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Nilpotent, Solvable, and Semisimple Lie Algebras

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Nilpotent, Solvable, and Semisimple Lie Algebras NILPOTENT, SOLVABLE, AND SEMISIMPLE LIE ALGEBRAS GEORGE H. SEELINGER 1. Introduction Just as in groups, Lie algebras can have additional structure which make them \nice." Due to the resemblence these structures have to their group theoretic counter-parts, we talk about solvable, nilpotent, and simple Lie algebras. The goal of this monograph is to primarily understand the most important theorems surrounding these structures, namely Engel's theorem, Lie's theorem, and Cartan's theorem. This monograph will borrow content freely from [Hum72] and makes no claim to originality of theorems, their statements, or their proofs. The reader should think of this monograph as companion notes to [Hum72, Sections 1{5]. Unless otherwise stated, we are always working over algebraically closed ground field F of characteristic 0, g is an arbitrary Lie algebra, and gl(V ) = End(V ) with the Lie bracket multiplication. 2. Basic Definitions 2.1. Definition. Consider a Lie algebra g. We say g is linear if [g; g] = 0. While rather boring since the Lie bracket does not provide much addi- tional structure, linear Lie algebras are important to keep in mind as easy examples. Linear Lie algebras play a similar role in Lie theory to abelian groups in group theory. 2.2. Theorem. Any 1-dimensional Lie algebra is linear. Proof. By definition, a Lie algebra g must have [g; g] = 0 for all g 2 g. Now, let g have basis element fxg. Then, [x; x] = 0 =) [g; g] = 0. 2.3. Definition. Consider a Lie algebra g. We say g is simple if g has no proper non-trivial ideals. Note that simple algebras are, in some sense, opposite to linear Lie alge- bras, since simple non-linear Lie algebras have the following property. 2.4. Proposition. Let g be a simple non-linear Lie algebra. Then, [g; g] = g. Date: May 2018. 1 Proof. Since [g; g] E g and [g; g] 6= 0 because g is non-linear, [g; g] = g by the simplicity of g. 2.5. Example. Consider g = sl2(C), the Lie algebra of 2 by 2 complex traceless matrices. Such a Lie algebra is simple and non-linear. This follows from the fact that the basis 0 1 0 0 1 0 e = ; f = ; h = 0 0 1 0 0 −1 has the relations [h; e] = 2e; [h; f] = −2f; [e; f] = h So, [g; g] = g. Now, suppose I is some nonempty ideal of sl2(C). Then, it contains some element ae + bf + ch for a; b; c 2 C. Then, [h; [h; ae+bf+ch]] = [h; a[h; e]+b[h; f]] = [h; 2ae−2bf] = 4ae+4bf =) c = 0 or h 2 I since 4(ae+bf+ch)−(4ae+4bf) 2 I. However, if h 2 I, then [e; h] = −2e 2 I and [f; h] = 2f 2 I and so I = g. So, it must be that c = 0. Now, consider since ae − bf 2 I by above, [e; ae − bf] = −bh 2 I =) b = 0 However, if ae 2 I, then [f; ae] = −ah 2 I and so we are done. It must be that I = sl2(C). Once more theory is developed, there are easier and more elegant ways to show sl2(C) is simple. 2.6. Definition. Let g be a Lie algebra and g(0) = g; g(1) = [g; g], and (i) (i−1) (i−1) (i) g = [g ; g ]. We call fg gi≥0 the derived series of g. 2.7. Definition. Let g be a Lie algebra. We say g is solvable if there exists (n) an n 2 N such that g = 0. Solvable Lie algebras have many of the same properties as solvable groups in regards to their behavior with ideals, homomorphisms, and short exact sequences. See [Hum72, p 11] for more information. 2.8. Example. The canonical example of a solvable Lie algebra is the Lie algebra of all upper triangular matrices over a field F . Let us denote such a Lie algebra as bn(F ). Let g = bn(F ). Then, we first seek to compute [g; g]. Let x; y 2 g. Then, x = dx + nx, where dx is the diagonal part of x and nx is the nilpotent part of x. We then check: [x; y] = xy − yx = (dx + nx)(dy + ny) − (dy + ny)(dx + nx) = dxdy + nxdy + dxny + nxny − dydx − nydx − dynx − nynx = nxdy + dxny + nxny − nydx − dynx − nynx since dxdy commute. 2 However, this final result must be strictly upper-triangular. Such matrices actually form another Lie algebra, denoted nn(F ). Thus, [g; g] ⊆ nn(F ). Let ei;j be the matrix with a 1 in the (i; j)th entry and 0 at all others. Then, bn(F ) is spanned by all ei;j such that j − i ≥ 0 and nn(F ) is spanned 0 by all ei;j such that j − i ≥ 1 = 2 . Furthermore, [ei;j; ek;`] = δj;kei;` − δ`;iek;j k−1 k−1 (k) Let us assume j − i ≥ 2 and ` − k ≥ 2 for some k 2 N and g is spanned by such ei;j. Then, [ei;j; ek;`] = δj;kei;` − δ`;iek;j 8 ei;` j = k; ` 6= i > <>e j 6= k; ` = i = k;j >(ei;i − ej;j) j = k; ` = i > :0 else However, if j = k and ` = i, then i − j ≥ 2k−1, which is a contradiction since j − i ≥ 2k−1. Thus, we actually have 8 e j = k; ` 6= i <> i;` [ei;j; ek;`] = ek;j j 6= k; ` = i :>0 else Moreover, if j = k, then `−i ≥ 2k−1 +k−i = 2k−1 +j−i ≥ 2k−1 +2k−1 = 2k, and similarly if ` = i, then j − k ≥ 2k. Thus, g(k+1) has a basis consisting k of all elements ei;j with j − i ≥ 2 . Using this fact, we can see that glog2 n+1 = 0 and thus g is solvable. 2.9. Example. A specific and important Lie algebra in this family of Lie al- gebras is n3(F ), which is isomorphic to the Heisenberg algebra, H, with basis ff; g; zg and relation [f; g] = z, as well as the properties that [H; H] ⊆ Z(H) and z 2 Z(H). The isomorphism between these two algebras is exhibited by f 7! e1;2; g 7! e2;3; z 7! e1;3 2.10. Proposition. Let g be an arbitrary Lie algebra. Then, g has a unique maximal solvable ideal. Proof. Let S E g be a maximal solvable ideal. We know that, if I;J E g are both solvable, then I + J is solvable. So, given another solvable ideal I E g, it must be that S + I is solvable, but since S is maximal, S + I = S. Thus, I ⊆ S. 2.11. Definition. If g is a Lie algebra, we call its unique maximal solvable ideal the radical of g denoted Rad g. 3 2.12. Definition. If g is a Lie algebra and Rad g = 0, we call g semisimple. 2.13. Remark. Since simple Lie algebras have no non-trivial ideals, their radical is 0 and thus any simple Lie algebra is also semisimple. 2.14. Proposition. Any non-abelian solvable Lie algebra has a non-trivial abelian ideal. Proof. Consider that the derived series of g has some minimal n 2 N such that g(n) = 0. Then, g(n−1) 6= 0 but [g(n−1); g(n−1)] = 0, so g(n−1) is abelian and is an ideal of g by repeated application of the fact that the Lie bracket of two ideals is an ideal of g. 2.15. Proposition. Let g be a Lie algebra. Then g= Rad g is semisimple. Proof. Consider the short exact sequence 0 ! Rad g ! g ! g= Rad g ! 0 If g 6= Rad g, then it must be that g= Rad g is not solvable, otherwise g would be solvable and thus g = Rad g. Now, consider an solvable ideal of g= Rad g must have the form I= Rad g for an ideal I E g. Then we would have short exact sequence of ideals 0 ! I \ Rad g ! I ! I= Rad g ! 0 and I would be solvable, so I = Rad g and I= Rad g = 0. Thus, g= Rad g is semisimple. 2.16. Definition. Let g be a Lie algebra. Then, the short exact sequence used in the proof above, namely 0 ! Rad g ! g ! g= Rad g ! 0 is the Levi decomposition of g. That is, g is the extension of a semisimple Lie algebra by a solvable algebra. 2.17. Definition. Let g be a Lie algebra, g0 := g; g1 := [g; g], and gi := i−1 i [g; g ]. We call fg gi≥0 the lower central series of g or the descending central series of g. 2.18. Definition. Let g be a Lie algebra. If there exists an n 2 N such that gn = 0, we say that g is nilpotent. Just like solvability, nilpotency behaves similarly to the group theoretic version with respect to homomorphisms, ideals, and short exact sequences. See [Hum72, p 12] for more details. 2.19. Example. Consider g = sl2(F ) where char F = 2. Such a Lie algebra is nilpotent since [h; e] = 2e = 0; [h; f] = −2f = 0; [e; f] = h and so g2 = hhi and thus g3 = 0. 4 2.20. Definition. Let g 2 g, a Lie algebra. Then, if adg is a nilpotent n endomorphism, that is, there is an n such that (adg) = 0, we say that g is ad-nilpotent.
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