Matching and geometry of 2-step nilpotent Lie groups
aa r X i v : . [ m a t h . DG ] J un Matching and geometry of 2-step nilpotent Liegroups
Babak HasanzadehDepartment of Mathematics, Faculty of ScienceAzarbaijan Shahid Madani UniversityTabriz, IranE-mail: [email protected] 24, 2018
Abstract
In this paper we study contact structure on 2-step nilpotent, Heisen-berg type Lie groups. We decompose this Lie groups to center and or-thogonal complement, then investigate properties of both orthogonal Liesubgroups. Finally, we provide a connection between matchings in groupsand field extensions and 2-step nilpotent Lie groups.
Keywords :Contact structure; 2-step nilpotent; Lie group; nonsingu-lar; skew adjoint; semi-symmetric. g is Liealgebra of G, for any X, Y ∈ g , we have[ X, Y ] = ∇ X Y − ∇ Y X. We define the linear map adX : g −→ g by Y [ X, Y ].Let G be a Lie group equipped by Riemannian left invariant metric, by usingonly these identities and combining a few permutations of variables obtain the
Mathematics Subject Classification : 58B20, 53D10, 17B62, 22E25. g ( ∇ X Y, Z ) = 12 { Xg ( Y, Z ) +
Y g ( X, Z ) − Zg ( X, Y ) (1)+ g ([ X, Y ] , Z ) + g ([ Z, X ] , Y ) − g ([ Y, Z ] , X ) } , for each nonzero left invariant vector fields X, Y, Z ∈ g . ∇ is covariant derivativein this paper first three terms are vanish. If (G,g) is a Lie group equipped byleft invariant metric, ad is skew adjoint if g ( adX ( Y ) , Z ) = − g ( Y, adX ( Z )) . If X ∈ Z ( g ) and Y, Z ∈ g ,we have Xg ( Y, Z ) = g ( ∇ Y + Z X, Y ) . In the next definition, we define the most basic concept that will be use in thispaper.
Definition 1
A nilpotent Lie group is a Lie group G which is connected andwhose Lie algebra is nilpotent Lie algebra g , that is, it’s Lie algebra has a se-quence of ideals of g by g = g , g = [ g , g ] , g = [ g , g ] ,..., g i = [ g , g i − ] . Also g is called nilpotent if g n = 0 for some n.[10] The following subgroup Z ( G ) = { x ∈ G : xy = yx, ∀ y ∈ G } , is called the center of G, it is a Lie subgroup with corresponding Lie subalgebra Z ( g ) = { X ∈ g : [ X, Y ] = 0 , ∀ Y ∈ g } . Let Z ⊥ ( g ) is orthogonal complement of Z ( g ) with left invariant metric g. Eachelement Z ∈ Z ( g ) defines a skew symmetric linear map j ( Z ) : Z ⊥ ( g ) → Z ⊥ ( g )given by j ( Z ) X = ( adX ) ∗ for all X ∈ Z ⊥ ( g ), where ( adX ) ∗ is the adjoint ofadX relative to g, equally and more usefully j ( Z ) is defined by the equation g ( j ( Z ) X, Y ) = g ([ X, Y ] , Z ) , (2)for all X, Y ∈ Z ⊥ ( g ).Following definition is the most practical concept in present paper. Definition 2
A finite dimensional Lie algebra g is 2-step nilpotent if g is notabelian and [ g , [ g , g ]] = 0 . A Lie group G is 2-step nilpotent if its Lie algebra g is 2-step nilpotent. In the other words, a Lie algebra g is 2-step nilpotent if [ g , g ] is non-zero and lay in the center of g .[8] heorem 3 Let G be a 2-step nilpotent Lie group of Heisenberg type. Then [ X, j ( Z ) X ] = | X | Z for all elements X ∈ Z ⊥ ( g ) and Z ∈ Z ( g ) .[8] Lemma 4
Let G be a 2-step nilpotent Lie group of Heisenberg type. Then [ X n , j ( Z c ) X n ] = | X n | Z c , for all elements X, Z ∈ g . [8] Because G is a Heisenberg Lie group from [8] we know g ( j ( Z ) X, j ( Z ′ ) X ) = g ( Z, Z ′ ) | X | . Some formulas and definitions are introduced, that will be used throughout thispaper. We may identify an element of g with a left invariant vector field on Gsince T e G may be identified with g . If X,Y are left invariant vector fields on G,then ∇ X Y is left invariant as well. One has the following formulas:(a) ∇ X Y = 12 [ X, Y ] , (b) ∇ X Z = ∇ Z X = − j ( Z ) X, (c) ∇ Z Z = 0 , for all X, Y ∈ Z ⊥ ( g ) and Z, Z ∈ Z ( g )[8].Next definitions are important concepts in present peper. Definition 5
A 2-step nilpotent Lie algebra g is nonsingular if adX : g → Z ( g ) is surjective for each X ∈ Z ⊥ ( g ) . A 2-step nilpotent Lie group G is nonsingularif its Lie algebra g is nonsingular.[8] Definition 6
A totally geodesic subgroup of G is a connected Lie subgroup N,such that N is a totally geodesic Lie subgroup as a submanifold of G. A subal-gebra n of g is totally geodesic if ∇ X Y ∈ n whenever X, Y ∈ n .[8] Let G denote a 2-step nilpotent Lie group with a left invariant metric g and g denote the Lie algebra of G. Write g = Z ( g ) ⊕ Z ⊥ ( g ) where Z ⊥ ( g ) its orthogonalcomplement of center Z ( g ), hence for any X ∈ g we have X = X c + X n , X c ∈ Z ( g ) and X n ∈ Z ⊥ ( g ). For any X, Y ∈ g , by straightforward calculationswe obtain following results and complete this section.( i ) ∇ X n Y n = 12 [ X, Y ] , ( ii ) ∇ X X = − j ( X c ) X n , ( iii ) ∇ X Y = 12 [ X n , Y n ] − j ( Y c ) X n − j ( X c ) Y n . Definition 7
A 2-step nilpotent Lie group is said to have the matching propertyif for any
A, B ⊆ G , | A | = | B | , there exist a matching from A to B. In this section we first introduce almost contact structure, and defined thisstructure on 2-step nilpotent Lie group, then we try to find new properties ofthese Lie groups as a smooth manifolds.Let M be an odd dimensional smooth manifold with a left invariant metric g .Denote by T M the Lie algebra of vector fields on M . Then M is said to be analmost contact metric manifold if there exists a tensor φ of type (1 , ξ called structure vector field and η , the dual 1-form of ξ satisfying thefollowing φ X = − X + η ( X ) ξ, g ( X, ξ ) = η ( X ) (3) η ( ξ ) = 1 , φ ( ξ ) = 0 , η ◦ φ = 0 (4) g ( φX, φY ) = g ( X, Y ) − η ( X ) η ( Y ) , (5)for any X, Y ∈ T M . In this case g ( φX, Y ) = − g ( X, φY ) , the fundamental 2-form Φ on M is given byΦ( X, Y ) = g ( X, φY )and the manifold is said to be contact metric manifold if Φ = dη . Every Heisen-berg Lie group is contact metric Lie group. Example.
Let { X , X , X , Z , Z } be basis of a 5-dimensional real vectorspace g , equip with a Lie algebra structure defined by the bracket relations[ X , X ] = Z , [ X , X ] = Z , [ X , X ] = Z . It’s trivial g is 2-step nilpotent and nonsingular, with left invariant metric g.By easy computation we have j ( Z ) X = X , j ( Z ) X = X , j ( Z ) X = X . Now we can write g ( j ( Z ) X , X ) = − g ( j ( Z ) X , j ( Z ) X )= g ( X , X ) . g is nonsingular, 2-step nilpotent Lie algebra.It is defined Nijenhuis torsion of φ [ φ, φ ]( X, Y ) = φ [ X, Y ] + [ φX, φY ] − φ [ φX, Y ] − φ [ X, φY ] , and N (1) ( X, Y ) = [ φ, φ ]( X, Y ) + 2 dη ( X, Y ) ξ,N (2) ( X, Y ) = ( L φX η )( Y ) − ( L φY η )( X ) ,N (3) = ( L ξ φ ) X,N (4) = ( L ξ η ) X. An almost contact structure ( φ, ξ, η ) is normal if and only if these four tensorsvanish. [6]In continue, we study a Lie group as an almost contact manifold and applyingthe Lie algebra and new results are obtained about the components of almostcontact structure and their features. In almost contact structure ( φ, ξ, η ), from N (3) we have the following theorem.Let N be a subgroup of G and n denote the Lie algebra of N, if n is an Abelian,totally geodesic subalgebra of N, then either n ⊆ Z ( g ) or n is an Abelian sub-space of Z ⊥ ( g ), [9]. The angle θ ( X ), 0 ≤ θ ( X ) ≤ π φX and n is calledthe Wirtinger angle of X. If the Wirtinger angle θ is constant, it’s called slantangle. If n be a slant Lie subalgebra, then H is called slant Lie subgroup, In theother words slant submanifold. If θ = 0, then N is called invariant subgroup.[7]Next theorem show important feature of 2-step nilpotent Lie groups. Theorem 8
Let (G,g) be an almost contact, 2-step nilpotent Lie group with aleft invariant metric and g denote the Lie algebra of G. Then a ) φ ( Z ( g )) = 0 ,b ) φ ( Z ⊥ ( g )) = 0 . Next theorem show that what is the relationship between Abelian subalgebraand invariant subalgebra with j.
Theorem 9
Let G be a 2-step nilpotent, Heisenberg, Lie group with a left in-variant metric, and N is a totally geodesic Lie subgroup of G. Let n and g denotetheir Lie algebras, respectively. If for any X ∈ n we have j ( X c ) X = X , then n is an Abelian Lie subalgebra or n ⊂ Z ⊥ ( g ) . The following theorem gives the necessary sufficient condition for 2-step nilpo-tent Lie groups having the matching property. See [4],[5] for more details.
Theorem 10
A Heisenberg Lie group G passess the matching property if andonly if G has no finite proper subgroup. eferences [1] M. Aliabadi, M. V. Janardhanan. On local matching property in Groupsand vector spaces , arXiv:1601.06322v2 [math.GR] 2 Feb 2016.[2] M. Aliabadi, M. Hadian, A. Jafari,
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