shown that every semisimple Lie algebra of finite dimension over C is a direct sum of simple Lie algebras. Definition 1 (Simple Lie Algebra). A Lie algebra g is Proposition 2. Let g be a finite dimensional semisimple called simple if g is not abelian and if g itself and f0g are Lie algebra over C and a ⊆ g an ideal. Then g = a ⊕ a? the only ideals in g. where the orthogonal complement is defined with respect to the Killing form K. Furthermore, the restriction of Definition 2 (Semisimple Lie Algebra). A Lie algebra g the Killing form to either a or a? is non-degenerate. is called semisimple if the only abelian ideal in g is f0g. Proof. a? is an ideal since for x 2 a?; y 2 a and z 2 g Definition 3 (Derived Series). Let g be a Lie Algebra. it holds that K([x; z; ]; y) = K(x; [z; y]) = 0 since a is an The derived series is the sequence of subalgebras defined ideal. Thus [a; g] ⊆ a. recursively by D0g := g and Dn+1g := [Dng;Dng]. Since both a and a? are ideals, so is their intersection Definition 4 (Solvable Lie Algebra). A Lie algebra g i = a \ a?. We show that i = f0g. Let x; yi. Then is called solvable if its derived series terminates in the clearly K(x; y) = 0 for x 2 i and y = D1i, so i is solv- trivial subalgebra, i.e. there exist an n 2 N such that able (by Cartan’s criterion for solvability). But since g Dng = f0g. is semisimple, f0g is the only solvable ideal in g, thus a \ a? = f0g. Definition 5 (Adjoint representation). The adjoint rep- The restriction of K to a is non-degenerate. Let x 2 resentation of a Lie algebra g is the Lie algebra homo- a with x 6= 0. Since K is non-degenerate on g, there morphism exists a y 2 g such that K(x; y) 6= 0. Assume on the contrary that there exists no such y with y 2 a, then x ad : g ! gl(g) ad(x)(y) := [x; y] is orthogonal to all elements of a, so x 2 a \ a? = f0g, from g into the Lie algebra gl(g), which is GL(g) with in contradiction with x 6= 0. A similar argument shows ? the commutator as the Lie bracket. that the restriction of K to a is non-degenerate. Since the restriction of K to a is non-degenerate, n Definition 6 (Killing from). The Killing form of a Lie there exists a K-orthonormal basis feigi=1 of a with algebra of finite dimension over C is the bilinear mapping K(ei; ej) = δij. Define the projection π : g ! a by ij ? π(x) := δ K(x; ei)ej. Obviously x − π(x) 2 a for all K : g × g ! C K(x; y) := tr(ad x ad y): x 2 g. Thus, g = a + a?. We have shown that g = a + a? and a \ a? = f0g, so Proposition 1. Let g be a Lie algebra. Then g is as a vector space it holds that g = a ⊕ a?. semisimple if and only if f0g is the only solvable ideal ? in g. For all x 2 a; y 2 a and z 2 g we have K(x; [y; z]) = 0 (since a? is an ideal), which is equivalent to K([x; y]; z) = Proof. 0. Since K is non-degenerate on g, it follows that [x; y] = ? ? ( An abelian ideal i is solvable since D1i = f0g. 0 for all x 2 a and y 2 a , so g = a ⊕ a also holds in the sense of Lie algebras. ) We show the contraposition: if g has a solvable ideal distinct from f0g, then g contains an non- Corollary 1. Let g be a finite dimensional semisimple trivial abelian ideal. Let a ⊆ g be a solvable ideal Lie algebra over C. Then g is a direct sum of simple Lie with a 6= f0g. If a is abelian, there is nothing to algebras. show. Otherwise, let n 2 N be the smallest num- ber such that Dn+1a = f0g. Then Dna 6= f0g is Proof. Let g be a finite dimensional semisimple Lie al- abelian, since [Dna;Dna] = Dn+1a = f0g. gebra over C. If g is simple, there is nothing to show. Otherwise, g contains a non-abelian ideal a (necessarily different from f0g). By the above proposition, it holds Theorem 1 (Cartan). Let g be a finite dimensional Lie that g = a ⊕ a? where both a and a? are semisimple algebra over R or C and denote by K the Killing form. again (since their Killing-forms are non-degenerate). If Then either of them is not simple, we can apply the same ar- 1. g is solvable if and only if for all x 2 g and all gument to decompose it into a direct sum of semisimple y 2 D1g it holds that K(x; y) = 0. Lie algebras. Since g has finite dimension, this process terminates and we end up with a decomposition of g into 2. g is semisimple if and only if the Killing form K is a direct sum of simple Lie algebras. non-degenerate..