NONCOMPLETE AFFINE STRUCTURES ON LIE ALGEBRAS OF MAXIMAL CLASS

Every affine structure on Lie algebra g defines a representation of g in aff(Rn). If g is a nilpotent Lie algebra provided with a complete affine structure then the corresponding representation is nilpotent. We describe noncomplete affine structures on the filiform Lie algebra Ln. As a consequence we give a nonnilpotent faithful linear representation of the 3-dimensional Heisenberg algebra. 2000 Mathematics Subject Classification: 53Cxx, 17Bxx.

1. Affine structure on a nilpotent Lie algebra 1.1.Affine structure on nilpotent Lie algebras Definition 1.1.Let g be an n-dimensional Lie algebra over R.An affine structure is given by a bilinear mapping for all X, Y , Z ∈ g.
If g is provided with an affine structure, then the corresponding connected Lie group G is an affine manifold such that every left translation is an affine isomorphism of G.In this case, the operator ∇ is nothing but the connection operator of the affine connection on G.
Let g be a Lie algebra with an affine structure ∇.Then the mapping f : g → End(g), (1.3) defined by and cannot correspond to an affine structure.

Classical examples of affine structures.
(i) Let g be the n-dimensional abelian Lie algebra.Then the representation defines an affine structure.
(ii) Let g be a 2p-dimensional Lie algebra endowed with a symplectic form For every X ∈ g we can define a unique endomorphism ∇ X by is an affine structure on g.
(iii) Following the work of Benoist [1] and Burde [2,3,4], we know that there exists a nilpotent Lie algebra without affine structures.

Faithful representations associated to an affine structure.
Let ∇ be an affine structure on an n-dimensional Lie algebra g.We consider the (n + 1)-dimensional linear representation given by It is easy to verify that ρ is a faithful representation of dimension n + 1.
We can note that this representation gives also an affine representation of g where A(X) is the matrix of the endomorphisms ∇ X : Y → ∇(X, Y ) in a given basis.
Definition 1.3.We say that the representation ρ is nilpotent if the endomorphisms ρ(X) are nilpotent for every X in g.Proposition 1.4.Suppose that g is a complex non-abelian indecomposable nilpotent Lie algebra and let ρ be a faithful representation of g.Then there exists a faithful nilpotent representation of the same dimension.
Proof.Consider the g-module M associated to ρ.Then, as g is nilpotent, M can be decomposed as where M λ i is a g-submodule, and the λ i are linear forms on g.For all X ∈ g, the restriction of ρ(X) to M i is in the following form: Let K λ i be the one-dimensional g-module defined by (1.17) The tensor product We prove that M is faithful.Recall that a representation ρ of g is faithful if and only if ρ(Z) ≠ 0 for every Z ≠ 0 ∈ Z(g).Consider X ≠ 0 ∈ Z(g).If ρ(X) = 0, then ρ(X) is a diagonal endomorphism.By hypothesis g ≠ Z(g) and there is i ≥ 1 such that X ∈ Ꮿ i (g), we have with Y j ∈ Ꮿ i−1 (g) and Z j ∈ g.The endomorphisms ρ(Y j )ρ(Z j )−ρ(Z j )ρ(Y j ) are nilpotent and the eigenvalues of ρ(X) are 0. Thus ρ(X) = 0 and ρ is not faithful.Then ρ(X) ≠ 0 and ρ is a faithful representation.In this case the descending sequence is

Affine structures on
and we have Example 2.2.The n-dimensional nilpotent Lie algebra L n defined by We can note that any Lie algebra of maximal class is a linear deformation of L n [5].

2.2.
On non-nilpotent affine structure.Let g be an n-dimensional Lie algebra of maximal class provided with an affine structure ∇.Let ρ be the (n + 1)-dimensional faithful representation associated to ∇ and we note that M = g C is the corresponding complex g-module.As g is of maximal class, its decomposition has one of the following forms or For a general faithful representation, we call characteristic the ordered sequence of the dimensions of the irreducible submodules.In the case of maximal class we have c(ρ) = (n + 1) or (n, 1) or (n − 1, 1, 1) or (n − 1, 2).In fact, the maximal class of g implies that there exists an irreducible submodule of dimension greater than or equal to n−1.More generally, if the characteristic sequence of a nilpotent Lie algebra is equal to (c 1 ,...,c p , 1) (see [5]) then for every faithful representation ρ we have c(ρ) = (d 1 ,...,d q ) with d 1 ≥ c 1 .
Theorem 2.3.Let g be the Lie algebra of the maximal class L n .Then there are faithful g-modules which are not nilpotent.
Proof.Consider the following representation given by the matrices ρ(X i ) where and for j ≥ 3 the endomorphisms ρ(X j ) satisfy where {e 1 ,...,e n ,e n+1 } is the basis given by e i = (X i , 0) and e n+1 = (0, 1).We easily verify that these matrices describe a nonnilpotent faithful representation.

2.3.
Noncomplete affine structure on L n .The previous representation is associated to an affine structure on the Lie algebra L n given by where L n is identified to the n-dimensional first factor of the (n + 1)-dimensional faithful module.This affine structure is complete if and only if the endomorphisms R X ∈ End(g) defined by are nilpotent for all X ∈ g (see [6]).But the matrix of R X 1 has the form

.10)
Its trace is 2a and for a ≠ 0 it is not nilpotent.We have proved the following proposition.
Proposition 2.4.There exist affine structures on the Lie algebra of maximal class L n which are noncomplete.
Remark 2.5.The most simple example is on dim 3 and concerns the Heisenberg algebra.We find a nonnilpotent faithful representation associated to the noncomplete affine structure given by where X 1 , X 2 , and X 3 are a basis of H 3 satisfying [X 1 ,X 2 ] = X 3 and ∇ X i the endomorphisms of g given by ∇ X i X j = ∇ X i ,X j . (2.12) The affine representation is written as (2.13)

Lie algebra of maximal class 2 . 1 . Definition Definition 2 . 1 .
An n-dimensional nilpotent Lie algebra g is called of maximal class if the smallest k such that Ꮿ k g = {0} is equal to n − 1.