Bull. Sci. math.•••(••••)•••–•••
www.elsevier.com/locate/bulsci
Sub-semi-Riemannian geometry of general H -type groups
✩Mauricio Godoy Molina
a,∗, Anna Korolko
b, Irina Markina
caCMAP, École Polytechnique, CNRS, France bPolytec R&D Institute, Norway
cDepartment of Mathematics, University of Bergen, Norway Received 2 September 2012
Abstract
We study a special class of nilpotent Lie groups of step 2, that generalizes the class of the so-called H(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of an inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of generalH-type groups. We also present some results on sectional curvature and the Ricci tensor of semi-Riemannian generalH-type groups.
©2013 Elsevier Masson SAS. All rights reserved.
MSC:53C50; 53B30; 53C17; 15A63
Keywords:H-type group; Sub-semi-Riemannian geometry; Geodesic; Composition of quadratic forms; Clifford algebra
1. Introduction
A. Kaplan in 1980 constructed the so-called Heisenberg-type algebras[19], whose commu- tators are intimately related to the existence of a Clifford algebra representation over an inner
✩ The work of the first author is partially supported by the ERC Starting Grant 2009 GeCoMethods and the DIGITEO- Région Ile-de-France project CONGEO. The work of the first and the third authors is partially supported by NFR- FRINAT grant #177355/V30.
* Corresponding author.
E-mail addresses:[email protected](M. Godoy Molina),[email protected] (A. Korolko),[email protected](I. Markina).
0007-4497/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.
http://dx.doi.org/10.1016/j.bulsci.2013.05.005
product space. He observed that the presence of a composition of two positive definite quadratic formsϕandλon two vector spacesHandV, respectively, allows one to introduce a Lie bracket [·,·]:H×H→V that induces a Lie algebra structure onH⊕V. He also showed that the re- quirement that the adjoint map on a Lie algebra(H⊕V ,[·,·])of step 2 is an isometry between the orthogonal complement to its kernel inH andV is a necessary and sufficient condition to recover a composition of certain quadratic forms. The presence of a composition is related to the existence of anH-representation of the Clifford algebra C(V ,−λ). We emphasize that in all mentioned constructions only positive definite quadratic forms and scalar products were used.
Nevertheless, since compositions are defined for arbitrary bilinear non-degenerate quadratic forms, the positivity restriction seems a priori unnecessary. Assuming only non-degeneracy, one can define in much the same way Lie brackets and, as a consequence, it is possible to construct generalH(eisenberg)-type algebras that include those constructed by Kaplan as a particular case.
To show that any generalH-type algebra arises as a result of this construction is one of our main results, see Theorem 1. As in the previous case, the construction is closely related to the existence of a representation onHof the Clifford algebra generated byV endowed with a non-degenerate scalarproduct.
We would like to mention that analogues of general H-type algebras and groups were in- troduced earlier under different names and definitions, see [9,10]. In [9], the author studied these Lie algebras from the point of view of existence of the Clifford modules and presented their natural classification according to the classification of Clifford algebras. In the work[10], generalH-type groups are particular examples of 2-step nilpotent groups with invariant semi- Riemannian metrics. In both cases the authors did not address clearly the connection between the introduced groups and the composition of quadratic forms. Our paper fills this gap and shows how these two situations are intimately related. In addition to these differences, the aforementioned studies focus mostly in semi-Riemannian geometry of the underlying Lie group, not dealing with the natural non-holonomic restrictions imposed by the structure of the Lie algebra.
The Heisenberg, and afterwards H-type, groups are core examples in the study of sub- Riemannian geometry. Recall that a sub-Riemannian manifold is a triplet(M,H, ρH), where M is a smooth manifold,His a smooth subbundle of the tangent bundle andρHis a smoothly varying inner product defined for vectors from Hm,m∈M. Under the bracket generating (or completely non-holonomic) condition onH, the sub-Riemannian manifold can be considered as a metric space where the distance function is induced by the metric tensorρH. We give a list of references that is far from being complete, where the fundamentals of sub-Riemannian geometry can be found[1,3,5,28,31].
In the same way as Riemannian and semi-Riemannian geometry are related, one can study an analogue of sub-Riemannian geometry, that we call sub-semi-Riemannian geometry. Namely, a sub-semi-Riemannian manifold is a triplet (M,H, H), where M andH are as above, but His a smoothly varyingscalar productdefined for vectors from Hm,m∈M. Examples and studies can be found in [7,10,13–15,17,18,22–24]. Our main interest is to consider the sub- semi-Riemannian geometry of general H-type Lie groups, where the smooth subbundleHis introduced by commutation relations in the corresponding Lie algebra. Since the generalH-type groups, introduced in the present work, carry a natural scalar product closely related to its Lie structure, we think that they will play an analogous cornerstone role in the study of sub-semi- Riemannian geometry. Apart of this mathematical interestH-type groups can find applications in affine control systems, relativity theory, Kaluza–Klein model or more general Yang–Mills theory and other subjects[2,16,28].
The present work is organized as follows. After the introduction we review the notion of com- position of quadratic forms and its relation to Lie algebras of step 2. In Section2.5the general H-type algebras are defined and the main result is proved. The rest of Section2is dedicated to relations between the structure constants of a generalH-type algebra and coefficients of the Clif- ford algebra representation. Section3contains examples, showing that there are generalH-type algebras that are not among the classical ones introduced by A. Kaplan. Moreover, we present an example showing that not all of generalH-type algebras can be obtained by taking the classical ones and changing the natural inner product to an arbitrary scalar product. Section4 is dedi- cated to the study of sub-semi-Riemannian manifold related to nilpotent Lie groups of step 2.
Using the Hamiltonian formalism, we write the geodesic equations and give general solutions in parametric form. For the generalH-type groups we present the closed parametric formulas for geodesics, which is possible due to the extra symmetries of the problem. The last Section5 collects some properties about the semi-Riemannian Levi-Civita connection on generalH-type groups, sectional curvature and the Ricci curvature tensor.
2. GeneralizedH-type algebras
2.1. Lie algebras of step2
LetHbe a Lie algebra andV a vector space. A central extension of the Lie algebraH byV is a new Lie algebra that can be obtained as follows. Consider a bilinear skew-symmetric map Ω:H×H→V satisfying
Ω
[h1, h2]H, h3 +Ω
[h2, h3]H, h1 +Ω
[h3, h1]H, h2
=0, (1)
for allh1, h2, h3∈H and where [·,·]H denotes the bracket in H. Such kind of map is called 2-cocycle. The vector spaceH⊕V endowed with the bracket
(h1, v1), (h2, v2)
=
[h1, h2]H, Ω(h1, h2) is the desired central extension.
IfH is abelian, then all brackets in (1)vanish, and by the bilinearity of Ω, Eq.(1) holds trivially. The new brackets take the form
(h1, v1), (h2, v2)
=
0, Ω(h1, h2)
, h1, h2∈H, v1, v2∈V .
Such kind of Lie algebras with abelianH are the main object of our work and we denote them byg=(H ⊕V ,[·,·]). It is easy to see thatgis a nilpotent Lie algebra of step 2. The subspace His known as the horizontal space, andV is known as the vertical space.
It is clear that the properties of the algebragare determined by properties of the 2-cocycleΩ. One of the possible ways of constructing suchΩ was proposed by A. Kaplan in the series of works[19–21], where compositions betweenpositive definite quadratic formswere used. Such kind of algebras are calledH-type algebras or Heisenberg-type algebras. In the present paper we propose to extend this construction by making use ofnon-degenerate quadratic formsadmitting a composition, regardless of the sign requirement. We start from reviewing the definition and properties of composition.
2.2. Composition of quadratic forms
LetHandUbe real vector spaces, and letϕ:H→Randλ:U→Rbe quadratic forms. For the rest of this article, we will assume that bothϕandλare non-degenerate, in the sense that the associated symmetric bilinear forms obtained by polarization
h1, h2ϕ=1 2
ϕ(h1+h2)−ϕ(h1)−ϕ(h2)
, h1, h2∈H, u1, u2λ=1
2
λ(u1+u2)−λ(u1)−λ(u2)
, u1, u2∈U, (2)
satisfy the condition that ifh1, h2ϕ =0, for allh1∈H, thenh2=0; and similarly for·,·λ. We also observe the trivial consequences of(2)
h, hϕ=ϕ(h), u, uλ=λ(u). (3)
We emphasize that the quadratic formsϕandλare not assumed to be positive definite.
Definition 1.A bilinear mapμ:U×H→H is called a composition ofϕandλif for anyu∈U and anyh∈Hthe equality
ϕ
μ(u, h)
=λ(u)ϕ(h) (4)
holds.
An old problem in the theory of quadratic forms asks for conditions for the existence of a composition of two given quadratic forms. The answer to this question, in the real case for sums-of-squares, is a classical non-trivial application of the theory of representation of Clifford algebras, see[25, pp. 133–139].
Example.LetU=H=R2withu=(y1, y2),h=(x1, x2)and
ϕa(h)=ϕa(x1, x2)=x12+ax22, λa(u)=λa(y1, y2)=y12+ay22, for anya∈R. The identity
y12+ay22
x12+ax22
=(y1x1+ay2x2)2+a(y1x2−y2x1)2 shows that the bilinear mapμa:R2×R2→R2defined by
μa(u, h)=μa
(y1, y2), (x1, x2)
:=(y1x1+ay2x2, y1x2−y2x1) (5) is a composition of the quadratic formsϕa=λa,a∈R.
Note that for a=0, Eq.(5) still gives a composition of ϕa andλa. Even though the non- degeneracy requirement plays no role in this example, non-degeneracy is of core importance in the general arguments that will follow.
2.3. Lie algebras and compositions
Assume there is a composition μ:U×H →H of the quadratic forms ϕ:H → R and λ:U →R. We will suppose that μ is normalized, in the sense that we choose u0∈U such thatλ(u0)=1, and
μ(u0, h)=h, h∈H.
This can always be done after a process of normalization, see[25, p. 134]. LetV ⊂U be the orthogonal complement of span{u0}, with respect to ·,·λ, and π:U→V the corresponding orthogonal projection. Note that the conditionλ(u0) =0 is essential here, since the requirement thatu0is not a null vector (λ(u0) =0) guarantees thatV =U⊕span{u0}, see[29, Lemma 2.23].
An important fact to have in mind is that the space End(H )of endomorphisms ofH ad- mits a representation of the Clifford algebra C(V ,−λ), i.e. there is an algebra homomorphism ρ: C(V ,−λ)→End(H ). More precisely, such algebra homomorphism is given by the map v→μ(v,·) defined on generators and then can be extended to the representation of whole Clifford algebra by standard methods. To see that this is indeed the case, note that the skew- symmetry, that will be proved inProposition 1, and composition formula give for any non-zero v∈V and arbitraryh∈H
μ
v, μ(v, h) , h
ϕ= −
μ(v, h), μ v, h
ϕ= −v, vλ
h, h
ϕ. (6)
We conclude thatμ2(v, h)def=μ(v, μ(v, h))= −v, vλhor, in other words, μ2(v,·):H→H is the identity map multiplied by the scalar−λ(v), which is exactly the image of the defining property of C(V ,−λ).
Remark 1.Up to an extra technical requirement, a converse of this result also holds, see[25, Chapter 5].
In the next step we use a compositionμto define a Lie algebra structure on the vector space H⊕V. To construct the corresponding Lie bracket, we first introduce a bilinear mapΦ:H× H→Uby means of the equality
u, Φ h, h
λ=
μ(u, h), h
ϕ, (7)
valid for allu∈Uand allh, h∈H. The mapΦ:H×H→Uis in general not anti-symmetric, for an example see Section3.1.2. Nevertheless, projected toV =span{u0}⊥, it has the following useful property.
Proposition 1. The map π ◦Φ :H ×H →V is an anti-symmetric bilinear map, i.e. π ◦ Φ(h, h)= −π◦Φ(h, h)for allh, h∈H.
Proof. Notice that Eq.(4)can be conveniently rewritten as μ(u, h), μ(u, h)
ϕ= u, uλh, hϕ, u∈U, h∈H, (8)
by using(3). Applying this identity tou+u∈U, we obtain the equality μ(u, h), μ
u, h
ϕ= u, u
λh, hϕ, (9)
by bilinearity. If in Eq.(9)we evaluateu=u0and assumeu=v∈V, then we see that μ(v, h), μ(u0, h)
ϕ=
μ(v, h), h
ϕ=0, (10)
due to the normalization imposed toμ. Thus we have 0=
μ
v, h+h , h+h
ϕ
=
μ(v, h), h ϕ
=0
+ μ
v, h , h ϕ
=0
+
μ(v, h), h
ϕ+ μ
v, h , h
ϕ,
forv∈V. Therefore the mapμ(v,·):H→H,v∈V, is a skew-symmetric map with respect to the scalar product·,·ϕ. This implies that forv∈V and for allh, h∈H we have
v, Φ h, h
λ= − v, Φ
h, h
λ,
which means thatπ◦Φ(h, h)= −π◦Φ(h, h). 2
As in Section2.1,Proposition 1allows us to define a Lie bracket onH⊕V by (h1, v1), (h2, v2)
=
0, π◦Φ(h1, h2)
. (11)
The resulting Lie algebrag=(H⊕V ,[·,·])is a Lie algebra of step two.
The next result is completely independent of the chosen scalar products; nevertheless, proving it with the non-degenerate products induced byϕandλ, helps us to obtain a very useful corollary.
Lemma 1.LetZ(g)denote the center of the Lie algebrag=(H⊕V ,[·,·])of the step2, where the bracket is defined by(11). Ifg∈H∩Z(g), theng=0.
Proof. Ifg∈H∩Z(g), theng=(h,˜ 0),h˜∈Hand (h,˜ 0), (h, v)
=
0, π◦Φ(h, h)˜
=(0,0) for any(h, v)∈g.
It implies thatΦ(h, h)˜ ∈kerπfor allh∈H, orΦ(h, h)˜ =ku0for somek∈R. Thus the equality μ(v,h), h˜
ϕ=
v, Φ(h, h)˜
λ=kv, u0λ=0 for allh∈H
and the non-degeneracy of the formϕyieldμ(v,h)˜ =0 for anyv∈V. Because of(8) 0=
μ(v,h), μ(v,˜ h)˜
ϕ= v, vλ ˜h,h˜ϕ, for anyv∈V ,
we conclude that ˜h,h˜ϕ=0. If the quadratic formϕis positive definite, then we conclude that h˜=0 and finish the proof. In the case of non-degenerate indefinite form we need more careful arguments.
Let us assume thath˜ =0 and ˜h,h˜ϕ=0. Using(8)we have for arbitraryh∈H μ(v,h˜+h), μ(v,h˜+h)
ϕ= v, vλ ˜h+h,h˜+hϕ
= v, vλ
2 ˜h, hϕ+ h, hϕ
. On the other hand, sinceμ(v,h)˜ =0 for allv∈V, we have that
μ(v,h˜+h), μ(v,h˜+h)
ϕ=
μ(v,h)˜ +μ(v, h), μ(v,h)˜ +μ(v, h)
ϕ
= v, vλh, hϕ.
Comparing both sides we see thatv, vλ ˜h, hϕ=0 for an arbitraryv∈V and anyh∈H. This leads to a contradiction with the fact thath˜ =0 due to the non-degeneracy of the formϕ. 2
As a corollary we immediately get the following.
Corollary 1.Ifμ:V×H→H is a composition of quadratic formsλandϕ, and for anyv∈V one hasμ(v, h)=0, thenh=0. Similarly ifμ(v, h)=0for anyh∈H, thenv=0.
Let us observe more properties of compositions.
1. Equality (10)shows that the mapμ(v,·):H→H transforms anyh∈H to a vectorh= μ(v, h)∈Horthogonal toh, for anyv∈V,v =0.
2. Formula (9) ensures that any null vectorh∈H (h =0, h, hϕ =0) is mapped to a null vectorh=μ(v, h)for anyv∈V.
3. The same formula(9)implies that ifh∈H is fixed andh, hϕ=1, then the mapμ(·, h) fromV to the image ofμ(·, h)inH is an isometry, and ifh, hϕ= −1 then the same map defines an anti-isometry.
4. For anyv∈V such thatv, vλ=1, Eq.(6)shows that the mapμ(v,·)is an almost complex structure onH, that is,μ2(v,·)= −IdH. Similarly, ifv, vλ= −1, thenμ(v,·)is a Cartan involution onH, i.e.,μ2(v,·)=IdH.
5. Lemma 1 implies that the center of the Lie algebra g=(H ⊕V ,[·,·])coincides with V and the bracket[·,·]defines a map[·,·]:H×H→V. Let us write adh(·)= [h,·], then adh
defines a map adh:H→V. The relation(7)can be written as v,adh
h
λ= v,
h, h
λ=
μ(v, h), h
ϕ, (12)
for v∈V, andh, h∈H. We see that for anyh∈H, the image ofμ(·, h)belongs to the orthogonal space to the kernel of adh, which we denote by ker⊥(adh). Moreover, the map μ(·, h)is the adjoint map to adh(·):(H, ϕ)→(V , λ).
6. Substitutingh=μ(v, h)withh =0,h, hϕ=0 in(12), we obtain v,adh
μ(v, h)
λ=
μ(v, h), μ(v, h)
ϕ= v, vλh, hϕ=0
for anyv∈V. The conclusion is thatμ(v, h)∈ker(adh)since the quadratic formλis non- degenerate.
7. Taking into account 5., we conclude that the image of the mapμ(·, h)is ker(adh)∩ker⊥(adh) for anyh =0 andh, hϕ=0.
2.4. General H(eisenberg)-type Lie algebras
Let us, among all Lie algebras of step 2 defined in Section2.1and carrying a scalar product
·,·, consider special ones that we call generalH-type Lie algebras since they generalize the definition given in[19]in the case when the scalar product·,·is an inner product (positive definite).
Let us consider an arbitrary Lie algebrag=(H ⊕V ,[·,·])of step two, whereV is the cen- ter of the Lie algebra andH is its complement. We assume thatH ⊕V is endowed with a non-degenerate scalar product·,· = ·,·H+ ·,·V such that·,·H is non-degenerate scalar
product onHand·,·V is a non-degenerate scalar product onV. The decompositionH⊕V be- comes orthogonal with respect to·,·. SinceV is the center of the Lie algebrag, the commutator is a map[·,·]:H×H→V. Leth∈H and we assume from now on that the kernel
ker(adh:H→V )=
h∈H:
h, h
=0
is non-degenerate subspace of (H,·,·H). We denote by Hh the orthogonal complement to ker(adh). ThenHh⊕ker(adh)=H,Hh∩ker(adh)= {0}, and the subspaceHh is also a non- degenerate subspace ofH, see[29].
Definition 2. We say that (g,·,·) is a Lie algebra of generalH-type if adh :Hh→V is a surjective isometry or anti-isometry for every vectorh∈H, such thath2H= h, hH= ±1.
If·,·is positive definite, thenDefinition 2coincides with the definition ofH-type groups given by A. Kaplan[19]. In this context, we have the following analogue of Theorem 1 in[19].
Theorem 1. Let g be the Lie algebra constructed in Section 2.3, by using a composition of quadratic formsϕandλ. Thengis a generalH-type Lie algebra with·,· = ·,·ϕ+ ·,·λ|V.
Conversely, for any given general H-type Lie algebra g=(H ⊕V ,[·,·],·,·) there exist a vector spaceU=V ⊕span{u0}, quadratic formsϕ on H and λ onU and a composition μ:U×H→Hofϕandλ, such thatgis built from the compositionμas in Section2.3.
Proof. Let g be a step 2 Lie algebra with underlying vector spaceH ⊕V, center V and a compositionμof the quadratic formsϕandλ. Then the commutator[·,·]:H×H→V is defined by Eq.(12). We see that if we define the scalar product·,·ongby·,· = ·,·ϕ+·,·λ|V, then the mapμ(·, h):V →H is a formal adjoint to adh:H→V with respect to the scalar product
·,·.
First, we need to prove that for eachh∈Hwith normh, hϕ= ±1 the map adh:Hh→V
h→ h, h
is an isometry or an anti-isometry. We start to show that it is a surjective map. Let v∈V and fixh∈H such that for instanceh, hϕ=1. We show thath=μ(v, h)satisfies adh(h)=v. According to(12) and (9), we have forh=μ(v, h)
v,adh
μ v, h
λ=
μ(v, h), μ v, h
ϕ= v, v
λh, hϕ= v, v
λ. Therefore
v,adh
μ v, h
−v
λ=0 for anyv∈V .
By non-degeneracy of the quadratic form λ, we have adh(μ(v, h))=v, which shows the sur- jectivity. If we would fixh∈Hwithh, hϕ= −1, then we need to choseh=μ(v,−h).
The map adh:Hh→V is clearly injective by the definition of Hh. We see that adh is an isomorphism betweenHhandV.
To check that adhis an (anti-)isometry, we need to check the equality adh
h ,adh
h
λ= ± h, h
ϕ for anyh, h∈Hh with some fixedh∈Hsuch thath, hϕ= ±1. Denote
adh
h
=v and adh
h
=v, then from the previous considerations we have
h=μ v, h
and h=μ v, h
or h=μ v,−h
and h=μ v,−h
, according to the valueh, hϕ=1 orh, hϕ= −1, respectively. This yields
h, h
ϕ= μ
v,±h , μ
v,±h
ϕ= v, v
λh, hϕ= ± adh
h ,adh
h
λ, where we used(9)in the second equality. We have finished the first part of the proof.
In the other direction, the problem is more subtle. Letg=H⊕V be a generalH-type algebra, with non-degenerate scalar product·,· = ·,·H+ ·,·V and Lie bracket[·,·]. At the first step we need to find a bilinear mapμ:V ×H→Hand then extend it to a map fromU×H toH.
We start from the following observation. The bilinear formBv:H×H→R Bv:H×H→R
h, h
→ v,
h, h
V
defined for anyv∈V \ {0}has the following property: ifh∈H is fixed andh, hH =0 then there ish˜∈H,h˜ =0, such thatBv(h,h)˜ =0. Indeed, fixh =0 inH withh, hH =0. Choose 0 =v∈V and since the scalar product·,·V is non-degenerate we find non-zerov∈V such thatv, vV =0. DenotehH
def=√
|h, hH|. Since the map ad h
hH :H h
hH →V is surjective, we find the unique non-zeroh∈H h
hH such that ad h
hH(h)=vandBv(h, h) =0 because of 0 =
v, v
V = v,ad h
hH
h
V = 1
hH
Bv h, h
.
Letv∈V andh∈H. We defineμ:V×H→H by the formula μ(v, h), h
H :=
v, h, h
V. (13)
It is easy to see the following properties ofμ.
a) The mapμis bilinear.
b) For anyv∈V andh∈H the elementμ(v, h)∈ker⊥(adh).
c) For any non-zerov∈V, the mapμ(v,·):H→H is skew adjoint with respect to·,·H: μ(v, h), h
H= − h, μ
v, h
H.
d) If we seth=h, then the last property immediately implies thatμ(v, h)is orthogonal toh for arbitrary choice ofhandv.
e) For fixed 0 =h∈H the map μ(·, h):V →H is the formal adjoint to adh:H →V with respect to the scalar product·,·ing.
Now we study the properties ofμ(·, h):V →H for someh∈H withh, hH =0. We will show
h, μ(v, h)
= h, hHv, h, hH =0, (14) and the formula
μ(v, h), μ v, h
H= v, v
Vh, hH, v, v∈V , h∈H. (15)
Letv∈V, then since the map adh:Hh→V is bijective we find the uniqueh˜∈Hhsuch that adh(˜h)=v. Then by the (anti-)isometry property we have for an arbitraryh∈H
h, hH
h, h˜
H = ±h, hH
ad h
hHh,˜ ad h
hHh
V = [h,h˜],
h, h
V
= v,
h, h
V =
μ(v, h), h
H. Thush, hHh˜=μ(v, h). Then
h, μ(v, h)
=
h,h, hHh˜
= h, hHv.
Now we move to show the composition formula(15). Letv, v∈V andh∈Hwithh2 =0.
Then
μ(v, h), μ v, h
H (13)=
v, h, μ
v, h
V
(14)= h, hH
v, v
V.
To show equalities(14) and (15)for h =0 withh2H=0 we use the continuity properties of linear maps. To proceed, we choose an orthonormal basis onH, see[29, p. 50], and consider coordinates with respect to this basis. It can be easily seen that the following arguments do not depend on this choice. Lethnbe a sequence of non-null vectors inHsuch thathn→hasn→ ∞ coordinate-wise. Thenμ(v, hn)→μ(v, h)asn→ ∞coordinate-wise inHfor any fixedv∈V. Since the Lie bracket and scalar product are continuous maps we conclude that(14)implies that for null vectorh∈H the image μ(v, h)belongs to ker adh for anyv∈V. The equality(15) shows that the imageμ(v, h)of a null vectorh∈His a null vector.
Having the equality(15)for allh∈H, we substitutehwith an arbitrary sumh+hand obtain μ(v, h), μ
v, h
H+ μ
v, h , μ
v, h
H=2 v, v
V
h, h
H. Applying skew-symmetry we get
− μ
v, μ v, h
, h
H− μ
v, μ v, h
, h
H=2 v, v
V
h, h
H
or in other words μ
v, μ v,·
+μ
v, μ(v,·)
= −2 v, v
VIdH(·), v, v∈V , where IdH is the identity map inH. Particularly, forv=vwe deduce
μ2(v,·)=μ
v, μ(v,·)
= −v, vVIdH(·).
Thus
μ(v, h), μ(v, h)
H= − μ
v, μ(v, h) , h
H = v, vVh, hH,
and we showed the composition formula(8)for μ:V ×H→H. With this we recover all the properties of the bilinear mapμ:V×H→H, listed in items 1.–7.
The next step is to extend the map μto a vector spaceU. SetU=V ⊕Rwith the scalar product·,·U = ·,·V + ·,·R, where·,·R is usual Euclidean product. Define an extended bilinear mapμ˜:U×H→Hby
˜
μ(v+α, h):=μ(v, h)+αh, v∈V , h∈H, α∈R. Then
μ(v˜ +α, h),μ(v˜ +α, h)
H =
μ(v, h), μ(v, h)
H+α2h, hH+2α
h, μ(v, h)
H.
The last term in the right hand side vanishes due to the property c). Applying the composition formula forμwe obtain
μ(v˜ +α, h),μ(v˜ +α, h)
H= h, hH
v, vV + α, αR ,
that shows the composition μ˜:U ×H → H of the quadratic forms λ(·)= ·,·U and ϕ(·)= ·,·H. As an element u0 in this case would be chosen u0 =(0,1) with λ(u0)= (0,1), (0,1)U= 0,0V + 1,1R=1.
Remark, that we can use a negatively definite product·,·−R inR, but in this case we define the scalar product onUby·,·U= ·,·V − ·,·−Rthat actually leads to the same product inU and the same final result. This finishes the proof. 2
The following corollary is implicitly contained in the proof ofTheorem 1.
Corollary 2.Let(g=H⊕V ,·,·)be a general H-type algebra with the centerV and the adjoint mapadh:Hh→V,h, h = ±1as was defined inTheorem1. Letμ:V×H→Hbe the restriction of the composition of quadratic forms·,·|H,·,·|V that rises to the algebrag. Then the mapsadh(·):Hh→V andμ(·, h):V →Hhare mutually inverse maps.
Proof. Since the algebragis of generalH-type the map adh(·):Hh→V is bijective and there- fore the inverse map exists. It was shown in the proof ofTheorem 1that the image of the map μ(·, h):V →Hbelongs toHhand moreover
adh
μ(·, h)
=IdV ifh, hH =1 and
adh
μ(·,−h)
=IdV ifh, hH= −1.
We claim thatμ(·, h):V →Hhis bijective. Indeed if we assume thatμ(·, h)is not surjective for h, hH =1, then there ish∈Hhthat is not in the image ofμ(·, h). Let adh(h)=v∈V then adh(μ(v, h))=vthat impliesh=μ(v, h)by injectivity of adhand leads to contradiction. If we now assume thatμ(·, h)is not injective then we findv, v∈V,v =v, such thatμ(v, h)= μ(v, h). But in this case
v=adh
μ v, h
=adh
μ v, h
=v
by bijectivity of adhand we again get a contradiction. The proof forh, hH= −1 is analogous and we conclude that adhandμ(·, h)are inverse maps to each other. 2
2.5. Structure constants of a generalH-type algebra
Consider the scalar product vector spaces(H,·,·H)and(V ,·,·V)of indicesνH andνV
respectively. Recall that the index of a scalar product vector space is the dimension of a maximal subspace where the scalar product is negative definite. Assume, from now on, that(h1, . . . , hn) and(v1, . . . , vm)are orthonormal bases of the vector spacesH andV, respectively. We assume for the rest of the paper that they are ordered in such a way that they satisfyhi, hjH=ενiHδij andvα, vβV =εανVδαβ, whereδlkis the Kronecker symbol andεkνis the sign symbol defined by
εkν=−1 ifkν, 1 otherwise.
In addition, denote byJH=(hi, hjH)=(ενiHδij)andJV=(vα, vβV)=(εναVδαβ)the Gram matrices of(H,·,·H)and(V ,·,·V)with respect to the chosen bases.
Let g=(H ⊕V ,[·,·],·,· = ·,·H + ·,·V)be a generalH-type algebra. As we saw in Section2.4, the Lie bracket defines an endomorphismμ(v,·):H →H for any non-vanishing v∈V by formula(13). We write
μ(vα, hi)= n j=1
Aαijhj, (16)
and
[hi, hj] = m β=1
Bijβvβ. (17)
Sinceghas step 2, the numbersBijβ are exactly the structure constants ofg.
Proposition 2.The coefficients Aαij and the structure constantsBijβ of the general H-type Lie algebra are related by
εjνHAαij=εναVBijα. Proof. We calculate
μ(vα, hi), hj
H=
vα,[hi, hj]
V = m β=1
Bijβvα, vβV = m β=1
BijβεανVδα,β=εναVBijα. (18) From the other side
μ(vαhi), hj
H= n k=1
Aαikhk, hjV = n k=1
AαikενkHδkj=ενjHAαij. (19) Comparing(18) and (19)we obtain the result. 2
3. Examples
3.1. General Heisenberg algebras
As one of our main motivating examples, we consider the case of the Heisenberg Lie algebra.
Consider the real nilpotent Lie algebraheis2n+1with generators X1, . . . , Xn, Y1, . . . , Yn, Z
satisfying the well-known commutator rules
[Xk, Yl] =δklZ, [Xk, Xl] = [Yk, Yl] = [Xk, Z] = [Yl, Z] =0, wherek, l∈ {1, . . . , n}. The horizontal subspace is defined by
H=span{X1, . . . , Xn, Y1, . . . , Yn} ⊂heis2n+1
and the vertical space is simplyV =span{Z}. We denote byH2n,νH,1the Lie algebraheis2n+1, endowed with the non-degenerate scalar product onHof indexνH. The super-index(2n, νH,1)
refers to the dimension ofH, its index and the dimension of the center ofheis2n+1. For example, the classical sub-Riemannian structure onheis2n+1is denoted byH2n,0,1.
The simplest non-trivial example of the objects studied in Section2is the general Heisenberg algebraH2,1,1endowed with a sub-Lorentzian metric[13,22]. To simplify the notation, consider the generatorsX, Y, Zsatisfying
[X, Y] =Z, [X, Z] = [Y, Z] =0.
As above, letH=span{X, Y} ⊂heis3and define a non-degenerate bilinear form on it by X, XH= −Y, YH=1, X, YH=0,
and then extending it linearly to all ofH. Additionally, consider the inner productZ, ZV =1 extended linearly to all ofV =span{Z}.
3.1.1. H2,1,1as a generalH-type algebra
To see thatH2,1,1 indeed satisfies the conditions as in Section2.4, note that ifh=αX+ βY∈H, forα, β∈R, then
h, hH = αX+βY, αX+βYH=α2−β2.
Assumeα2−β2=1, thusα =0. The case in whichα2−β2= −1 can be treated analogously.
Note that
[αX+βY, αX+βY] =αβZ−αβZ=0, αX+βY, βX+αYH=αβ−αβ=0, therefore
ker adh=span{αX+βY} and Hh=span{βX+αY}. An immediate consequence of the above calculations is that
adh(βX+αY )= [αX+βY, βX+αY] =
α2−β2 Z=Z is an isometry between the vector spacesHhandV.
3.1.2. H2,1,1in terms of composition of quadratic forms
In this sense, the above construction has a very clear interpretation, given by formula(5)for a= −1. Proceeding as in Section2.3, we have that ifH =U=R2, and ϕ=λ are such that ϕ(h)=ϕ(x1, x2)=x12−x22andλ(h)=ϕ(y1, y2)=y12−y22, then
ϕ(x1, x2)λ(y1, y2)=ϕ μ
(y1, y2), (x1, x2) ,
whereμ((y1, y2), (x1, x2))=(y1x1−y2x2, y1x2−y2x1). Let us fix u0=(1,0)∈U, which satisfiesλ(u0)=1. Then
μ(u0, h)=μ
(1,0), (x1, x2)
=(x1, x2),
forh=(x1, x2)∈H, thusμ(u0,·)is simply the identity map ofH. The orthogonal complement of span{u0}isV =span{v}, wherev=(0,1). Fixπ:U→V to be the orthogonal projection. It is easy to check that the mapΦ:H×H→Udetermined by Eq.(7)is given by
Φ
(x1, x2), x1, x2
=
x1x1 −x2x2, x2x1 −x1x2 ,
and therefore the spaceH⊕V inherits the Lie algebra structure given by (x1, x2), v1
, x1, x2
, v2
= 0,
x2x1 −x1x2 v
,
for anyh=(x1, x2)∈H,h=(x1, x2)∈H and anyv1, v2∈V. Taking a basis ofH⊕V given byX=((1,0),0),Y=((0,1),0)andZ=((0,0), v), we see that
[X, Y] =Z, [X, Z] = [Y, Z] =0.
3.1.3. Care needs to be taken
Picking incompatible quadratic forms on H andU can have undesirable consequences. For example, consider the problem of finding a composition of the quadratic forms
˜
ϕ(x1, x2)=x12−x22 and λ(y˜ 1, y2)=y12+y22,
defined onH =U=R2, respectively. In order to solve the problem, we need to determine the coefficientsa, b, c, d, α, β, γ , δ∈Rfor a bilinear form
˜ μ
(y1, y2), (x1, x2)
=(ay1x1+by1x2+cy2x1+dy2x2, αy1x1+βy1x2+γ y2x1+δy2x2) to satisfy the composition rule
˜
ϕ(x1, x2)˜λ(y1, y2)= ˜ϕ
˜ μ
(y1, y2), (x1, x2) . If such a map exists, then the following equations must hold
a2−α2=b2−β2= −c2+γ2= −d2+δ2=1, (20) ab−αβ=ac−αγ=ad−αδ=bc−βγ =bd−βδ=cd−γ δ=0. (21) From Eq.(20), it follows thata, b, γ , δ =0. This in turn impliesα, β, c, d =0, by using the first and last equation in(21). Thus none of the coefficients ofμ˜ can vanish. Sinceab=αβ, the first equation in(20)can be rewritten as
α2β2
b2 −α2=α2
β2−b2 b2
= −α2 b2 =1.
But thenα2+b2=0, which gives the desired contradiction.
3.1.4. Extensions to higher dimensions
It is interesting to observe that a sort of “product” construction holds for the general Heisen- berg algebrasH2n,n,1. Consider non-degenerate bilinear form given by
Xk, XlH= −Yk, YlH=δk,l, Xk, YlH=0,
for k, l∈ {1, . . . , n}, and then extending it linearly to all ofH. The inner productZ, ZV =1 is extended linearly to all ofV =span{Z}. It is a simple exercise to note that the arguments in Section3.1.1can be generalized to this case.
Similarly to what was done in Section3.1.2, we consider the vector spaces H =R2n and U=R2with quadratic forms
ϕ(h)=ϕ(x1, y1, . . . , xn, yn)=x12−y12+ · · · +xn2−yn2, λ(u)=λ(u1, u2)=u21−u22,
