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]

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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

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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.

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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. Suppose that a e ker d and that a = 0,3 for some /3 E A. Then 0 = (da,,3) = (a, 0/3) = (/3, 0/3). Hence a = 0, as required.

We now present a dual version of Theorem 2.3. First, define 6 A -* A by defining it on A' to be 6 = (-l)n(i+l)+l * d*, and set A = SD + D6. Then analogous to Lemma 2.2, one finds in the same way that: Lemma 2.4. 6 is the adjoint of D, A is self-adjoint and ker A = ker D n ker 6. Just as in the proof of Theorem 2.3, Lemma 2.4 can then be used to establish: Theorem 2.5. A = Im AE9ker A = Im D EDIm6ED ker A and ker D = Im D ker A. We can now prove Theorem 2.1. By Theorem 2.3,

H* (g, R) = ker d/ Im d = (Im d (Dker A))/ Im d rv ker A. Similarly, by Theorem 2.5, one has HL (g, R) - ker A. Theorem 2.1 then follows immediately from the fact that * ker A -- ker A is an isomorphism.

3. UNIMODULAR EXTENSIONS

It is well known that if a E A' is d-closed, then ker a is an ideal. Let a = ker nf ker -y. We have:

Lemma 3.1. If ai E A' is D-closed, then a is an ideal. Proof. Suppose that a is D-closed. Let x E g and y E a. Since ker y is an ideal, we have [x, y] E ker y. Since Da = 0 and y E a, one has a([x, y]) = da(x, y) = y Aa(x, y) = 0. So [x, y] E a. Hence a is an ideal. LI

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Lemma 3.2. If a e A' and a + 0, then a is D-exact if and only if ker a = ker y. Proof. Indeed, a is D-exact if and only if a = k-y for some k e R.

Now consider an extension a c*> --* g/a. The natural projection q$ g - g*/a induces a "pull-back" map q* A(g/a)* -* Ag*: if a E Ai(g/a)*, then the i-form qO*ais defined by

(q*a)(x1, . . . Xi) = a(qxi, q*xi),, for all x1, ... ,xi e g. It is well known that q* commutes with d and thus induces a homomorphism #: H* (g/a, R) -* H* (g, R). Lemma 3.3. If a Z --* g/a is a unimodular extension, then q* commutes with D.

Proof. Let the canonical 1-forms of g and g/a be denoted -yg and 'yg/a, respectively. Since the extension is unimodular, one has 'yg q (Yg/a) So D(qO*a)= d($*a) - -ygA (qO*a)= q*(da) - A a) = Xs (Da). [D

By the above lemma, q* induces a homomorphism q# : Hb (g/a, R) -* HL (g, R). Lemma 3.4. If a ---> -- g /a is a unimodular extension, then the induced map : 9# HD'(g/a, R) -* HID (g, R) is infective. Proof. Suppose that a e A1(g/a)* and that qO*ais D-exact. Then qO*a= k-yg for some k E R. So q*(a - kyg/a) = 0 and hence a = k-yg/a and a is D-exact. D

4. PROOF OF THEOREM 1.1

First suppose that there exists a unimodular extension a -*g --* aff. Notice that g can't be unimodular, since aff isn't. Hence cd(g) < n. Since H1(aff, 1) $ 0 one has by Poincareduality that HDj(aff, 1) $&0. Hence, by Lemma3.4, HD (g, 1) $ 0. So by Poincare duality again, Hn-1 (g, 1) $ 0 and hence cd(g) = n - 1. Conversely, suppose that cd(g) = n-1. It follows by Poincare duality that there exists a D-closed 1-form a E A which is not D-exact. Let a = ker a n ker -y. By Lemma 3.1, a is an ideal, and by Lemma 3.2, a has codimension 2. It remains to show that g/a rv aff and that the extension a --> -* aff is unimodular. Choose elements x, y E g such that ker -y = a EDRy and ker a = a EDRx, with -y(x) = 1 and a(y) = 1. To see that g/a _' aff, it suffices to show that y([x,y]) = 0 and a ([x, y]) = 1. But -y([x, y]) = 0 since ker y is an ideal. And since Da = 0 one has a([x, y]) = da(x, y) = Aa(x, y) = 1.

Finally, notice that for all w E g one has

tr(ad(w)Ia) = tr(ad(w)) - -y([w, x]) - a([w, y]) = -y(w) - y([w, x]) - ([w, y]). In particular, since a is an ideal and since a c ker y, one has tr(ad(w)Ia) = 0 for all w E a. Furthermore,

tr(ad(y) Ia) = Y(Y)- Y([Y,x]) = 0, tr(ad(x) Ia) = y(x) - a([x, y]) = 1 - a([x, y]) = 0. We conclude that the extension is unimodular. This proves Theorem 1.1.

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5. REMARKS By Theorem 1.1, Lie algebras of cohomological codimension 1 are unimodular extensions of aff. Recall that an extension a -- -- aff is inessential if g possesses a subalgebra complementary to a. Notice that an inessential extension a - g - aff is completely determined by the related representation aff -l Der(a) of aff in the Lie algebra of derivations of degree 0 of a. Such representations are easy to construct and hence give examples of Lie algebras of cohomological codimension 1. We remark that the representations of aff are of two types: they are either faithful, or they are determined by a single derivation of a. For example, in the latter case, if a is an arbitrary unimodular Lie algebra, and if $ is a trace-free derivation of a, then one obtains a Lie algebra g of cohomological codimension 1 by introducing two additional generators x, y and imposing the following relations: [x,)y] = y, [x, w] = qb(w) for all w E a, and [y,w] = 0 for all w E a. By construction, this is an inessential unfaithful unimodular extension of aff. Notice however that a given Lie algebra g may be expressible in different ways as a unimodular extension of aff, as the following example shows: Example 5.1. Consider the Lie algebra g of dimension 8, with cd(g) = 7, having the generators {x, y, z, w, a, b, c, d} and the following relations: [x, y] = y + c, [x, z] =z [x,,w] = w, [x, a] = -a, [x, c] = c, [x, dl = -2d, [w, a] = b, [w, b] = z, [w, d] = a. Notice that g is simultaneously: (a) an essential unimodular extension of aff by the ideal (w, z, a, b, c, d), (b) an inessential unfaithful unimodular extension of aff by (w, y, a, b,c, d), (c) an inessential faithful unimodular extension of aff by (y, z, a, b, c, d).

We remark that if dim Hdim g-1 (g, R) = 1, then g can be written as a unimodular extension in only one way. Proposition. Suppose that g is a finite dimensional Lie algebra with cd(g) = dimg - 1 and dimHdiml-1(g, R) = 1. If 7r1:, aff and iTr2: -* aff are two unimodular extensions, then ker 7r1= ker ir2. Proof. As we saw in the proof of Theorem 1.1, the unimodular extensions g -- aff are parameterized by HL (g, R). So the Proposition follows from Theorem 2.1. O Finally, we give an equivalent alternative presentation of Theorem 1.1. First note that every finite dimensional Lie algebra g is isomorphic to a codimension 1 subalgebra of a unimodular Lie algebra . Indeed, if g is unimodular, we just take = g R. Otherwise, let u be the kernel of the canonical 1-form -y, and choose x e g\u such that -y(x) = 1. Then one obtains the desired algebra g by introducing an additional generator z and imposing the following relations: [z, x] = z and [z, w] = 0 for all w E U. Notice, in particular, that if m denotes the Lie algebra of the group of isometrics of the Minkowski plane, m = (x, y, z I[x, y] = y, [x, z]=-z),

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 714 GRANT F. ARMSTRONG, GRANT CAIRNS, AND GUNKY KIM then aff is isomorphic to the subalgebra t = (x, y). More generally, Theorem 1.1 gives:

Corollary. Let g be a finite dimensional Lie algebra. Then cd(g) = dim g - 1 if and only if there exists a finite dimensional unimodular Lie algebra g and an epimorphism ir: -* m such that g is isomorphic to irV (t). Proof. One direction is immediate from Theorem 1.1. Conversely, suppose that cd(g) = dim g - 1. By Theorem 1.1, g is a unimodular extension of a codimension 2 ideal a; so we may assume that there are elements x, y E g\a such that: (a) u = a E Ry is a unimodular ideal. (b) [x, y] = y (mod ac). Now g is isomorphic to a codimension 1 subalgebra of the unimodular Lie algebra g constructed above. Furthermore, a is an ideal of g and g/a -m. L

REFERENCES 1. J-J. Koszul, Homologie et cohomologie des algebres de Lie, Bull. Soc. math. France 78 (1950), 65-127. MR 12:120g 2. J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. in Math. 21 (1976), 293-329. MR 54:12970 3. H. Tasaki and M. Umehara, An invariant on 3-dimensional Lie algebras, Proc. Amer. Math. Soc. 115 (1992), 293-294. MR 92i:17009

SCHOOL OF MATHEMATICS, LA TROBE UNIVERSITY, MELBOURNE, AUSTRALIA 3083 E-mail address: matgf a~lure. latrobe. edu. au

E-mail address: G. Cairns(Olatrobe. edu. au

E-mail address: G. Kimclatrobe. edu. au

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