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Lie Algebras of Cohomological Codimension One

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Lie Algebras of Cohomological Codimension One University of Wollongong Research Online Faculty of Engineering and Information Faculty of Engineering and Information Sciences - Papers: Part A Sciences 1999 Lie algebras of cohomological codimension one Grant F. Armstrong Latrobe University Grant Cairns Latrobe University Gunky Kim University of Wollongong, [email protected] Follow this and additional works at: https://ro.uow.edu.au/eispapers Part of the Engineering Commons, and the Science and Technology Studies Commons Recommended Citation Armstrong, Grant F.; Cairns, Grant; and Kim, Gunky, "Lie algebras of cohomological codimension one" (1999). Faculty of Engineering and Information Sciences - Papers: Part A. 2686. https://ro.uow.edu.au/eispapers/2686 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Lie algebras of cohomological codimension one Abstract We show that if g is a finite dimensional ealr Lie algebra, then g has cohomological dimension cd(g) = dim(g) - 1 if and only if g is a unimodular extension of the two-dimensional non-Abelian Lie algebra aff. Keywords cohomological, lie, one, algebras, codimension Disciplines Engineering | Science and Technology Studies Publication Details Armstrong, G. F., Cairns, G. & Kim, G. (1999). Lie algebras of cohomological codimension one. Proceedings of the American Mathematical Society, 127 (3), 709-714. This journal article is available at Research Online: https://ro.uow.edu.au/eispapers/2686 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 3, March 1999, Pages 709-714 S 0002-9939(99)04562-1 LIE ALGEBRAS OF COHOMOLOGICAL CODIMENSION ONE GRANT F. ARMSTRONG, GRANT CAIRNS, AND GUNKY KIM (Communicated by Roe Goodman) ABSTRACT. We show that if g is a finite dimensional real Lie algebra, then g has cohomological dimension cd(g) = dim(g) -1 if and only if g is a unimodular extension of the two-dimensional non-Abelian Lie algebra aff. 1. INTRODUCTION Throughout this paper, g denotes a finite dimensional real Lie algebra. The cohomological dimension of g is the largest integer cd(g) such that Hcd(g) (S. R) $ 0, where H* (,DR) denotes the cohomology of g with trivial coefficients. It is well known that cd(g) = dim(g) if and only if g is unimodular [1]; that is, if the adjoint map ad(x) y 4 [x, y] has zero trace for all x E g. So in the space of Lie algebras of any given dimension, the subset of Lie algebras with cd(g) < dim(g) constitutes a large open set. For example, in dimension 3 there are infinitely many non- isomorphic real Lie algebras, but only 6 isomorphism classes having cd(g) = dim(g) (see [2]). Interestingly, the algebras with cd(g) = dim(g) - 1 are also quite rare. The simplest Lie algebra with cd(aff) = dim(g) - 1 is the 2-dimensional affine Lie algebra aff = (x, y [x, y] = y). Definition. An extension a * -> b, of a Lie algebra b by a Lie algebra a, is said to be unimodular if ad(x) la has zero trace for all x E Z. Theorem 1.1. cd(g) = dim(g) - 1 if and only if g is a unimodular extension of aff. It follows from Theorem 1.1 that the only Lie algebra of dimension 3 with cd(g) = dim(g) - 1 is IRED aff. In the classification of 3-dimensional Lie algebras, IRE aff is the algebra with X = ox, where X is the Tasaki-Umehara invariant [3]. The paper is organized as follows. In Section 2 we establish some duality results which we require in the proof of Theorem 1.1. In Section 3 we give some elementary results for unimodular extensions. Theorem 1.1 is then proved in Section 4. The paper finishes with some remarks, in Section 5. Let us introduce some notation and basic definitions. For convenience, we write dim(g) = n. When there is no risk of confusion, we will use the notation Ai to denote the i-th exterior product Ai" of the dual space I*. We denote 3% A" by A. Recall that a linear map q$ A - A is a derivation of degree k if qO(aA 3) = (qoa) A /3 + ( l)kioa A (q$3), for all a E Ai,/3 E A. The differential d in A is Received by the editors May 13, 1997 and, in revised form, July 7, 1997. 1991 Mathematics Subject Classification. Primary 17B56. Key words and phrases. Lie algebra, cohomology, cohomological dimension. (?)1999 American Mathematical Society 709 This content downloaded from 130.130.37.84 on Mon, 18 Aug 2014 04:22:55 UTC All use subject to JSTOR Terms and Conditions 710 GRANT F. ARMSTRONG, GRANT CAIRNS, AND GUNKY KIM the derivation of degree 1 whose restriction to Al is the dual of the Lie product [, ]: A2g - Ag. The cohomology H*(A, R) of g is the cohomology of d. The canonical 1-form -y E Al is defined by -y(x) = tr(ad(x)) for all x Ega 2. TWISTED DUALITY Let D: A - A be defined by D(a) = da --y A a for all a E A. Note that -y is d-closed and for all a e An-, one has doa= -y A a. So Da = O for alla eA Notice that D is not a derivation, but instead one has (t) D(a A i3) = (Da) A ,3 + (-1)'a A (dco)= (doa)A ,3 + (-1)'a A (DiO), for all a E Ai,,3 e A. One also sees easily that D2 = 0 and so we can define the twisted cohomology groups HD (g, R) = ker D/ Im D. The main purpose of this present section is to prove: Theorem 2.1 (Poincare duality for Lie algebras). For all i E {O, ... , n}, one has Hi (a R) _Hn-i(o R). To prove this result we first recall the Hodge star-operator * A -> A. Choose a basis {x 1,... ,x n} for g and consider the dual basis {a1,... ,an } for A'. Then for each i E {0,. ..., n}, set ... ... *(a, (1) A A a,(i)) = (-1)sgn(a)aa(i+1) A A aa(n) i for every permutation o E Sn. Then * extends to a unique linear isomorphism * : MAi An-' and ** = (-l)i(n-i). Notice that the bilinear map (): A x A JR defined by (a,,,3) = *(a A *,3) is just the inner product on A with respect to which the basis elements of the form ajl A .. .Aaj, are orthonormal. Next define a: A -> A by defining it on A' to be a = (-l)n(i+l)+l * D*, and set A = Ad + da. One has: Lemma 2.2. (a) a is the adjoint of d; that is, (da, /3) = (a, a/3), for all a, ,3 E A. (b) A is self-adjoint; that is, (Aa, /3) = (a, A/3), for all a, ,3 E A. (c) ker A = ker df nkera. Proof. (a) Let i E {O, ... , n} and let a e Ai,,3 E Ai+l. We are required to show that *((daA*i3) = *(aA*ai3) or equivalently daA*/3 = aA*a/3. Since aA*,3 E An-l, we have D(a A */3) = 0 and so by (t), a A *a/3 = a A *(-l)n(i+2)+1 * D * a = (_l)n(i+2)+l+i(n- ) A D *, = -(-l)ia A D *,3 = da A *,3. (b) By part (a), (Aaf3) = (daa,f3) + (adca,f3)= (aa,QQ) + (da,dc3) = (a,dca3) + (a, dc,3)= (a, A3). (c) Clearly ker c ker a C ker A. Conversely, let a E ker A. Then by part (a), 0 = (Aa, a) = (di~a + adca,a) = (aa, aa) + (dca,dca), which implies that aa = 0 and dca= 0, as required. El Theorem 2.3 (The Hodge decomposition theorem for Lie algebras). (a) A = Im A E ker A. (b) A = Im d E Im a E kerA. (c) kerd=ImcdEker A. This content downloaded from 130.130.37.84 on Mon, 18 Aug 2014 04:22:55 UTC All use subject to JSTOR Terms and Conditions LIE ALGEBRAS OF COHOMOLOGICAL CODIMENSION ONE 711 Proof. (a) First, we claim that Im A n ker A = 0. To show this, suppose that /3 E ImAf n ker A. Then /3 = Aa for some a and 0 = A/3 = A2a. Hence by Lemma 2.2(b), 0 = (/2a, a) = (Aa, Aa). So /3= Aa = 0. Thus Im i n ker A = 0 and so A = Im zA ker A for dimension reasons. (b) Note that Im A C Im d + Im 0, by the definition of A. So, by (a) it suffices to establish the following 3 conditions: (i) Im d n Im a = 0, (ii) Im d n ker A = 0, (iii) ima n ker A = 0. (i) Suppose that a = d,3 and a = O-yfor some ,3 and -y. Then by Lemma 2.2(a), (a, a) = (d,3, O-y)= (d2,3,-y) = 0. Hence a = 0 as required. (ii) Suppose that a = d,3 for some ,3 and that Aa = 0. Then by Lemma 2.2(c), aa = 0. So 0 = (Oa,,3) (Od/3,/3)= (d3, d/3), which gives d/3 = 0. Hence a = 0 as required. (iii) Suppose that a = 0,3 for some /3 and that Aa = 0. Then by Lemma 2.2(c), da = 0. So 0 = (da,/3) = (d03,3) = (0/3,0/3), which gives 0,3 = 0. Hence a = 0 as required. This establishes (b). (c) By part (b), since Im dD ker A C ker d, it suffices to show that ker d n Im a = 0.
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