Research report: "Geometry induced on R4n" doc

Induced geometry on R4n Nguyen Viet Hai (a) Abstract The present paper is a continuation of Nguyen Viet Hai's ones [4], [5] In this the author give a method to construct hypersymplectic structures on affine-symplectic data on R 2n R4n from Preliminaries A hypersymplectic structure on a where J, E 4n-dimensional manifold are endomorphisms of the tangent bundle of M M is given by (J, E, δ) such that J = −1, E = 1, JE = −EJ, δ is a neutral metric (that is, of signature (2n, 2n)) satisfying δ(X, Y ) = δ(JX, JY ) = −δ(EX, EY ) for all vector fields X, Y on M and the following associated 2-forms are closed ω1 (X, Y ) = δ(JX, Y ), ω2 (X, Y ) = δ(EX, Y ), ω3 (X, Y ) = δ(JEX, Y ) In [5] we have determined the flat torsion-free connections on the 2-dimensional Lie algebras which are compatible with a symplectic form and obtained their equivalence classes We showed all the flat torsion-free connections that preserve a symplectic form on the 2-dimensional Lie algebras, namely, on R2 and on aff(R) Those importante results used in the 4-dimensional case In [4] we presented a method to contruct four-dimensional Lie algebras carrying a hypersymplectic structure from two 2-dimensional Lie algebras equipped with compatible flat torsion-free connections and symplectic forms Using this method we obtained the classification, up to equivalence, of all left-invariant hypersymplectic structures on 4-dimensional Lie groups All those Lie groups are exponential type The purpose of this paper is to give a procedure to construct hypersymplectic structures on R4n with complete and not necessarily flat associated neutral metrics Nhận ngày 24/11/2006 Sửa chữa xong ngày 14/12/2006 The idea behind the construction will be to consider the canonical flat hypersymplec- R4n and then translate it by using an appropriate group acting simply 4n 4n 2n and transitively on R This group will be a double Lie group (R , R ×{0}, {0}× R2n ) constructed from affine data on R2n 4n The paper is organized as follows In §2 we give to R a structure of a nilpotent 2n Lie group Starting with a fixed symplectic structure ω on R which is parallel with tic structure on respect to a pair of affine structures we form the associated double Lie group (R4n , R2n × {0}, {0} × R2n ) §3 we consider canonical symplectic R , constructed from the given ω on R2n We analyze in §4 the geome- and show that it is at most 3-step nilpotent In structures on 4n try of the homogeneous metric obtained by using the double Lie group structure given to R4n to translate the standard inner product of signature (2n, 2n) on R2n ⊕ R2n The resulting metric is hypersymplectic, complete and not necessarily flat Three-step nilpotent group structure on R4n We shall begin by recalling some definitions which will be used throughout this article An affine structure (or a left symmetric algebra structure) on connection n n : R × RR → R , that is, a bilinear map n Rn is given by a satisfying the following conditions xy x y = = y x, y (1) (2) x x, y ∈ Rn If ω is a non-degenerate skew-symmetric bilinear form on Rn , the affine structure is compatible with ω if for all ω( x y, z) = ω( We notice that affine structures x z, y), on R2n x, y, z ∈ Rn compatible with (3) ω satisfy a condition stronger than (2), namely, x y = 0, x, y ∈ R2n (4) The last equation follows from ω( x y z, w) = ω( w x, z y) = −ω( = −ω( Let and w x, y) z z x, y) w = −ω( R2n be two affine connections on furthermore that and = y w, x z) = −ω( ω compatible with x y z, w) and assume satisfy the following compatibility condition x = y y x for all x, y ∈ R2n From (5) and the compatibility of the connections with y x x y = y =− x, for all x, y for all (5) ω , we obtain the following x, y ∈ R2n (6) x, y ∈ R2n (7) Indeed, (6) follows from ω( x y z, w) = −ω( y z, x w) = −ω( = −ω( x x w, z) y = = −ω( y w, z) x z, y w) = ω( z w y, x) y x z, w), y x z, w) and (7) follows from ω( x y z, w) = ω( = −ω( w y x z, w) z y, x) = ω( = −ω( w y, x w, z x) y z) = −ω( = −ω( x y z, w) = = −ω( In the next theorem ­we shall show that two affine structures and satisfying (5) and (6) give rise to a Lie group structure on the manifold on R2n R4n such that (R4n , R2n × {0}, {0} × R2n ) is a double Lie group We recall that a double Lie group is given by a triple (G, G+ , G− ) of Lie groups such that G+ , G− are Lie subgroups of G and the product G+ × G− → G, (g+ , g− ) → g+ g− is a diffeomorphism (see [8]) The next result shows that the additional condition (3) of and with a fixed ω imposes restrictions on the Lie group obtained Theorem 2.1 form ω Let and be two affine structures on and satisfying also (5) Then R2n × R2n R2n compatible with a symplectic with the product given by (x, x ) · (y, y ) = (x + α(x , y), β(x , y) + y ) (8) where x, x , y, y ∈ R2n and α(x , y) = y + yx − yx y β(x , y) = x − , x y− x x y (9) is a 3-step nilpotent double Lie group Furthermore, the associated Lie bracket on its Lie algebra R2n ⊕ R2n is [(x, x ), (y, y )] = ( yx − xy , xy − yx ) (10) R2n := R2n × {0} and R2n := {0} × R2n We denote α(x , y) = − + 2n αx (y) and β(x , y) = βy (x ) for x ∈ R− , y ∈ R2n + 2n 2n 2n 2n 2n 2n The maps α : R− × R+ −→ R+ and β : R− × R+ −→ R− satisfy the conditions Chøng minh Let us set α0 = α(0, R2n ) = 1, + αx (0) = 0, β0 = β(0, R2n ) = 1, − αx +y = αx ◦ αy (11) βx+y = βy ◦ βx (12) βy (0) = 0, x , y ∈ R2n − , x, y ∈ R2n + , where we denote αx (y) := α(x , y) and βy (x ) := β(x , y) for x ∈ R2n , y ∈ R2n The above relations show that α is a left action of R2n − + − 2n 2n 2n on R+ and β is a right action of R+ on R− These maps satisfy also the following for all compatibility conditions αx (x + y) = αx x + αβx x y, βx (x + y ) = βαy x x + βx y According to [8], the product given in (8) defines a Lie group structure on R4n such 4n 2n 2n that (R , R+ , R− ) is a double Lie group Note that the neutral element of this group 4n structure is (0, 0) and the inverse of (x, x ) ∈ R is (α(−x , −x), β(−x , −x)) Let us determine now the associated Lie algebra Linearizing the above actions, we obtain representations µ : R2n −→ End(R2n ) and ρ : R2n −→ End(R2n ) given by − + + − µx (y) = For α and β d (dαtx )0 (y), dt ρy (x ) = d (dβty )0 (x ) dt given in (9), we obtain that (dαx )0 = + x , (dβy )0 = − y Hence, µx (y) = x y, ρy (x ) = − yx , showing that the bracket of its Lie algebra is the one given in (10) If and ad(x,x ) , x, x ∈ R2n stands for the transformation given by (10), then, using that are torsion-free (see (1)) one has  ad(x,x ) From (4) applied to both     =    − and − x x x x        , we obtain   ad2 ) (x,x    =    − x x − x x x x x x        Finally, using (7), ad3 ) = (x,x Hence R2n ⊕ R2n is a 3-step nilpotent Lie algebra (i.e ad3 = 0) or R2n × R2n is a 3-step nilpotent Lie group, as claimed We set the notation to be used in what follows Since the construction of the Lie group structure on R4n in Theorem 2.1 depends on the affine structures we will denote this Lie group by H and , The corresponding Lie algebra will be denoted R2n ⊕ {0}, {0} ⊕ R2n will be denoted h+ , h− , respectively We note that (h , h+ , h− ) is a double Lie algebra, that is, h+ and h− are Lie subalgebras of h and h , = h+ ⊕ h− as vector spaces h and the abelian Lie subalgebras Invariant symplectic structures on In this section we show that the symplectic form ω in R 2n h R4n carries three symplectic structures, obtained from compatible with and These forms, defined at the Lie algebra level, give rise to left-invariant symplectic forms on the corresponding R4n inherits symplectic structures which are invariant by this nilpotent group First, we recall that a symplectic structure on a Lie algebra g is a non-degenerate skew-symmetric bilinear form ω satisfying dω = 0, where Lie group h Hence, d ω(x, y, z) = ω(x, [y, z]) + ω(y, [z, x]) + ω(z, [x, y]) (13) x, y, z ∈ g A given symplectic form ω on R2n allows us to define the following non 2n degenerate skew-symmetric bilinear forms on R ⊕ R2n :  ω1 ((x, x ), (y, y )) := ω(x, y) + ω(x , y ),  (14) ω ((x, x ), (y, y )) := −ω(x, y ) + ω(y, x ),   ω3 ((x, x ), (y, y )) := ω(x, y) − ω(x , y ) for We show below that the above forms are closed with respect to the Lie bracket given in Theorem 2.1 Therefore, they define symplectic structures on Proposition 3.1 The 2-forms ω1 , ω2 and ω3 are closed on h h R2n ⊕ {0} and {0} ⊕ R2n are abelian subalgebras of h that the forms ωi , i = 1, 2, 3, are closed if and only if Chøng minh Since , it follows (dωi )((x, 0), (y, 0), (0, z )) = (dωi )((0, x ), (0, y ), (z, 0)) = for all x, y, z, x , y , z ∈ R2n But (dωi )((x, 0), (y, 0), (0, z )) = ωi ([(y, 0), (0, z )], (x, 0)) + ωi ([(0, z ), (x, 0)], (y, 0)) = ωi ((− yz , yz ), (x, 0)) + ωi (( xz ,− xz ), (y, 0)) and (dωi )((0, x ), (0, y ), (z, 0)) = ωi ([(0, y ), (z, 0)], (0, x )) + ωi ([(z, 0), (0, x )], (0, y )) = ωi (( Using the expressions of zy ,− zy ), (0, x )) + ωi ((− zx , zx ωi , i = 1, 2, given in (14), we compute  (dω1 )((x, 0), (y, 0), (0, z )) = (dω3 )((x, 0), (y, 0), (0, z )) =   = −ω(− z , x) + ω( z , y), y x (dω1 )((0, x ), (0, y ), (z, 0)) = −(dω3 )((0, x ), (0, y ), (z, 0)) =    = −ω( z y , x ) + ω( z x , y ) ), (0, y )) and (dω2 )((x, 0), (y, 0), (0, z )) = ω(x, (dω2 )((0, x ), (0, y ), (z, 0)) = −ω( Since and ) − ω(y, x z ), z y , x ) + ω( z x , y ) yz satisfy (1) and (3), we obtain that It follows from the definitions of the forms dωi = 0, i = 1, 2, ωi , i = 1, 2, that: The restrictions of ω1 and ω3 to h+ and h− are symplectic forms on these subalgebras 2n 2n−1 Let the form ω on R be given by ω = e ∧ e + e ∧ e + · · · + e ∧ e2n , 2n 2n where {e1 , , e2n } is a fixed basis of R and {e , , e } denotes the dual basis Let us set ej := (ej , 0) and fj := (0, ej ), j = 1, , 2n (see also [4]) Hence {e1 , , e2n , f1 , , f2n } is a basis of R2n ⊕ R2n and the forms ωi , i = 1, 2, 3, can be written as ω1 = e1 ∧ e2 + · · · + e2n−1 ∧ e2n + f ∧ f + · · · + f 2n−1 ∧ f 2n , ω2 = −e1 ∧ f − · · · − e2n−1 ∧ f 2n + e2 ∧ f + · · · + e2n ∧ f 2n−1 , ω3 = e1 ∧ e2 + · · · + e2n−1 ∧ e2n − f ∧ f − · · · − f 2n−1 ∧ f 2n h a double Lie algebra, the endomorphism E given by E(x, y) = (x, −y) 2n for x, y ∈ R is a product structure (see [4]) on h , that is, E = and E is integrable, Being in the sense that it satisfies the condition E[(x, x ), (y, y )] = [E(x, x ), (y, y )] + [(x, x ), E(y, y )] − E[E(x, x ), E(y, y )] for all x, x , y, y ∈ R2n The symplectic form ω2 satisfies ω2 (E(x, x ), E(y, y )) = −ω2 ((x, x ), (y, y )) Induced geometry on for all x, x , y, y ∈ R2n R4n In this section we analyze the properties of the metric on the manifold tained by left-translating by the Lie group ture (15) 2n (2n, 2n) on R ⊕ R 2n this metric on previously R ob- , the standard inner product of signa- We show that this metric is always complete and it is flat if and only if the Lie group 4n h R4n h is 2-step nilpotent (see Theorem 4.2) Furthermore, is hypersymplectic with respect to the structures J and E defined 4.1 The bilinear form η on h Let us define a bilinear form η on h by η((x, x ), (y, y )) = −ω(x, y ) + ω(x , y) (16) (x, x ), (y, y ) ∈ h It is clearly symmetric and non degenerate With respect to the basis {e1 , , e2n , f1 , , f2n }, η can be written as for all η = −e1 · f − · · · − e2n−1 · f 2n + e2 · f + · · · + e2n · f 2n−1 , where · denotes the symmetric product of 1-forms Moreover, η satisfies the two fol- lowing conditions η(J(x, x ), J(y, y )) = η((x, x ), (y, y )) η(E(x, x ), E(y, y )) = −η((x, x ), (y, y )) for (17) (18) x, x , y, y ∈ R2n Indeed, η(J(x, x ), J(y, y )) = η((−x , x), (−y , y)) = −ω(−x , y) + ω(x, −y ) = η((x, x ), (y, y )) and η(E(x, x ), E(y, y )) = η((x, −x ), (y, −y )) = ω(x, y ) − ω(x , y) = −η((x, x ), (y, y )) η is a Hermitian metric on h with respect to both structures J and E We note that the subalgebras h+ and h− are both isotropic subspaces of h with respect to η and this metric has signature (2n, 2n) Moreover, it is easy to verify that the 2-forms ω1 , ω2 and ω3 can be recovered from g and the endomorphisms J and E Thus, Indeed we have  ω1 ((x, x ), (y, y )) = η(J(x, x ), (y, y )),  ω ((x, x ), (y, y )) = η(E(x, x ), (y, y )),   ω3 ((x, x ), (y, y )) = η(JE(x, x ), (y, y )) (19) The endomorphisms metric J and E h of 2-forms ω1 , ω2 and ω3 and the by left translations Hence, h is equipped , as well as the η can be extended to the group h with: A complex structure J and a product structure A (pseudo) Riemannian metric E JE = −EJ ; such that η such that η(J(x, x ), J(y, y )) = η((x, x ), (y, y )), η(E(x, x ), E(y, y )) = −η((x, x ), (y, y )) for all x, x , y, y ∈ Γ(T(h )); Three symplectic forms ω1 , ω2 and ω3 which satisfy (19) To summarize, we have obtained Theorem 4.1 The nilpotent Lie group ture given by the h carries a left-invariant hypersymplectic struc- 3-tuple {J, E, η} 4.2 The completeness of Since η η η is left-invariant, the Levi-Civita connection invariant vector fields, i.e., on the Lie algebra that h can be computed on left- After a computation one finds  η (x,x )    =     x + x x One can verify, using the above expression of η + x , that J        and (20) E are parallel with respect to the Levi-Civita connection R denotes the curvait is easily seen (using (4)) that R((x, 0), (y, 0)) = R((0, x ), (0, y )) = We will show next that this connection need not be flat If ture of η , Moreover, R((x, 0), (0, y )) = η (x,0) η (0,y ) − η (0,y ) η (x,0) − η (− xy , xy ) and using (20) together with (1), (5) and (7) one obtains   x    R((x, 0), (0, y )) =     Since y x     = −4 ad[(x,0),(0,y )]    y R and the Lie bracket are skew-symmetric, one finally obtains R((x, x ), (y, y )) = −4 ad[(x,x )(y,y )] , η thus showing that will be flat if and only if R=0 for all h is if and only if x 2-step nilpotent Note also that y =0 x, y ∈ R2n η We end this section studying the completeness of η (21) It follows from [2, 4] that will be complete if and only if the differential equation on h x(t) = adη x(t) ˙ x(t) (22) x(t) ∈ g defined for all t ∈ R Here adη means the adjoint of the x transformation adx with respect to the metric η It is easy to verify that the right η η hand side of (22) is given by ad x(t) x(t) = − x(t) x(t) for all t in the domain of x and admits solutions thus we have to solve the equation η x(t) x(t) x(t) = − ˙ (23) x(t) on h can be written as x(t) = (a(t), b(t)), where a(t), b(t) ∈ R2n 2n are smooth curves on R Hence, using (20), equation (23) translates into the system The curve a=− ˙ ˙ b=− − ab − aa b a, (24) b b Let us differentiate the first equation of the system above We have a = ă =2 = 0, aa a − aa ˙− ab +2 a ˙ ba ab + a ab + a bb + b aa + b ab using (4), (5), (6) and (7) In the same fashion, we differentiate the second equation of (24) and obtain ă= b = ab a ab ba −2 a bb + ˙ bb + b aa + b ab +2 b ab +2 bb b = 0, Using again (4), (5), (6) and (7) Thus, there exist constant vectors R 2n A, B, C, D ∈ such that a(t) = At + B, b(t) = Ct + D The explicit solution of the system (24) with initial condition x(0) = (a0 , b0 ) is given by a(t) = (− Therefore, a a0 − a0 b0 )t + a0 , b(t) = (− a0 b0 − x(t) is defined for all t ∈ R and, in consequence, b0 b0 )t η + b0 is complete Thus, we have obtained Theorem 4.2 Hypersymplectic metrics on h are always complete tài liệu tham khảo [1] A Andrada, S Salamon, Complex product structures on Lie algebras, to appear in Forum Math [2] Guediri, M Sur la completude des pseudo-metriques invariantes a gauche sur les groupes de Lie nilpotentes Rend Sem Mat Univ Pol Torino 52 (1994), 371- -376 [3] Nguyen Viet Hai, Quantum co-adjoint orbits of M D4 -groups, Vietnam J Math Vol 29, IS 02/2001, pp.131-158 [4] Nguyen Viet Hai, Four-dimensional Lie algebras carrying a hypersymplectic structure, Journal of science-Vinh University, XXXIV, 4A, 2005, 29-39 [5] Nguyen Viet Hai, Symplectic flat torsion-free connections on the 2-dimensional Lie algebras, Journal of science- Hanoi university of education, 1, 2006, 13-20 [6] A A Kirillov, Elements of the Theory of Representation, Springer Verlag, Berlin - New York - Heidelberg, 1976 [7] S Kaneyuki, Homogeneous symplectic manifolds and dipolarizations in Lie algebras Tokyo J Math 15 (1992), 313 325 [8] J.-H Lu, A Weinstein,Poisson Lie groups, dressing transformations and Bruhat decompositions J Diff Geom 31 (1990), 501 526 [9] A I Malcev, On a class of homogeneous spaces Reprinted in Amer Math Soc Translations, Series 1, (1962), 276 307 [10] D Segal, The structure of complete left-symmetric algebras Math Ann 293 (1992), 569 578 summary Hình học cảm sinh R4n Bài báo tiếp tục [4], [5] tác giả Trong báo tác giả trình bày phương pháp xây dựng cấu tróc siªu-symplectic trªn affine-symplectic cã trªn R2n (a) Department of Mathematics, Haiphong University R4n tõ d÷ liƯu ... The above relations show that α is a left action of R2n − + − 2n 2n 2n on R+ and β is a right action of R+ on R− These maps satisfy also the following for all compatibility conditions αx (x + y)... the Levi-Civita connection invariant vector fields, i.e., on the Lie algebra that h can be computed on left- After a computation one finds  η (x,x )    =     x + x x One can verify,... notation to be used in what follows Since the construction of the Lie group structure on R4n in Theorem 2.1 depends on the affine structures we will denote this Lie group by H and , The corresponding