On the Classification of Metabelian Lie Algebras(')

The classification of 2-step nilpotent Lie algebras is attacked by a generator-relation method. The main results are in low dimensions or a small number of relations. Introduction. According to a theorem of Levi, in characteristic zero a finitedimensional Lie algebra can be written as the direct sum of a semisimple subalgebra and its unique maximal solvable ideal. If the field is algebraically closed, all semisimple Lie algebras and their modules are classified [15]. Around 1945, Malcev [20] reduced the classification of complex solvable Lie algebras to several invariants plus the classification of nilpotent Lie algebras. The latter problem is investigated in this paper. In §1, I introduce a generator-relation method for attacking the classification of nilpotent Lie algebras. Most of the results achieved in the remainder of the paper concern metabelian (i.e. 2-step nilpotent) Lie algebras and are obtained by specializing the results of § 1. If A is a metabelian Lie algebra ([[A, A], A] = 0) then g = dim A/[A, A] is the least number of elements required to generate A. Let V be a g-dimensional vector space. In §2 it is shown that isomorphism classes of g-generator metabelian Lie algebras are in one-to-one correspondence with orbits of subspaces of A2 V acted on by GL(K). This correspondence lends itself to the rapid and natural development of a duality theory for these algebras having the same fundamental properties as Scheuneman's duality [24]. That is, to an algebra such as A, we associate another such algebra A0, the dual, satisfying (A0)0 at A, A, 3£ A2 if and only if A,0 at N2°, and if dim A = g + (f) p, then dim N° = g + p. Furthermore, a canonical isomorphism is exhibited between A2 V and Alt(F*) (the space of alternating forms on V*) which induces a bijection between the orbit spaces A2 K/GL(F) and Alt(K*)/GL(K*). Thus, the classification problem can be viewed as a problem of obtaining a "simultaneous canonical form" for a space of alternating forms on V*. In particular, determining orbits of 1dimensional subspaces is equivalent to computing ranks of alternating forms. Due to classical results of Weierstrass and Kronecker on so-called pencils of matrices, a canonical form is obtained for a 2-dimensional space of alternating Received by the editors November 9, 1971. AMS (MOS) subject classifications (1970). Primary 17B30; Secondary 17B40, 15A21, 15A72, 15A75. (') Part of the research presented here is contained in the author's doctoral thesis at Notre Dame where he was supported by an NSF fellowship. Copyright © 1973, American Mathematical Society 293 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 294 MICHAEL A. GAUGER forms. These results are applied in §7 to give a complete classification of metabelian Lie algebras of dimension < 7 and nearly complete results for dimension 8. By an algebro-geometric argument it is shown that there are infinitely many metabelian Lie algebras for each dimension « > 9. Over a field of characteristic zero, Jacobson [17] showed that a Lie algebra having an injective derivation was nilpotent. The converse was later disproven by Dixmier and Lister [10]. However, we do exhibit a large class of nilpotent Lie algebras, including metabelian Lie algebras, which do have injective derivations. As a consequence, these algebras are algebraic and can be faithfully represented on a space whose dimension exceeds that of the algebra itself by one. The first two chapters of Jacobson [15] provide an excellent reference for the basic facts about Lie algebras, while Chapter 1 of Mumford [23] contains most of the necessary algebraic geometry. Results cited on Grassmann varieties can be found in [14]. Immeasurable thanks are due to Carl Riehm for his inspiration and contributions. In particular, though I was aware of a duality theory, it was he who first suggested the "canonical" approach through the natural duality of A2 V and A (V*). He also provided many calculations of the low-dimensional algebras. The dimension argument of Theorem 7.8 was suggested to me by Mario Borelli. Finally, it was unforeseeable that the referee would add so much to the exposition and theory itself as he did. In many places he improved results by suggesting different and better references. The proof of Lemma 6.2 is due to him. It is not only simpler and more elegant than my original proof (good only in characteristic zero) but it works in all characteristics other than 2. The profound benefit is that the 2-relation isomorphism Theorem 6.15 is also extended to all characteristics except 2. Using his suggestion to postpone low-dimension calculations until the 2-relation problem was solved, a great deal of tedious, ad hoc calculations were replaced by easy applications of Theorem 6.15. Furthermore he offered a proof of the identity of my own and Scheuneman's duality theories. This proof was identical to one I gave myself in a paper under preparation. 1. Generators and relations for nilpotent Lie algebras. Let F be an arbitrary field. With one exception, certain free algebras, all algebras in the following are assumed to be finite-dimensional Lie algebras over K. The notation A Sé B will be reserved for isomorphic algebras A and B, while V ~ W will be used to indicate vector space isomorphisms. If x, y, z,..., w are elements of a Lie algebra L, we will usually write [x,y,z,..., w] for the more cumbersome [[... [[•*,>'], z] • ■ • ],w]. Define L" to be the subspace generated by all elements of the type [xx,... ,x„] where the x¡ belong to L. Clearly Ln+X E L", and in fact the subspaces L" are ideals. The algebra is said to be nilpotent if L" is zero for some positive integer «. If F' ^ 0 = L'+x we will say F is /-step nilpotent. The first result gives a good indication of how to generate such an algebra in the most economical fashion. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON THE CLASSIFICATION OF METABELIAN LIE ALGEBRAS 295 Lemma 1.1. Suppose f : M —* A is a homomorphism of Lie algebras. Then f(M") C N" and if f is surjective the equality obtains. Proposition 1.2. Let N be a nilpotent Lie algebra. A subset S of N generates A if and only if the cosets {s + N2 \ s E S] span A/A2. Proof. The result is well known and follows from an easy induction argument. Corollary 1.3. Let g = dim A/A2 and suppose A is nilpotent. A subset {yx,... ,yg} generates A if and only if{y¡ + N2}f=x is a basis of A/A2. Definition. If A is as in the corollary we will say A has g-generators. Note that A can be generated by g elements, but by no fewer than g. Fixing / and g, we propose to study the /-step nilpotent Lie algebras with ggenerators. Every such algebra can be viewed as a quotient of a certain "universal" algebra N(l,g), and two quotients will be isomorphic when the corresponding defining ideals in N(l,g) are congruent under the action of its automorphism group. Let 9 be the free Lie algebra on g-generators yx,... ,yg [15, p. 167]. 9 is infinite-dimensional. Let 9„ denote the subspace of 9 generated by all elements of the type [yh,... ,yin] where /, E {1,2,... ,g}. 9 is graded with 9„ as the homogeneous component of degree n, and furthermore 9" = 0 2>>n &jLet N(l,g) = 9/9'+1 and let x, denote the image of y¡ under the canonical surjection 9 -^ N(l,g). Then the x, generate N(l,g). Since 9/+l is homogeneous, N(l,g) inherits a grading from 9 : A(/,g) = © 2j=i N(Ag), where N(l,g)j is the subspace spanned by all elements [x„,... ,x,_] and [N(l,g)¡,N(l,g)j] C A(/,g),+/. Universal mapping property of N(l,g). For any k-step nilpotent Lie algebra A7 with k < I, and any g-elements mx,..., mg of M, the correspondence x, -> w, extends uniquely to a homomorphism. Proof. Extend^, -> m¡ to a homomorphism 9 : 9 -h> M. Since k < I, 9/+l is in the kernel of 9; hence 9 factors through N(l,g) by a homomorphism t taking x¡ —» m¡. The uniqueness follows since the x, generate N(l,g). According to a result of Witt [15, p. 194], if the characteristic of K is zero then (1.1) dim 9„ = (1/n) 2 P(d)gn/d d\n where p is the Moebius function. Thus, since N(l,g) = 9/9/+l ^9, © ... © 9, we have 0-2) dim A(/,g) = 2 (l/n)(2 p(d)g»/d). n=l V|b / Proposition 1.4. N(l,g) is an l-step nilpotent Lie algebra with g-generators. Any other nilpotent Lie algebra with the invariants l, g is a quotient of N(l,g). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 296 MICHAEL A. GAUGER Proof. By construction N(l,g) s¿ 9, © ... © D, so it is finite-dimensional. Furthermore, N(l,g)< ~ 9, ^ 0 while N(l,g)l+X = 0. Since A(/,g)2 ^ 92 © ... © 9„ N(l,g)/N(l,g)2 sí 9, and dim 9, = g. If M is any other /-step nilpotent Lie algebra with g-generators, pick generators m,,..., mg and apply the mapping property of N(l,g) to obtain a homomorphism N(l,g) -h> M which takes x¡ to m¡. The image of this map contains a set of generators so it is surjective. Remark. If A = N(l,g)/I is an /-step nilpotent Lie algebra with g-generators, the ideal / can be viewed as the relations among xx,..., xg of N(l,g) which define A. The dimension of / will be called the number of relations defining A. Definition. Let ú(l,g) be the set of all ideals / of N(l,g) such that N(l,g)/I is /step nilpotent with g-generators. Proposition 1.5. An ideal I of N(l,g) belongs to ú(l,g) if and only if: (i)A(/,g)' % I; (ii)/ C A(/,g)2. Proof. Observe that (N(l,g)/I)n s (A(/,g)" + /)//. Therefore (N(l,g)/I)1 =£ 0 if and only if A(/,g)' $ /. Also (N(l,g)/I)/(N(l,g)/I)2 as N(l,g)/(N(l,g)2 + I). But dim N(l,g)/N(l,g)2 = g, so the latter space is g-dimensional if and only if / E N(l,g)2. If / = A(/,g)2, condition (i) is violated; hence / c N(l,g)2. Definition. If / and J are in 0(l,g) we say they are equivalent if N(l,g)/I s A(/,g)//. This equivalence relation on ü(l,g) is completely described by the following propositions. Let Aut(A) represent the group of automorphisms of an algebra A. Proposition 1.6. Suppose I and J belong to ú(l,g). Then I is equivalent to J if and only if there is a 9 in Aut(A(/,g)) satisfying 9(1) = J. Proof. (<=) Easy. (=>)Let A, and N2 represent quotients of N(l,g) by / and J respectively and suppose t : A, a A2 is an isomorphism. Consider the diagram