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Home , Semisimple Lie algebra, Simple Lie algebra

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 of finite dimension over C is a direct sum of simple Lie algebras.

Definition 1 (). 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 {0} 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 K. Furthermore, the restriction of Definition 2 (). 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 {0}. Proof. a⊥ is an ideal since for x ∈ a⊥, y ∈ a and z ∈ 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 (). A Lie algebra g i = a ∩ a⊥. We show that i = {0}. Let x, yi. Then is called solvable if its derived series terminates in the clearly K(x, y) = 0 for x ∈ i and y = D1i, so i is solv- trivial subalgebra, i.e. there exist an n ∈ N such that able (by Cartan’s criterion for solvability). But since g Dng = {0}. is semisimple, {0} is the only solvable ideal in g, thus a ∩ a⊥ = {0}. Definition 5 (). The adjoint rep- The restriction of K to a is non-degenerate. Let x ∈ 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 ∈ g such that K(x, y) 6= 0. Assume on the contrary that there exists no such y with y ∈ a, then x ad : g → gl(g) ad(x)(y) := [x, y] is orthogonal to all elements of a, so x ∈ a ∩ a⊥ = {0}, 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 {ei}i=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) ∈ a for all K : g × g → C K(x, y) := tr(ad x ad y). x ∈ g. Thus, g = a + a⊥. We have shown that g = a + a⊥ and a ∩ a⊥ = {0}, so Proposition 1. Let g be a Lie algebra. Then g is as a it holds that g = a ⊕ a⊥. semisimple if and only if {0} is the only solvable ideal ⊥ in g. For all x ∈ a, y ∈ a and z ∈ 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 = {0}. 0 for all x ∈ a and y ∈ 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 {0}, 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= {0}. If a is abelian, there is nothing to algebras. show. Otherwise, let n ∈ N be the smallest num- ber such that Dn+1a = {0}. Then Dna 6= {0} is Proof. Let g be a finite dimensional semisimple Lie al- abelian, since [Dna,Dna] = Dn+1a = {0}. gebra over C. If g is simple, there is nothing to show. Otherwise, g contains a non-abelian ideal a (necessarily different from {0}). 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 ∈ g and all gument to decompose it into a direct sum of semisimple y ∈ 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.