Affine Actions on Nilpotent Lie Groups

To any connected and simply connected nilpotent Lie group N , one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N , via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G) = dim(N) ≤ 5. Finally, we specialize to the case of abelian simply transitive affine actions on a given connected and simply connected nilpotent Lie group. It turns out that such a simply transitive abelian affine action on N corresponds to a particular Lie compatible bilinear product on the Lie algebra n of N , which we call an LR-structure. 1. NIL-affine actions In 1977 [12], J. Milnor asked whether or not any connected and simply connected solvable Lie group G of dimension n admits a representation ρ : G → Aff(R) into the group of invertible affine mappings, letting G operate simply transitively on R. For some time, most people were convinced that the answer to Milnor’s question was positive until Y. Benoist ([1], [2]) proved the existence of a simply connected, connected nilpotent Lie group G (of dimension 11) not allowing such a simply transitive affine action. These examples were generalized to a family of examples by D. Burde and F. Grunewald ([6]), also in dimension 10 ([4]). To be able to construct these counterexamples, both Benoist and Burde–Grunewald used the fact that the notion of a simply transitive affine action can be translated onto the Lie algebra level. In fact, if G is a simply connected, connected nilpotent Lie group with Lie algebra g, the existence of a simply transitive affine action of G is equivalent to the existence of a certain Lie algebra representation φ : g → aff(n) = R ⋊ gl(n,R). As the answer to Milnor’s question turned out to be negative, one tried to find a more general setting, providing a positive answer to the analogue of Milnor’s problem. One such a setting is the setting of NIL-affine actions. To define this setting, we consider a simply Date: February 2, 2008. 1991 Mathematics Subject Classification. 22E25, 17B30. The first author thanks the KULeuven Campus Kortrijk for its hospitality and support. The second author expresses his gratitude towards the Erwin Schrödinger International Institute for Mathematical Physics. Research supported by the Research Programme of the Research Foundation-Flanders (FWO): G.0570.06. Research supported by the Research Fund of the Katholieke Universiteit Leuven. 1 2 D. BURDE, K. DEKIMPE, AND S. DESCHAMPS connected, connected nilpotent Lie group and define the affine group Aff(N) as being the group N ⋊Aut(N), which acts on N via ∀m,n ∈ N, ∀α ∈ Aut(N) : n = m · α(n). Note that this is really a generalization of the usual affine group Aff(R) = R ⋊ GL(n,R), where GL(n,R) is the group of continuous automorphisms of the abelian Lie group R. In [7] it was shown that for any connected and simply connected solvable Lie group G, there exists a connected and simply connected nilpotent Lie group N and a representation ρ : G → Aff(N) letting G act simply transitively on N . This shows that in this new setting, any connected and simply connected solvable Lie group does appear as a simply transitive group of affine motions of a nilpotent Lie group (referred to as NIL-affine actions in the sequel). Apart from the existence of such a simply transitive NIL-affine action for any simply connected solvable Lie group G not much is known about such actions. As a first approach towards a further study of this topic, we concentrate in this paper on the situation where both G and N are nilpotent. In the following section we show that in this case any simply transitive action ρ : G → Aff(N) is unipotent. This result was known in the usual affine case too. Then we obtain a translation to the Lie algebra level, which is again a very natural generalization of the known result in the usual affine situation. Thereafter we present some examples of simply transitive NIL-affine actions in low dimensions and finally, we specialize to the situation where G is abelian. 2. Nilpotent simply transitive NIL-affine groups are unipotent Let N be a connected and simply connected nilpotent Lie group with Lie algebra n. It is well known that N has a unique structure of a real algebraic group (e.g., see [13]) and also Aut(N) ∼= Aut(n) carries a natural structure of a real algebraic group. It follows that we can consider Aff(N) = N ⋊Aut(N) as being a real algebraic group. The aim of this section is to show that any nilpotent simply transitive subgroup of Aff(N) is an algebraic subgroup and in fact unipotent. This generalizes the analogous result for ordinary nilpotent and simply transitive subgroups of Aff(R) proved by J. Scheuneman in [15, Theorem 1]. Throughout this section we will use A(G) to denote the algebraic closure of a subgroup G ⊆ Aff(N) and we will refer to the unipotent radical of A(G), by writing U(G). Using these notations, the aim of this section is to prove the following theorem: Theorem 2.1. Let N be a connected and simply connected nilpotent Lie group and assume that G ⊂ Aff(N) is a nilpotent Lie subgroup acting simply transitively on N , then G = A(G) = U(G). We should mention here that with some effort this theorem can be reduced from the corresponding theorem in the setting of polynomial actions as obtained in [3, Lemma 5]. For the readers convenience we will repeat the necessary steps here and adapt them to our specific situation. First we recall two basic technical results which will be needed later on: Lemma 2.2. [3, Lemma 3] Let T be a real algebraic torus acting algebraically on R, then the set of fixed points (R) is non-empty. AFFINE ACTIONS ON NILPOTENT LIE GROUPS 3 Any connected and simply connected nilpotent Lie group N can be identified with its Lie algebra n using the exponential map exp. We can therefore speak of a polynomial map of N , by which we will mean that the corresponding map on the Lie algebra n is expressed by polynomials (with respect to coordinates to any given basis of n). For instance, it is well known that the multiplication map N × N → N : (n1, n2) 7→ n1n2 is polynomial. We use such polynomial maps in the following lemma. Lemma 2.3. [3, Lemma 2 (a)] Let N be a connected and simply connected nilpotent Lie group. Assume that θ : N ×Rn → R is an action which is polynomial in both variables. Let v0 be a point in R . Then the isotropy group of v0 is a closed connected subgroup of N . Now we adapt [3, Proposition 1] to our situation: Proposition 2.4. Let G ⊂ Aff(N) be a solvable Lie group, acting simply transitively on N . Then the unipotent radical U(G) of A(G) also acts simply transitively on N . Proof. As G is a connected, solvable Lie subgroup, its algebraic closure A(G) is solvable also and Zariski connected. Therefore it splits as a semi-direct product A(G) = U(G) ⋊ T where T is a real algebraic torus. From the fact that G acts simply transitively, we immediately get that A(G) acts transitively on N . By Lemma 2.2 and the fact that N is diffeomorphic to R (for n = dimN), we know that there exists a point n0 ∈ N which is fixed under the action of T . It follows that N = (U(G) ⋊ T ) · n0 = U(G) · n0, showing that U(G) acts transitively on N . By [14, Lemma 4.36], we know that dimU(G) ≤ dimG. On the other hand dimG = dimN , as G is acting simply transitively on N , from which we deduce that dimU(G) = dimG = dimN . It follows that U(G) acts with discrete isotropy groups (because stabilizers have dimension 0). Lemma 2.3 then implies that U(G) acts with trivial isotropy groups, allowing us to conclude that U(G) acts simply transitively on N . The last step we need before we can prove Theorem 2.1 is the following (compare with [3, Lemma 4]) Lemma 2.5. Let G ⊂ Aff(N) be a solvable Lie group acting simply transitively on N and let U(G) be the unipotent radical of A(G). Then the centralizer of U(G) in A(G) coincides with the center of U(G). Proof. Using Proposition 2.4 we know that U(G) acts simply transitively on N , and A(G) splits as a semi-direct product A(G) = U(G)⋊ T where T is a real algebraic torus. The centralizer C of U(G) in A(G) is also an algebraic group and therefore, for every element c ∈ C, the unipotent part cu and the semisimple part cs of c are also in C. Now assume that C does not belong completely to U(G), then there is an element c of C with a nontrivial semisimple part cs 6= 1, and cs also belongs to C. This semisimple part cs, belongs to the algebraic torus T . So, as in lemma 2.2, we can conclude that (N)s is non-empty. However, as cs centralizes U(G), the set (N) cs is U(G)-invariant. This contradicts the transitivity of U(G) on N . It follows that C is included in U(G). We are now ready to prove the main result of this section. Proof of theorem 2.1: As G is nilpotent, its algebraic closure A(G) is also nilpotent. A nilpotent algebraic group splits as a direct product A(G) = U(G) × S(G), where S(G) denotes the set of semi-simple elements of A(G). In this case S(G) centralizes U(G) and by 4 D. BURDE, K. DEKIMPE, AND S. DESCHAMPS Lemma 2.5, we can conclude that S(G) has to be trivial, so A(G) = U(G). From the fact that both G and A(G) = U(G) are acting simply transitively on N (Proposition 2.4), we can conclude that G = A(G) = U(G). 3. Translation to the Lie algebra level In this section we will show that we can completely translate the notion of a simply transitive NIL-affine action of a nilpotent Lie group G to a notion on the Lie algebra level. As before, let G and N be connected, simply connected Lie groups. We will use g to denote the Lie algebra of G and n to denote the Lie algebra of N . Recall that the Lie algebra corresponding to the semi-direct product Aff(N) = N ⋊ Aut(N) is equal to the semi-direct product n ⋊Der(n), where Der(n) is the set of all derivations of n. This semi-direct product n ⋊Der(n) is a Lie algebra via