The Fifth International Workshop on Analele Universit˘ a¸tii din Timi¸soara Differential Geometry and Its Applications Vol. XXXIX, 2001 September 18–22, 2001, Timisoara Seria Matematic˘a-Informatic˘ a Romania Fascicul˘ a special˘a-Matematic˘ a
Affine structures on Lie algebras by Michel Goze and Elisabeth Remm Abstract We present some results obtained in the PhD dissertation of E. Remm concerning the existence problem of left invariant flat affine connections on a Lie group. In particular we introduce an algebraic approach based on Lie admissible algebras.
We give the classical notion of left invariant affine connection on a Lie group, that is affine structure on a Lie algebra. We illustrate this notion studying the abelian case and we propose an algebraic construction starting with the notion of admissible algebra. In fact a Lie algebra provided with an affine structure carries up a second algebra structure called Vinberg algebra or left symmetric algebra. All these algebras belong to the category of Lie admissible algebras. We present in this talk this point of view.
1
Affine structures on Lie algebras
The Lie group Af f (Rn ) is the group of affine transformations of Rn . It is constituted of matrices µ ¶ A b 0 1 AMS Subject Classification: 22E60, 17B60, 53C05 Keywords and phrases: Lie algebra, affine structure, rigid affine structure, invariant affine structure, Vinberg algebra, Lie admissible algebra, left invariant flat affine connection on a Lie group
251
e n by with A ∈ GL(n, R), b ∈ Rn . It acts on the real affine space R µ ¶µ ¶ µ ¶ A b v Av + b = , 0 1 1 1 e n. where (v, 1)t ∈ R Its Lie algebra, denoted by af f (Rn ), is the linear algebra ½µ n
af f (R ) =
A b 0 0
¶
¾ n
; A ∈ gl(n, R), b ∈ R
.
Definition 1.1 An affine structure on a Lie algebra g is a morphism Ψ : g→af f (Rn ) of Lie algebras. Remark. Let us consider an affine representation of a Lie group G , that is an homomorphism ϕ : G→Af f (Rn ). For every g ∈ G , ϕ(g) is an affine e n . This representation induces an affine transformation on the affine space R structure on the Lie algebra g of G. Proposition 1.1 The Lie algebra g is provided with an affine structure if and only if the underlying vector space A(g) of g is a left symmetric algebra, that is there exists a bilinear mapping A(g)×A(g) → A(g), (X, Y ) 7→ X.Y, satisfying 1) X. (Y.Z) − Y. (X.Z) = (X.Y ) .Z − (Y.X) .Z, 2) X.Y − Y.X = [X, Y ] , for every X, Y, Z ∈ A(g). If Ψ is a morphism giving an affine structure on g, then the left symmetric product on A(g) is defined as follows: µ ¶ A (X) b (X) ∀X ∈ g, Ψ (X) = 0 0 and we put X.Y = b−1 (A (X) .b (X)) , 252
where b : A(g) →Rn is supposed bijective. The fact that Ψ is a representation implies that X.Y is a left symmetric product. Conversely, if X.Y is a left symmetric product on A(g), and if LX indicates the left representation : LX (Y ) = X.Y, then the map µ ¶ LX X X 7→ 0 0 defines an affine structure on the Lie algebra g. If Ψ is an affine structure on g, it defines a representation µ ¶ A (X) b (X) Ψ (X) = 0 0 and a left symmetric product X.Y . This last induces an affine representation µ ¶ LX X , 0 0 which is equivalent to Ψ (and equal if b = Id). Definition 1.2 An affine structure on g is called complete if the endomorphism θX : A(g) → A(g), Y 7→ Y + Y.X , is bijective for every X ∈ A(g). This is equivalent to one of the following properties: a) RX : A(g) → A(g), Y 7→ Y.X , is nilpotent for all X ∈ A(g); b) tr(RX ) = 0 for all X ∈ A(g). Remarks 1. In [G.R] we have given examples of non complete affine structures on some nilpotent Lie algebras of maximal class. In particular, the following affine structure associated to the representation a(x1 + x2 ) a(x1 + x2 ) 0 x1 a(x1 + x2 ) a(x1 + x2 ) 0 x2 αx1 + (β − 1)x2 βx1 + (α + 1)x2 0 x3 0 0 0 0 on the 3-dimensional Heisenberg algebra h3 is non-complete. 2. The complete affine structure on g corresponds to the simply-transitive affine action of the connected corresponding Lie group G. 253
2 2.1
Affine structures on an abelian Lie algebra Affine structures on an abelian Lie algebra
Let g be a real abelian Lie algebra. If g is provided with an affine structure then the left symmetric algebra A(g) is a commutative and associative real algebra. In fact we have X.Y − Y.X = 0 = [X, Y ] and X. (Y.Z) − Y. (X.Z) = ([X, Y ] .Z) = 0 gives X. (Z.Y ) = (X.Z) .Y and A(g) is associative. Let Ψ1 and Ψ2 be two affine structures on g. They are affinely equivalent if and only if the corresponding commutative and associative algebras are isomorphic. Then the classification of affine structures on abelian Lie algebras corresponds to the classification of commutative and associative (unitary or not) algebras. If the affine structure is complete, the endomorphisms RX are nilpotent. As A(g) is commutative, then A(g) is a nilpotent associative commutative algebra. The classification of complete affine structures on abelian Lie algebras corresponds to the classification of nilpotent associative algebras. In this field, we can quote the works of Gabriel [Ga], Mazzola [M], who study the varieties of unitary associative complex laws and give the classification for the dimension less than 5, and the works of Makhlouf or Cibils, who study the deformations of these laws and propose classifications for nilpotent associative complex algebras [Ma].
2.2
Rigid affine structures
Let us consider a fixed basis {e1 , ..., en } for the vector space Rn . An associative law on Rn is given by a bilinear mapping µ : Rn × Rn → Rn satisfying µ(ei , µ(ej , ek )) = µ(µ(ei , ej ), ek ). If we put µ(ei , ej ) = then the structural constants Cijk satisfy X l s Cils Cjk − Cijl Clk = 0, s = 1, ..., n. (1) l
254
P
Cijk ek ,
Moreover if µ is commutative, we have (2) Cijk = Cjik . Then the set of associative laws on Rn is identified with the algebraic real 3 embedded set in Rn , defined by the polynomial equations (1) and (2). We note this set Ac (n). The law µ is unitary if there exists e ∈ Rn such that µ(e, x) = x. The set of unitary laws of Ac (n) is a subset denoted by Ac1 (n). The linear group GL(n, R) acts on Ac (n) : GL(n, R) × Ac (n) → Ac (n), (f, µ) 7→ µf , where µf (ei , ej ) = f −1 µ(f (ei ), f (ej )). We denote by θ(µ) the orbit of µ related to this action. It is isomorphic R) , where G = {f ∈ GL(n, R) / µ = µ} . to the homogeneous space GL(n, µ f Gµ 3
The topology of Ac (n) is the induced topology of Rn . Definition 2.1 The law µ ∈ Ac (n) is rigid if θ(µ) is open in Ac (n). Let µ be a real associative algebra and let us denote by µC the corresponding complex associative algebra. If µ is rigid in Ac (n), then or µC is rigid in the scheme Assn of complex associative law, or µC admits a deformation µ eC which is never the complexification of a real associative algebra. This topological approach of the variety of associative algebras allows to introduce a notion of rigidity for affine structures. Definition 2.2 An affine structure Ψ on an abelian Lie algebra g is called rigid if the corresponding associative algebra A(g) is rigid in Ac (n). Cohomological approach It is well known that a sufficient condition for an associative algebra a to be rigid in A(n) is H 2 (a, a) =0 , where H ∗ (a, a) is the Hochschild cohomology of a. Suppose that a is commutative. If µ is the bilinear mapping P i defining the product of a, then a is rigid if every deformation µt = µ + t ϕi of µ is isomorphic to µ. The commutativity of µ allows to assume that every ϕi is a symmetric bilinear mapping.Then considering only the symmetric cochains ϕ, we can define the commutative Harrison cohomology of µ by putting : Zs2 (µ, µ) = {ϕ ∈ Sym(R3 × R3 , R3 )
|
δµ ϕ = 0},
where δµ ϕ(x, y, z) = µ(x, ϕ(y, z) − ϕ(µ(x, y), z) − µ(ϕ(x, y), z) + ϕ(x, µ(y, z)). We can note that for every f ∈ End(R3 ), the coboundary δµ f ∈ Sym(R3 × R3 , R3 ) and then δµ f ∈ Zs2 (µ, µ). 255
2
Zs (µ,µ) Proposition 2.1 If Hs2 (µ, µ) = B = {0}, then the corresponding affine 2 s (µ,µ) 3 structure on R is a rigid structure.
2.3 2.3.1
Affine structures on the 2-dimensional abelian Lie algebra Classification of commutative associative real algebras
• Suppose that the algebra A is commutative and unitary. Then its law is isomorphic to µ1 (e1 , e1 ) = e1 µ2 (e1 , e1 ) = e1 µ3 (e1 , e1 ) = e1 µ1 (e1 , e2 ) = e2 µ2 (e1 , e2 ) = e2 µ3 (e1 , e2 ) = e2 ; ; . µ (e , e ) = e µ (e , e ) = e µ3 (e2 , e1 ) = e2 1 2 1 2 2 2 1 2 µ1 (e2 , e2 ) = e2 µ2 (e2 , e2 ) = 0 µ3 (e2 , e2 ) = −e1 • Suppose that A is nilpotent (and not unitary). The law is isomorphic to ½
µ4 (e1 , e2 ) = e2 µ4 (e2 , e1 ) = e2
;
µ5 = 0.
• Suppose that A is non-nilpotent and non-unitary. Then its law is isomorphic to µ6 (e1 , e1 ) = e1 (in this list, the non-defined products are equal to zero). 2.3.2
Description of the affine structures
Proposition 2.2 There exist 6 non-affinely equivalent affine structures on the 2-dimensional abelian Lie algebra. In the following table we give the affine structures on the 2-dimensional abelian Lie algebra, the corresponding action and we specify the property to
256
be complete or not of these structures.
A1
A2
A3
A4
A5
A6
2.3.3
affine a b 0 a b 0 a b 0 0 a 0 0 0 0 a 0 0
structure 0 a a+b b 0 0 0 a a b 0 0 −b a a b 0 0 0 a 0 b 0 0 0 a 0 b 0 0 0 a 0 b 0 0
affine action ea 0 ea − 1 ea (eb − 1) ea eb ea (eb − 1) 0 0 1 a e 0 ea − 1 bea ea bea 0 0 1 a e cos b −ea sin b 1 − ea cos b ea sin b ea cos b ea sin b 0 0 1 1 0 a a 1 a2 + b 2 0 0 1 1 0 a 0 1 b 0 0 1 a e 0 ea − 1 0 1 b 0 0 1
complete NO
NO
NO
YES
YES
NO
On the group of affine transformations
To each affine structure Ai corresponds a flat affine connection without torsion ∇i on the abelian Lie group G. The set of all affine transformations of (G, ∇i ) is a Lie group, denoted by Af f (G, ∇i ). Its Lie algebra is the set of complete affine vector fields, that is complete vector fields satisfying £ ¤ X, ∇iY Z = ∇i[X,Y ] Z + ∇iY [X, Z] . If ∇i is complete (in our case i = 4, 5), then the Lie algebra of Af f (G, ∇i ) is the Lie algebra of affine vector fields af f (G, ∇i ) and Af f (G, ∇i ) acts transitively on G. Let us consider the corresponding affine action considered in the previous section. The translation part defines an open set Ui ⊂ R2 . For the complete case Ui = R2 . For the non-complete case we have © ª U1 = U2 = U6 = (x, y) ∈ R2 / x > −1 , © ª U3 = (x, y) ∈ R2 , (x, y) 6= (1, 0) . 257
Let φ be an affine transformation which leaves Ui invariant. For i = 1, 2, 6 the matrix of φ has the following form a 0 et − 1 b c u . 0 0 1 The group Af f (G, ∇i ) is the maximal group included in the semigroup generated by the previous matrices. Then the group Af f (G, ∇i ) = B2 × R, where B2 is the subgroup of GL(2, R) constituted of triangular matrices. 2.3.4
Rigid affine structures on the 2-dimensional abelian Lie algebra
Lemma 2.3 Every infinitesimal deformation of an unitary associative algebra in Ac (n) is unitary. The proof is based on the study of perturbation of idempotent elements made in [Ma]. Let X be an idempotent element in an associative algebra of law µ. The operators lX
Y p−−− −→ XY and rX
Y p−−− −→ Y X are simultaneously diagonalizable and the eigenvalues are respectively (1,1,1,..., 1,0,...,0) and (1,1,1,...,1,0,...,0). This set of eigenvalues is called bisystem associated to X. If X corresponds to the identity, the bisystem is {(1, ..., 1) , (1, ..., 1)} . From [Ma], if µ0 is a perturbation of µ, there exists in µ0 an idempotent element X 0 such that b(X 0 ) = b(X). Consider the perturbation of the identity in µ0 . As the bisystem corresponds to {{1, ..., 1} , {1, ..., 1}} we can conclude that X 0 is the identity of µ0 . Consequence. The set of unitary associative algebras is open in Ac (n). Let us denote by θ(µ) the orbit of the law µ in Ac (n). From the previous classification, we see that µi ∈ θ(µ1 ) for i = 2, 5, 6. Then we have Theorem 2.4 The affine structures A1 and A3 on the 2-dimensional abelian Lie algebra are rigid. The other structures can be deformed into A1 or A3 . Consequence. Every complete affine structure is not rigid. 258
2.3.5
Invariant affine structures on T2
The compact abelian Lie group T2 is defined by T2 = R2 /Z2 , identifying (x, y) with (x + p, y + q), (p, q) ∈ Z2 . Proposition 2.5 Only the structures A4 and A5 induce affine structures on the torus T2 . Proof. It is easy to see that the affine action associated to A1 , A2 , A3 and A6 are incompatible with the lattice defined by Z2 . Then only the complete structures give affine structures on T2 . For A4 we obtain the following affine transformations ¡ ¡ ¢¢ (x, y) 7→ x + p, px + y + q + p2 . This structure on T2 is not Euclidean. For A5 , the affine structure on T2 , which is Euclidean, corresponds to the transformations (x, y) 7→ (x + p, y + q) . Remark 2.1 In this Proposition we find again, in the particular case of the torus, a classical result of Kuiper [K], giving the classification of affine structures on surfaces.
2.4 2.4.1
Affine structures on the 3-dimensional abelian Lie algebra Classification of 3-dimensional commutative associative algebras
Let us begin by describing the classification of real associative commutative algebras. Let a be a 3-dimensional (not necessary unitary) real associative algebra. If a is simple, then a is, following Wedderburn theorem, isomorphic to (M1 (R))3 , M1 (R)⊕M1 (C), where Mn (D) is a matrix algebra on a division algebra on R, that is D = R or C. This gives the following algebras i = 1, 2, 3, µ1 (e1 , ei ) = µ1 (ei , e1 ) = ei , µ1 (e2 , e2 ) = e2 , , µ1 (e3 , e3 ) = e3 , i = 1, 2, 3, µ2 (e1 , ei ) = µ2 (ei , e1 ) = ei , µ2 (e2 , e2 ) = e2 , µ2 (e3 , e3 ) = e2 − e1 . 259
If a is not simple, then a = J(a) ⊕ s, where s is simple and J(a) is the Jacobson ideal of a. If s = (M1 (R))2 , we obtain ½ µ3 (e1 , ei ) = µ3 (ei , e1 ) = ei , i = 1, 2, 3, µ3 (e2 , e2 ) = e2 . If s = M1 (C) ½
µ4 (e1 , ei ) = µ4 (ei , e1 ) = ei , µ4 (e3 , e3 ) = e2 ,
i = 1, 2, 3,
where J(a) is not abelian or {µ5 (e1 , ei ) = µ5 (ei , e1 ) = ei ,
i = 1, 2, 3,
where J(a) is abelian. Suppose that a is not unitary. As the Levi’s decomposition is also true in this case, we have the following possibilities: a ' (M1 (R))2 ⊕ J(a) or M1 (C) ⊕ J(a). This gives the following algebras ½ µ7 (e1 , e1 ) = e1 µ6 (e1 , e1 ) = e1 µ7 (e1 , e2 ) = µ7 (e2 , e1 ) = e2 , , µ6 (e2 , e2 ) = e2 µ7 (e2 , e2 ) = −e1 ½ {µ8 (e1 , e1 ) = e1 ,
µ9 (e1 , e1 ) = e1 , µ9 (e1 , e2 ) = e2
½
µ10 (e1 , e1 ) = e1 µ10 (e2 , e2 ) = e3
.
If a is nilpotent it is isomorphic to one of the following algebras: ½ ½ µ11 (e1 , e1 ) = e2 µ12 (e1 , e1 ) = e2 , , µ12 (e3 , e3 ) = −e2 µ11 (e3 , e3 ) = e2 ½ {µ13 (e1 , e1 ) = e2
,
{µ15 (ei , ej ) = 0,
µ14 (e1 , e1 ) = e2 µ14 (e1 , e2 ) = µ14 (e2 , e1 ) = e3
,
i, j ∈ {1, 2, 3} .
Theorem 2.6 Every 3-dimensional real commutative associative Lie algebra a is isomorphic to one of the algebras ai , i = 1, 2, ..., 15. If a is nilpotent, a is isomorphic to ai , i = 11, ..., 15. 260
2.5
Affine structures on R3
For each associative algebra of the previous classification we can describe the affine action on R3 . Theorem 2.7 There exist 15 invariant affine structures not affinely equivalent on the 3-dimensional abelian Lie algebra. They are given by : A1
A2
A3
A4 A5
A6
A7 A8
A9
A10
A11 A12
α e x¡ + eα −¢ 1, eα eβ − 1 x + eα+β y, (x, y, z) 7→ α γ α+γ z + eα (eγ − 1) e α(e − 1) x +α e sin¢γ + eα cos γ − 1, xe ¡ αcos γ − ze α+β −e cos γ + e x + eα+β y + zeα sin γ − eα cos γ + eα+β , (x, y, z) 7→ α α sin γ + ze cos γ + eα sin γ xe α e x¡ + eα −¢ 1, ¡ ¢ eα eβ − 1 x + eα+β y + eα eβ − 1 , (x, y, z) 7→ α x + eα z + γeα γe α e x − 1 + eα , βx + eα y + γz + β, (x, y, z) 7→ + eα z + γ γx α e x − 1 + eα , βx + eα y + β, (x, y, z) 7→ + eα z + γ γx α e x − 1 + eα , eβ y − 1 + eβ , (x, y, z) 7→ z +α γ xe cos β − yeα sin β − 1 + eα cos β, xeα sin β + yeα cos β + eα sin β, (x, y, z) 7→ zα+ γ e x − 1 + eα , y + β, (x, y, z) 7→ zα+ γ e x − 1 + eα , βx + eα y + β, (x, y, z) 7→ zα+ γ e x − 1 + eα , y + β, (x, y, z) 7→ βy + z + γ x + α, αx + y + γz + β, (x, y, z) 7→ z+γ x + α, αx + y − γz + β, (x, y, z) 7→ z+γ
261
A13
A14
A15
x + α, αx + y + β, (x, y, z) 7→ z+γ x + α, αx + y + β, (x, y, z) 7→ βx + αy + z + γ x + α, y + β, (x, y, z) 7→ z+γ
Only the structures Ai , i = 11, .., 15, are complete.
2.6
Rigid affine structures
Recall that an affine structure on R3 is called rigid if the corresponding commutative and associative real algebra is rigid in the algebraic variety Ac (3). As the laws µ1 and µ2 are semi-simple associative algebras, their second cohomology group is trivial. These structures are rigid. Consider the other laws µi . We can easily compute the linear space 2 Hs (µ, µ) and exhibit the results in the following boxes : Unitary case : laws µ3 µ4 µ5
dimHs2 (µ, µ) basis for Hs2 (µ, µ) 1 ϕ(e ½ 3 , e3 ) = e2 − e1 ϕ1 (e2 , e2 ) = e2 2 ½ ϕ2 (e2 , e3 ) = e3. ½ ϕ1 (e2 , e2 ) = e2 ϕ3 (e3 , e3 ) = e2 4 ; ϕ2 (e2 , e3 ) = e3. ϕ4 (e3 , e3 ) = e3.
Non-unitary case : laws µ6 µ7
dimHs2 (µ, µ) 1 1
µ8
6
µ9
3
µ10
1
basis for Hs2 (µ, µ) ϕ1 (e3 , e3 ) = e3 ϕ ½1 (e3 , e3 ) = e3 ½ ½ ϕ1 (e2 , e2 ) = e2 ϕ3 (e3 , e3 ) = e2 ϕ5 (e2 , e3 ) = e2 ; ; ϕ (e , e ) = e ϕ (e , e ) = e ϕ 3. 4 3 3 3. 6 (e3 , e3 ) = e3. ½ 2 2 2 ϕ1 (e2 , e2 ) = e1 ; ϕ3 (e3 , e3 ) = e3 ϕ2 (e2 , e3 ) = e2. ϕ1 (e2 , e3 ) = e2 , ϕ1 (e3 , e3 ) = e3 .
262
Nilpotent and complete case : laws
dimHs2 (µ, µ)
µ11
3
µ12
4
µ13
7
µ14
3
µ15
18
basis for Hs2 (µ, µ) ½ ½ ϕ2 (e1 , e3 ) = e3 © ϕ1 (e1 , e3 ) = e1 ; ; ϕ3 (e3 , e3 ) = e3 . ϕ1 (e2 , e3 ) = e2. ϕ2 (e3 , e3 ) = e1. ½ ϕ4 (e1 , e1 ) = e3 ϕ1 (e1 , e2 ) = e2 . ϕ3 (e1 , e3 ) = e3 ϕ (e , e ) = e1 ; ; ϕ2 (e3 , e3 ) = e3. ϕ3 (e3 , e3 ) = e1. 4 1 3 ϕ 4 (e2 , e3 ) = 2e2 . ½ ϕ1 (e1 , e2 ) = e1 ϕ (e , e ) = e ϕ5 (e1 , e3 ) = e3 . 3. ½ 3 1 2 ϕ1 (e2 , e2 ) = e2. ; ϕ4 (e1 , e3 ) = e1 ϕ6 (e3 , e3 ) = e2 . ϕ (e , e ) = e . ϕ7 (e3 , e3 ) = e3 . ϕ (e , e ) = e . 2 4 2 3 2 2 1 2 ϕ (e , e ) = e 1 1 3 1 ϕ2 (e1 , e3 ) = e2 ½ ϕ1 (e2 , e2 ) = e1 ϕ3 (e1 , e3 ) = e3 ϕ2 (e2 , e2 ) = e2 ; ; ϕ (e , e ) = e ϕ 1 2 3 2 3 (e2 , e2 ) = e3. ϕ2 (e2 , e3 ) = e3 . ϕ1 (e3 , e3 ) = e3 .
In this box we suppose that ϕ(ei , ej ) = ϕ(ej , ei ) and the non-defined ϕ(es , et ) are equal to zero. We will denote by µi → µj when µi ∈ O(µj ). We obtain the following diagram % % ↑ µ14
% µ12
→ % µ10 ↑ µ11 ↑ µ13
µ1 ↑ µ6 ↑ ↑ µ8
µ2 µ4 ↑ µ5
% %
& →
µ3 ↑ µ7 ↑ µ9
Theorem 2.8 There exist two rigid affine structures on the 3-dimensional abelian Lie algebra. They are the structures associated to the semisimple associative algebras µ1 and µ2 .
2.7
Invariant affine structures on T3
We can see that the affine actions A1 to A10 are compatible with the action of Z3 on R3 if the exponentials which appear in the analytic expressions of the affine transformations are equal to 1. This gives the identity for A1 and A3 . As A2 is incompatible with the action of Z3 for every values of the parameters α, β, γ, the affine structures corresponding to the unitary cases are given by A4 and A5 for α = 0. This corresponds to the following affine 263
structures on the torus T3 :
x px + y + qz + p R3 /Z3 3 (x, y, z) 7→ qx + z + q
and
x 3 3 px + y + p R /Z 3 (x, y, z) 7→ qx + z + q
,
,
with p, q ∈ Z. For the actions A6 to A10 , they induce affine actions on the torus if α = β = 0 for A6 and α = 0 for A7 to A10 . But these appear as particular cases of A11 , A13 and A14 . Let us examinate the complete and nilpotent cases; we find the following affine structures on T3 : x + p, 3 3 px + y + rz + q, R /Z 3 (x, y, z) 7→ z+r x + p, 3 3 px + y − rz + q, R /Z 3 (x, y, z) 7→ z+r x + p, 3 3 px + y + q, R /Z 3 (x, y, z) 7→ z+r x + p, 3 3 px + y + q, R /Z 3 (x, y, z) 7→ qx + py + z + r x + p, 3 3 y + q, R /Z 3 (x, y, z) 7→ z+r Theorem 2.9 There exist 7 affine structures on the torus T3 . They correspond to the following affine crystallographic subgroups of Af f (R3 ) : Γ1 = Γ3 =
1 p q 0 1 p 0 0
0 1 0 0 0 1 0 0
0 q 1 0 0 r 1 0
0 p q 1 p q r 1
Γ2 = Γ4 = 264
1 p q 0 1 p 0 0
0 1 0 0 0 1 0 0
0 0 0 p 1 q 0 1 0 p −r q 1 r 0 1
Γ5 = Γ7 =
1 p 0 0 1 0 0 0
0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0
p q r 1 p q r 1
1 0 0 p 1 0 Γ6 = q 0 1 0 0 0
p q r 1
Remark 2.2 The nilpotent (complete) and unimodular cases are completely classified in [F.D].
3 3.1
Lie admissible algebras Definition
Let A = (A, µ) be a finite dimensional algebra over a commutative field K of characteristic zero. In this notation, µ is the law of A, that is an operation µ:A⊗A→A on the vector space A. Definition 3.1 The algebra A = (A, µ) is called Lie admissible algebra if the law µ satisfies X (−1)ε(σ) [µ(Xσ(1) , µ(Xσ(2) , Xσ(3) )) − µ(µ(Xσ(1) , Xσ(2) ), Xσ(3) )] = 0 (∗) σ∈S3
for every X1 , X2 , X3 in A, where S3 is the group of permutations of order 3. This definition is equivalent to say that the mapping [, ]µ : A ⊗ A → A defined by [X, Y ] = µ(X, Y ) − µ(Y, X) is a Lie bracket. We will denote by ALie the Lie algebra (A, [, ]µ ).
3.2
Actions of the symmetric group S3
Let aµ : V ⊗ V ⊗ V → V be the associator operator of the algebra law µ : aµ (X, Y, Z) = µ(µ(X, Y ), Z) − µ(X, µ(Y, Z)). For each σ ∈ S3 consider the action aµ ◦ σ(X1 , X2 , X3 ) = aµ (Xσ(1) , Xσ(2) , Xσ(3) ). 265
Let G be a subgroup of S3 . We will say that the algebra law is invariant under G if X (−1)ε(σ) aµ ◦ σ(X1 , X2 , X3 ) = 0 σ∈G
for all X1 , X2 , X3 in A. Of course, such algebra is a Lie admissible algebra. Let us consider the set of subgroups of S3 . We have G1 = {Id}, G2 = {Id, τ12 }, G3 = {Id, τ23 }, G4 = {Id, τ13 }, G5 = {Id, (231), (312)}, G6 = S3 , where τij is the transposition between i and j and (231) the cycle of order 3. We deduce the following types of Lie admissible algebras : 1. If µ is invariant under G1 , then µ is an associative law. 2. If µ is invariant under G2 , then µ is a law of Vinberg algebra (also called left-symmetric algebra) : µ(X, µ(Y, Z)) − µ(Y, µ(X, Z)) = µ(µ(X, Y ), Z) − µ(µ(Y, X), Z). 3. If µ is invariant under G3 , then µ is a pre-Lie algebra law (also called right-symmetric algebra): µ(µ(X, Y ), Z) − µ(X, µ(Y, Z)) = µ(µ(X, Z), Y ) − µ(X, µ(Z, Y )). 4. If µ is invariant under G4 , then µ satisfies (X.Y ).Z − X.(Y.Z) = (Z.Y ).X − Z.(Y.X). 5. If µ is invariant under G5 , then µ satisfies the generalized Jacobi condition : (X.Y ).Z + (Y.Z).X + (Z.X).Y = X.(Y.Z) + Y.(Z.X) + Z.(X.Y ). If moreover the law is antisymmetric, then it is a Lie algebra law. This shows that the definition of G5 -invariance gives another generalization of ”noncommutative” Lie algebra than the notion of Leibniz algebra. We will propose a special study of this class in a subsequent paper. 6. If µ is invariant under G6 = S3 , then µ is a Lie admissible law. We will say that an algebra, whose law has an associator invariant under Gi , is Gi -associative. Example. Let us consider the vector space C ∞ (R,R) of real smooth functions with values in R. We can endow this space with the following algebra structures : 1. µ1 (f, g) = f.g and (C ∞ (R,R),µ1 ) is associative. 2. µ2 (f, g) = f.g 0 and (C ∞ (R,R),µ2 ) is a Vinberg algebra. 266
3. µ3 (f, g) = f 0 .g and (C ∞ (R,R),µ3 ) is a pre-Lie algebra. 4. µ4 (f, g) = f 0 .g + f.g 0 and (C ∞ (R,R),µ4 ) is invariant under G4 . Remarks. Every associative algebra is a Vinberg algebra and a pre-Lie algebra. A Vinberg algebra is a pre-Lie algebra if and only if all the associators are equal: (µ(Xσ(1) , µ(Xσ(2) , Xσ(3) )) − (µ(µ(Xσ(1) , Xσ(2) ), Xσ(3) )) = (µ(Xυ(1) , µ(Xυ(2) , Xυ(3) )) − (µ(µ(Xυ(1) , Xυ(2) ), Xυ(3) )) for every υ and σ in S3 . A Lie algebra is associative (respectively a Vinberg algebra, respectively a pre-Lie algebra) if and only if it is 2-step nilpotent.
3.3
Geometrical interpretation
Let g be a real Lie algebra provided with an affine structure. The associated covariant derivative 5 satisfies ½ 5X Y − 5Y X − [X, Y ] = 0, 5X 5Y − 5Y 5X = 5[X,Y ] . Then the product µ(X, Y ) = 5X Y endows the vector space g as a leftsymmetric algebra (Vinberg algebra). As we have [X, Y ] = µ(X, Y ) − µ(Y, X), every Lie algebra provided with an affine structure is subordinated to a Lie admissible algebra, which is in this case a Vinberg algebra. More generally, every Lie algebra is subordinated to a Lie admissible algebra. In fact it is sufficient to consider the Levi-Civita connection (which always exists). As the torsion is vanishing, its covariant derivative satisfies the first of the previous equations and µ(X, Y ) = 5X Y is a law for a Lie admissible algebra such that [X, Y ] = [X, Y ]µ . This allows to consider the set of Lie admissible algebras as the set of invariant linear connections with vanishing torsion on Lie algebras. In fact if µ is a law of a Lie admissible algebra, putting 5X Y = µ(X, Y ), 5 defines a linear connection if the Bianchi’s identities are satisfied. Let us denote by R(X, Y ) the curvature tensor corresponding to 5. As µ is an admissible law, we have R(X, Y ).Z + R(Y, Z).X + R(Z, X).Y = 0, so that the first identity is realized. For the second one, we can prove that (5X R)(Y, Z) + (5Y R)(Z, X) + (5Z R)(X, Y ) = 0. Using the relations (5X R)(Y, Z) = 5X (R(Y, Z))−R(5X Y, Z)−R(Y, 5X Z), and 5[X,Y ] = 55X Y − 55Y X , we deduce the second identity. 267
3.4 3.4.1
Cohomology of Lie admissible algebras The Nijenhuis-Gerstenhaber product
Let f and g be an n and m-linear mapping on a vector space V respectively. We define the (n + m − 1)-linear mapping f ¯6 g by P P f ¯6 g(X1 , ..., Xn+m−1 ) = i=1,..,n σ∈Sn+m−1 (−1)ε(σ) (−1)(i−1) f (Xσ(1) , .., , Xσ(i−1) , g(Xσ(i) , .., Xσ(i+m−1) ), Xσ(i+m) , .., Xσ(n+m−1) ) where Sp refers to the p−symmetric group. if f = g = µ and n = m = 2, then µ ¯6 µ(X1 , X2 , X3 ) = P For example, ε(σ) (−1) {µ(µ(X σ(1) , Xσ(2) ), Xσ(3) ) − µ(Xσ(1) , µ(Xσ(2) , Xσ(3) ))} and µ is σ∈S3 a Lie admissible law if and only if µ ¯6 µ = 0. Likewise if µ is a bilinear mapping and f an endomorphism of V , then µ ¯6 f (X1 , X2 ) = µ(f (X1 ), X2 ) − µ(f (X2 ), X1 )−µ(X1 , f (X2 ))+µ(X2 , f (X1 )) and f ¯6 µ(X1 , X2 ) = f (µ(X1 , X2 )) − f (µ(X2 , X1 )) = f ([X1 , X2 ]µ ) as soon as µ ¯6 µ = 0. We will denote by Cn (V ) the vector space of n-linear mappings on V . As each Gi -associative algebra (i = 1, ..., 5) is a particular case of a Lie admissible algebra, we can also define an adapted Nijenhuis-Gerstenhaber product. 3.4.2
G1 −associative algebras (associative case)
f ¯1 g(X1 , ..., Xn+m−1 ) = Xi+m , .., Xn+m−1 ) .
P
(i−1) f (X1 , .., Xi−1 , g(Xi , .., Xi+m−1 ), i=1,..,n (−1)
This product is denoted in [N] by f og. 3.4.3
G2 −associative algebras (Vinberg algebras)
P P f ¯2 g(X1 , ..., Xn+m−1 ) = σ∈Sn+m−2 (−1)ε(σ) { i=1,..,n−1 (−1)(i−1) f (Xσ(1) , .., Xσ(i−1) , g(Xσ(i) , .., Xσ(m+i−1) ), Xσ(m+i) , ., Xσ(n+m−2) , Xn+m−1 ) + (−1)(n−1) f (Xσ(1) , .., Xσ(n−1) , g(Xσ(n) , .., Xσ(m+n−2) , Xn+m−1 )}. In [N] the set of cochains is ⊕Hom(∧k (V ) ⊗ V, V ) and the product f ¸cg concerns these cochains. We can note that if f (resp. g) is in Hom(∧n−1 (V )⊗ V, V ) [resp. Hom(∧m−1 (V )⊗V, V )], then f ¯2 g = (n−1)f ¸cg. In this work we prefer to consider the same space of cochains for all Gi − associative algebras. 268
3.4.4
G3 −associative algebras (pre-Lie algebras)
P f ¯3 g(X1 , ..., Xn+m−1 ) = σ∈Sn+m−2 (−1)ε(σ) {f (g(X1 , Xσ(2) , .., Xσ(m) ), P Xσ(m+1) , .., Xσ(m+n−1) ) + i=2,..,n−1 (−1)(i−1) f (X1 , Xσ(2) , .., Xσ(i) , g(Xσ(i+1) , .., Xσ(m+i) ), ., Xσ(n+m−1) )} . If we restrict the set of cochains to ⊕Hom(V ⊗ ∧k (V ), V ), then we find again a product associated to the cohomology of the right-symmetric algebra proposed in [Dz]. 3.4.5
G4 −associative algebras
P P f ¯4 g(X1 , ..., Xn+m−1 ) = σ∈Sn+m−2 i=0,..,n−1 (−1)ε(σ)+i f (Xσ(1) , X2 , Xσ(3) , ..., Xσ(i) , g(Xσ(i+1) , .., Xσ(m+i) ), ..., Xσ(n+m−1) ). 3.4.6
G5 −associative algebras (version of noncommutative Lie algebras)
P P f ¯5 g(X1 , ..., Xn+m−1 ) = σ∈An+m−1 i=0,..,n−1 (−1)i f (Xσ(1) , Xσ(2) , Xσ(3) , ..., Xσ(i) , g(Xσ(i+1) , .., Xσ(m+i) ), ..., Xσ(n+m−1) ) , where Ap refers to the alternated group.
3.5
Cohomology of Lie admissible algebras
Let us consider the following bracket [f, g]¯6 = f ¯6 g − (−1)(n−1)(m−1) g ¯6 f. It satisfies : 1. [g, f ]¯6 = (−1)(n−1)(m−1)+1 [f, g]¯6 ; 2. (−1)(n−1)(p−1) [[f, g]¯6 , h]¯6 + (−1)(m−1)(n−1) [[g, h]¯6 , f ]¯6 + (−1)(p−1)(m−1) [[h, f ]¯6 , g]¯6 = 0, where h is a p-linear mapping on V . Definition 3.2 Let φ be an n-linear mapping on the Lie admissible algebra A whose law is µ. We define δµ φ = −[µ, φ]¯6 . 269
n
Let C n = {φ : A⊗ → A}. Then δµ : C n → C n+1 and δµ (δµ φ) = 0. We will denote by H ∗ (Aµ , Aµ ) the corresponding cohomology. Definition 3.3 Let g be a Lie algebra. Let us denote by Aµ this Lie algebra considered as a Lie admissible algebra. Then H ∗ (Aµ , Aµ ) is called admissible cohomology of the Lie algebra g. Let ϕ : Aµ × Aµ → Aµ be a symmetric bilinear mapping. Lemma 3.1 δµ ϕ = 0. Proof. We have δµ ϕ (X1 , X2 , X3 ) =
X
¡ ¡ ¢ ¢ ϕ Xσ(1) , Xσ(2) , Xσ(3) ¡ ¡ ¢¢ −µ Xσ(1) , ϕ Xσ(2) , Xσ(3) ) X ¡ ¡ ¢ ¢ ε(σ) + (ϕ µ Xσ(1) , Xσ(2) , Xσ(3) P (−1) σ∈ 3 ¡ ¡ ¢¢ −ϕ Xσ(1) , µ Xσ(2) , Xσ(3) ) . ε(σ) (µ P (−1) σ∈ 3
As ϕ is symmetric, the first part is reduced to zero. As µ is antisymmetric, one gets ¡ ¡ ¢ ¢ ¡ ¡ ¢ ¢ ϕ µ Xσ(1) , Xσ(2) , Xσ(3) = −ϕ µ Xσ(2) , Xσ(1) , Xσ(3) ¡ ¡ ¢¢ = −ϕ Xσ(3) , µ Xσ(2) , Xσ(1) ¡ ¡ ¢¢ = ϕ Xσ(3) , µ Xσ(1) , Xσ(2) and the second part is also zero.
¤
Lemma 3.2 If µ is a Lie algebra law and ϕ a symmetric bilinear mapping, then µ + ϕ is a Lie admissible law such that Aµ+ϕ = Aµ . Proposition 3.3 The deformation µ + ϕ of µ is a Lie admissible law. It is a Vinberg law if and only if the symmetric bilinear mapping satisfies 2ϕ (µ (X2 , X1 ) , X3 ) + µ (X1 , ϕ (X2 , X3 )) + ϕ (X1 , µ (X2 , X3 )) +ϕ (X1 , ϕ (X2 , X3 )) − µ (X2 , ϕ (X1 , X3 )) − ϕ (X2 , µ (X1 , X3 )) −ϕ (X2 , ϕ (X1 , X3 )) + µ (µ (X2 , X1 ) , X3 ) = 0. 270
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[Ma] Makhlouf N., The irreducible components of nilpotent associative algebras. Revista Matematica Univ. Complutense, 6 (1), 27-40, 1993. [N] Nijenhuis A., Sur une classe de propri´et´es communes ´a quelques types diff´erents d’alg`ebres. Enseignement Math. (2) 14, (1970), 225-277. [N.Y] Nagano T., Yagi K., The affine structures on the real two torus. Osaka J. Math. 11, 181-210, 1974. Michel Goze Universit´e de Haute Alsace Facult´e des Sciences et Techniques 4, rue des Fr`eres Lumi`ere 68093 Mulhouse Cedex, France
[email protected]
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Elisabeth Remm Universit´e de Haute Alsace Facult´e des Sciences et Techniques 4, rue des Fr`eres Lumi`ere 68093 Mulhouse Cedex, France
[email protected] 