Geometric analysis on H -type groups related to division algebras
Ovidiu Calin∗1, Der-Chen Chang∗∗2,andIrina Markina∗∗∗3
1 Department of Mathematics, Eastern Michigan University, Ypsilanti, MI, 48197, USA
2 Department of Mathematics, Georgetown University, Washington DC 20057-0001, USA
3 Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway Received 27 March 2007, accepted 23 March 2008
Published online 15 December 2008
Key words Hamiltonian formalism,H-type groups, geodesics, division algebras, the Siegel upper half space MSC (2000) Primary: 53C17; Secondary: 53C22, 35H20
The present article applies the method of Geometric Analysis to the studyH-type groups satisfying theJ2 condition and finishes the series of works describing the Heisenberg group and the quaternionH-type group.
The latter class ofH-type groups satisfying theJ2condition is related to the octonions. The relations between the group structure and the boundary of the corresponding Siegel upper half space are given.
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
We would like to start from a nice description of four normed division algebras: real numbers (R), complex numbers (C), quaternions (H), and octonions (O) given by Baez [1]. “The real numbers are the dependable breadwinner of the family, the completed ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncom- mutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.” These division algebras generate a special class among theH-type homogeneous groups, the class satisfying theJ2Clifford algebras condition [12]. Three asso- ciative algebras (R,C,H) give the origin to the most exciting generalization incorporating the geometric concept of the direction, so-called Clifford algebras, which Clifford himself called “geometric algebras” [15, 19]. The homogeneous groups satisfying theJ2condition act as translations on the corresponding hyperbolic Siegel upper half spaces and this action can be extended up to the boundary. We present precise formulas of these actions.
The elements of the groups can be associated with the points of the boundary of the corresponding hyperbolic spaces though the action at the origin. The Lie algebras of the corresponding groups can be associated with left invariant vector fields on the tangent bundle to the boundary of the Siegel upper half spaces. The nonvanishing commutative relations define the sub-Riemannian geometry on the boundary of those spaces. The corresponding sub-Laplace operators are closely related with the boundary behavior of holomorphic functions defined on the corresponding Siegel upper half spaces [11, 21].
In the present article we describe the construction ofH-type homogeneous groups associated with the above four division algebras. The Heisenberg group, the quaternion and the octonionH-type groups, have the maximal dimension of their center, and satisfy theJ2condition. These groups can be identified as groups of actions on the Siegel upper half spaces. We give a geometric description between these groups and the boundary of the Siegel upper space via a generalized Cayley transformation. Using Hamiltonian formalism which was first developed by Beals, Gaveau and Greiner on the Heisenberg groups [5] (see also the recent paper by Greiner [14]), we obtain the exact formulas of geodesics connecting two arbitrary points of the group. The cardinality of geodesics depending
∗ e-mail:[email protected], Phone: +1 734 487 1292, Fax: +1 734 487 2489
∗∗ Corresponding author: e-mail:[email protected], Phone: +1 202 687 5609, Fax: +1 202 687 6067
∗∗∗ e-mail:[email protected], Phone: +47 555 82853, Fax: +47 555 89672
on the location of end points. We also use the Lagrangian formalism to obtain the classical action. Then the complex action is presented. The critical points of the complex action recover all the geodesics.
The final version of this paper was in part written while the second and the third authors visited the National Center for Theoretical Sciences and National Tsing Hua University during January 2007. They would like to express their profound gratitude to Professors Jing Yu and Shu-Cheng Chang for their invitation and for the warm hospitality extended to them during their stay in Taiwan.
2 Definitions
We start from the basic definitions that reader can find, for instance, in [12]. LetGbe a real Lie algebra, equipped with the Lie bracket[·,·], which can be written as an orthogonal direct sum,
G=V1⊕V2, [V1, V1]⊆V2, [V1, V2] = [V2, V2] = 0.
Suppose thatG is endowed with a scalar product·,·. Define the linear mappingJ : V2 → End(V1)by the formula
JZX, X=Z,[X, X], for all X, X∈V1, for all Z ∈V2, (2.1) whence
JZT =−JZ, for all Z∈V2. (2.2)
We say thatGisH-type if
JZ2 =−|Z|2U (2.3)
for all Z inV2, where U denotes the identity mapping. TheH(eisenberg)-type groups were introduced by Kaplan [16] in 1970-s and have been studied extensively by many mathematicians, see for instance [12, 17, 18, 20]. The conditions (2.2), (2.3) imply
JZJZ+JZJZ =−2Z, ZU, for all Z, Z∈V2, (2.4) see [12]. If there is a linear mappingJ :V2→End(V1)satisfying (2.2) and (2.3), thenV1is called the Clifford module overV2. The algebraG(or the Clifford module associated withG)satisfies theJ2condition if, whenever X ∈V1andZ, Z∈V2withZ, Z= 0, then there existsZinV2such that
JZJZX =JZX. (2.5)
We present here a result from [12] giving the classification ofH-type algebras satisfying theJ2condition. Denote byG0nthe Euclideann-dimensional space, byG1nthen-dimensional Heisenberg algebra, byG3nthen-dimensional quaternionH-type algebra, and byG71the octonionH-type algebra. The lower index corresponds to the topo- logical dimension of V2 and the upper index reflects the real, complex, quaternion and octonion topological dimensions ofV1.
Theorem 2.1 ([12]) Suppose thatGis anH-type algebra satisfying theJ2condition. ThenGis isometrically isomorphic toG0n,G1n,G3nor toG71.
Before we describe the general construction of groupsG0n, Gn1,G3n, and G71, we would like to remind the Cayley–Dickson construction of division algebrasR(real numbers),C(complex numbers),H(quaternion num- bers), andO(octonion numbers). The Cayley–Dickson construction explains why each one of the algebras fits neatly inside the next. Recall that the division algebra means that each nonzero element has a unique inverse element. The Cayley-Dickson construction is nicely given in [1]. We present it here for the completeness of this article.
The complex number, as well-known, can be thought of as a pair(a, b)of real numbersa, b∈R. We define the conjugate to a real number asa∗=aand the conjugate to the pair as
(a, b)∗= (a∗,−b). (2.6)
Then theCayley–Dickson productis defined by
(a, b)(c, d) = (ac−db∗, a∗d+cb). (2.7)
Now we can think a pair(a, b)as a quaternion, wherea, b ∈ C. The conjugate is defined as in (2.6) and the product as in (2.7). We obtain the quaternion numbersH that form a noncommutative algebra with respect to (2.7). Finally, we define an octonion as a pair(a, b)witha, b∈H, the conjugate as in (2.6), and the product as in (2.7). The octonions with the operation (2.7) makes up a noncommutative, nonassociative algebra. Actually, we can continue the Cayley–Dickson construction doubling the dimension and getting a bit worse algebras. During the procedure we have lost the fact that every element is own conjugate, then we lost commutativity, associativity, and continuing we lost the division algebra property.
3 Constructions of H -types groups satisfying J
2condition
We present a general construction of theH-types algebras, satisfyingJ2condition. Using the Cayley–Dickson product, we first describe the following groups: Euclideann-dimensional spaceGn0 =Rn, the Heisenberg group Gn1, the quaternionH-type groupGn3, and the octonionH-type groupG17. Then we obtain the corresponding algebrasGn0,G1n,G3n, andG71as infinitesimal representations of the groups.
3.1 The Euclidean space
The space Gn0 = Rn is a trivial example of anH-type group since all commutative relations vanish. The underlying spaceV1is identified withRnvia the exponential map which is identity in this case. The centerV2is the empty set.
3.2 The Heisenberg groupGn1
We start fromn= 1and then generalize for an arbitraryn= 2k,k∈N. Complex numbers has 2 unities, whose absolute value of square equals1:
real 1 = (1,0), and imaginary i= (0,1),
such that12= 1,i2=−1. Take a complex numberz= (x1, x2),x1, x2∈R, and a real numbert. Define a new noncommutative law between elementsh= [z, t]∈C×Randp= [z, t]∈C×Rby
hp= [z, t][z, t] =
z+z, t+t+1 2(zi)·z
, (3.1)
where we first take the Cayley–Dickson productzi= (x1, x2)(0,1)and then the scalar product “·” of vectors z, z∈R2. This multiplication law can be deduced from the matrix product of upper triangular3×3-matrices [8].
If we use the representation ofias the(2×2)matrix i=
0 1
−1 0
, then the group low can be written as
hp= [z, t][z, t] =
z+z, t+t+1 2(zi) ·z
.
Using the algebraic form of a complex numberz=x1+ix2=Rez+iImz, we can write (3.1) in the form hp= [z, t][z, t] =
z+z, t+t+1
2Im(z∗z)
,
wherez∗zis a Cayley–Dickson product ofz∗byz. The non-commutativity of new multiplication law inC×R is seen for the last variabletfrom the one dimensional real space, that corresponds to the existence of only one
imaginary unit. It can be shown thatC×Rwith new non-commutative multiplication forms a Lie group with the identity element[0,0], with the left translationLh(p) =hp= [z, t][z, t], and the inverse toh= [z, t]element h−1= [−z,−t].
In order to present ann-dimensional analogue of the Heisenberg group we take n-dimensional vectors of complex numbersw = (z1, . . . , zn),w = (z1, . . . , zn). The matrixiis changed by a block diagonal matrix J = diagiwithn matricesion the diagonal. The multiplication law between the elementsh = [w, t] and p= [w, t]∈Cn×Ris transformed into the following one
hp= [w, t][w, t]
=
w+w, t+t+1 2
n l=1
(zli) ·zl
=
w+w, t+t+1
2(wJ)·w
=
w+w, t+t+1
2Im(w∗w)
,
wherew∗w =n
l=1z∗lzl.
The Heisenberg algebraG1n,n = 2k,k∈N, is identified with the left invariant vector fields on the tangent space at the identity element of the group. It splits into the direct sumV1⊕V2, where
V1= span{X11, X12, X21, X22, . . . , X1n, X2n} with a basis given by
X1l=∂x1l−1
2x2l∂t, and X2l=∂x2l+1
2x1l∂t, l= 1, . . . , n.
The vector fieldX = (X11, . . . , X2n) =
∇x+ 12(xJ)∂t withx= (x11, . . . , x2n),∇x = (∂x11, . . . , ∂x2n), is a natural analogue of the Euclidean gradient onR2n. The subspaceV2is one dimensional and generated by Z =∂t. Since[X1l, X2l] =Zand other commutators vanish, we verify the condition (2.1). The endomorphism JZ is represented by the matrixJwhich possesses properties (2.2), (2.3). TheJ2condition holds trivially, since only oneJZ is different from0.
3.3 Quaternion groupGn3
As in the previous case, we start from 1-dimensional case and then consider the multidimensional analogue.
Quaternion numbers, which we think of as a pair of complex numbers, has one real unity1 = (1,0),12= 1and three imaginary unities
i1= (i,0), i2= (0,1), i3= (0, i), such that i21=i22=i23=i1i2i3=−1.
The Cayley–Dickson product is no longer commutative, for example,
i1i2=−i2i1=−i3, i2i3=−i3i2=−i1, i3i1=−i1i3=−i2. (3.2) In order to design the quaternionH-type groupG13, we take a quaternionq= (z1, z2),z1, z2∈C, and three real numberst1, t2, t3that reflects the three dimensional setting of the space of the imaginary quaternions. Define a new non-commutative law between elementsh= [q, t1, t2, t3]∈H×R3andp= [q, t1, t2, t3]∈H×R3by
hp= [q, t1, t2, t3][q, t1, t2, t3]
=
q+q, t1+t1+1
2(qi1)·q, t2+t2+1
2(qi2) ·q, t3+t3+1
2(qi3)·q
, (3.3)
whereqik,k= 1,2,3is the Cayley–Dickson product for the quaternions and “·” is the scalar product inR4. As in the case of the Heisenberg group we can use the matrix representation of imaginary unities
i1=
⎡
⎢⎢
⎣
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
⎤
⎥⎥
⎦, i2=
⎡
⎢⎢
⎣
0 0 1 0
0 0 0 −1
−1 0 0 0
0 1 0 0
⎤
⎥⎥
⎦, i3=
⎡
⎢⎢
⎣
0 0 0 1
0 0 1 0
0 −1 0 0
−1 0 0 0
⎤
⎥⎥
⎦,
and rewrite the group law (3.3) in the form hp=
q+q, t1+t1+1
2(qi1)·q, t2+t2+1
2(qi2)·q, t3+t3+1
2(qi3) ·q
. Using the imaginary unities we can represent a quaternion in the algebraic form as
q=α+i1β+i2γ+i3δ=α+i1Im1q+i2Im2q+i3Im3q.
Then the multiplication law (3.3) takes the form hp= [q, t1, t2, t3][q, t1, t2, t3]
=
q+q, t1+t1+1
2Im1(q∗q), t2+t2+1
2Im2(q∗q), t3+t3+1
2Im3(q∗q)
, (3.4) whereq∗qis the Cayley–Dickson product ofq∗byq.
To give ann-dimensional analogue of the quaternionH-type group, we taken-dimensional vectors of quater- nion numbersw= (q1, . . . , qn),w = (q1, . . . , qn). Each of the matricesim,m= 1,2,3, is changed by the block diagonal matrixMm= diagimwithn(4×4)-dimensional matricesimon the main diagonal. The multiplication law between the elementsh= [w, t1, t2, t3],p= [w, t1, t2, t3]∈Hn×R3is the following
hp= [w, t1, t2, t3][w, t1, t2, t3]
=
w+w, t1+t1+1 2
n l=1
(qli1)·ql, t2+t2+1 2
n l=1
(qli2)·ql, t3+t3+1 2
n l=1
(qli3)·ql
=
w+w, t1+t1+1
2(wM1) ·w, t2+t2+1
2(wM2)·w, t3+t3+1
2(wM3) ·w
=
w+w, t1+t1+1
2Im1(w∗w), t2+t2+1
2Im2(w∗w), t3+t3+1
2Im3(w∗w)
, wherew∗w=n
l=1q∗l ql.
The quaternion algebraG3n,n= 4k,k∈N, is the direct sum ofV1⊕V2, where V1= span(X11, X21, X31, X41, . . . , X1n, X2n, X3n, X4n)
with
X1l(w, t) =∂x1l+1 2
−x2l∂t1−x3l∂t2−x4l∂t3 ,
X2l(w, t) =∂x2l+1 2
x1l∂t1+x4l∂t2−x3l∂t3 ,
X3l(w, t) =∂x3l+1 2
−x4l∂t1+x1l∂t2+x2l∂t3 ,
X4l(w, t) =∂x4l+1 2
x3l∂t1−x2l∂t2+x1l∂t3 ,
l= 1, . . . n, (3.5)
andw= (q1, . . . , qn) = (x11, x21, x31, x41, . . . , x1n, x2n, x3n, x4n). The latter system of vector fields can be written as
X = (X11, . . . , X4n) =
∇x+1 2
3 m=1
(xMm)∂tm
,
withx= (x11, . . . , x4n),∇x= (∂x11, . . . , ∂x4n). The subspaceV2is spanned by{Z1, Z2, Z3}withZk=∂tk. The following commutator relations
[X1l, X2l] =Z1, [X1l, X3l] =Z2, [X1l, X4l] =Z3, [X2l, X3l] =Z3, [X2l, X4l] =−Z2, [X3l, X4L] =Z1,
hold for l = 1, . . . , nand others vanish. Thus, the condition (2.1) is verified. The endomorphismsJZm are represented by matricesMm,m= 1,2,3. TheJ2condition holds by the relation (3.2) and it is independent of elementsX ∈V1.
Remark 3.1 If we involve into the construction only two imaginary unities, then we obtain the quaternion H-type group with two dimensional centerV2. Taking into consideration one of theik,k = 1,2,3, we get a group isomorphic to the Heisenberg groupGn1.
3.4 OctonionH-type groupG17
Octonion numbers, that we think of as a pair of quaternion numbers, has one real unity1 = (1,0),12= 1and 7 imaginary unities
j1= (i1,0), j2= (i2,0), j3= (i3,0),
j4= (0,1), j5= (0, i1) j6= (0, i2), j7= (0, i3),
whose squares equal−1. The rule of multiplication is presented in Table 1. The product of octonions is not
j1 j2 j3 j4 j5 j6 j7
j1 −1 −j3 j2 −j5 j4 j7 −j6 j2 j3 −1 −j1 −j6 −j7 j4 j5 j3 −j2 j1 −1 −j7 j6 −j5 j4 j4 j5 j6 j7 −1 −j1 −j2 −j3 j5 −j4 j7 −j6 j1 −1 j7 −j6 j6 −j7 −j4 j5 j2 −j7 −1 j5 j7 j6 −j5 −j4 j3 j6 −j5 −1 Table 1 Rules of multiplication ofjm
associative, for example,
j1(j2j4) =−j7, (j1j2)j4=j7.
We take an octonionw = (q1, q2)withq1, q2 ∈ H and 7 real numberstk,k = 1, . . . ,7, that correspond to 7-dimensional space of imaginary octonions. Define a new non-commutative law for elementsh = [w, t] = [w, t1, . . . , t7],p= [w, t] = [w, t1, . . . , t7]∈O×R7by
hp= [w, t][w, t]
= [w, t1, . . . , t7][w, t1, . . . , t7]
=
w+w, t1+t1+1
2(wj1)·w, . . . , t7+t7+1
2(wj7)·w
=
w+w, t1+t1+1
2Im1(w∗w), . . . , t7+t7+1
2Im7(w∗w)
,
(3.6)
wherewjm,m= 1, . . . ,7, andw∗ware the Cayley–Dickson product and “·” is the scalar product inR8. There is no matrix representation ofjk since the multiplication betweenjk is not associative, but the matrix multiplication is so. Nevertheless, it is possible to associate a matrixJmwith any imaginary unitjmthat will represent the corresponding endomorphismJZm,m = 1, . . . ,7. The matricesJmare given in the Appendix.
UsingJmwe write the multiplication law (3.6) as follows hp= [w, t][w, t] =
w+w, t1+t1+1
2(wJ1)·w, . . . , t7+t7+1
2(wJ7)·w
. (3.7)
The matricesJmsatisfy the following properties:
J2m=−U, JTm=−Jm, J−1m =Jm, m= 1, . . . ,7, (3.8) whereU is the(7×7)identity matrix. The product of the matricesJmdoes not correspond to the product of the corresponding imaginary unitiesjm, for example,
j1j2=−j3, but J1J2=−J3.
The matricesJmdo not represent the unit imaginary octonions, but they can be used to write the group law and the left invariant basis of the corresponding algebra.
The octonionH-type algebraG71is the direct sumV1⊕V2, whereV1= span(X1, . . . , X8)with Xl(w, t) =∂xl+1
2 7 m=1
(xJm)l∂tm, l= 1, . . . ,8, (3.9) wherew= (x1, . . . , x8)and(xJm)lis thel-th coordinate of the vectorxJm. We give the coefficients(xJm)lin the Table 2.
∂t1 ∂t2 ∂t3 ∂t4 ∂t5 ∂t6 ∂t7 X1 −x2 −x3 −x4 −x5 −x6 −x7 −x8 X2 x1 x4 −x3 x6 −x5 −x8 x7 X3 −x4 x1 x2 x7 x8 −x5 −x6 X4 x3 −x2 x1 x8 −x7 x6 −x5 X5 −x6 −x7 −x8 x1 x2 x3 x4 X6 x5 −x8 x7 −x2 x1 −x4 x3 X7 x8 x5 −x6 −x3 x4 x1 −x2 X8 −x7 x6 x5 −x4 −x3 x2 x1 Table 2 The productxJm
The subspaceV2 is spanned by{Z1, . . . , Z7}withZm = ∂tm. The non-vanishing commutators are given in Table 3.4 showing that the condition (2.1) holds.
Using the normal coordinates(w, t)for the elements, we identify the elements of the group with the elements of the algebra via the exponential map:
exp 8
k=1
xkXk+ 7 m=1
tmZm
∈G17.
TheJ2condition says that givenX = (α, β)andZ, Z ∈V2withZ, Z= 0 (for instance corresponding to the multiplication byj1andj2), there existsZinV2, such that
JZJZX =JZX.
X1 X2 X3 X4 X5 X6 X7 X8
X1 0 Z1 Z2 Z3 Z4 Z5 Z6 Z7
X2 −Z1 0 Z3 −Z2 Z5 −Z4 −Z7 Z6 X3 −Z2 −Z3 0 Z1 Z6 Z7 −Z4 −Z5 X4 −Z3 Z2 −Z1 0 Z7 −Z6 Z5 −Z4 X5 −Z4 −Z5 −Z6 −Z7 0 Z1 Z2 Z3 X6 −Z5 Z4 −Z7 Z6 −Z1 0 −Z3 Z2 X7 −Z6 Z7 Z4 −Z5 −Z2 Z3 0 −Z1 X8 −Z7 −Z6 Z5 Z4 −Z3 −Z2 Z1 0 Table 3 Non-vanishing commutators
LetZ = (a, b). In order to findaandb, we have to solve the linear system of 8 equations with 8 unknown variables.
Example 3.2 LetZ =j1,Z =j2,X = (α, β). We look for the elementZ= (a, b)corresponding to the actionJ1J2in the equation
J1J2(α, β) = (i1,0)((i2,0)(α, β)) = (a, b)(α, β). (3.10) Using the Cayley–Dickson product we write the left- and right-hand side of (3.10) in coordinates. If X = (α, β) = (1,0, . . . ,0), we deduce thata =
(0,0)(0,−1) andb =
(0,0)(0,0) or(a, b) = −j3. IfX = (α, β) = (0,0, . . . ,0,1), thena=
(0,0)(0,1) andb=
(0,0)(0,0) or(a, b) =j3.
4 Octonion H -type group G
17The Heisenberg group has been studied extensively by many mathematicians, see for instance [5, 8, 12, 17, 18].
The quaternionH-group was studied in [9, 10]. We concentrate our attention on the octonionH-group following the ideas developed in [2, 6, 7, 8]. There is an essential difference between the casesGn3 andG17. Even for Gn3 theJ2condition (2.5) is rather trivial, since it does not depend onX ∈V1. In the case ofG17it essentially depends onX ∈V1as it was shown in the example. The endomorphismsJm,m= 1, . . . ,7, are represented by matricesJm. But the composition action of two endomorphismsJlJk does not correspond to the action of the product of the corresponding matricesJlJk. The multiplication law (3.7) defines the left translationLq(p)of the elementpby the elementq. The Lie algebra is identified with the set of left invariant vector fields whose basis is given by(X, Z) = (X1, . . . , X8, Z1, . . . , Z7). A basis of one-forms dual to(X, Z)isdx= (dx1, . . . , dx8), and dϑ= (dϑ1, . . . , dϑ7)with
dϑm=dt−1
2(xJm·dx).
The subspaceT Hq of the tangent spaceTq,q∈G17defined by the formulaker(dϑ) = 0is called thehorizontal subspace. Sincedϑ(X) = 0, the horizontal subspace at q ∈ G17 isV1(q) = span{X1(q), . . . , X8(q)}. We say that an absolutely continuous curvec(s) : [0,1] → G17 is horizontalif the tangent vectorc(s)˙ satisfies
˙
c(s) =8
l=1al(s)Xl(c(s)). The definition of the horizontal space gives the following horizontality conditions.
Proposition 4.1 A curvec(s) =
x(s), t(s) is horizontal if and only if t˙m=1
2xJm·x.˙ (4.1)
P r o o f. Consider a curvec(s) =
x(s), t(s) . Then the velocityc(s)˙ can be written as
˙ c(s) =
8 l=1
˙
xl(s)∂xl+ 7 m=1
t˙m(s)∂tm
=
l
˙ xl
∂xl+1 2
m
(xJm)l∂tm−1 2
m
(xJm)l∂tm
+ 7 m=1
t˙m(s)∂tm
=
l
˙
xlXl+
m
t˙m−1
2
l
˙
xl(xJm)l
∂tm
=
l
˙
xlXl+
m
t˙m−1
2xJm·x˙m
∂tm
Hencec˙belongs to the distribution spanned by the vector fieldsX1, . . . , X8if and only if t˙m=1
2xJm·x,˙ m= 1, . . . ,7.
These are the non-holonomic constraints of the velocity, which will be used to set up a Lagrangian in Section 8.
The following properties of horizontal curves can be easily obtained (see also [9]).
(i) If a curvec(s) = (x(s), t(s))is horizontal, then
˙ c(s) =
8 l=1
˙
xl(s)Xl(c(s)). (4.2)
(ii) Left translationLqof a horizontal curvec(s)is a horizontal curve˜c=Lq(c)with the velocity
˙˜
c(c) = 8
l=1
˙
xl(s)Xl(˜c(s)).
(iii) The acceleration vector¨c(s)of a horizontal curvec(s)is a horizontal vector such that
¨ c(s) =
8 l=1
¨
xl(s)Xl(c(s)).
The following equalities
xJm·w=−x·wJm for all x, w∈R8, m= 1, . . . ,7 (particularly xJm·x= 0) (4.3) are used in the proof of the last assertion. SinceZl, Zm= 0,Zl, Zm∈V2,m=l, the actionJmJlpossesses the following property
JmJl+JlJm= 0 for all m, l= 1, . . . ,7, m=l, (4.4)
by (2.4).
Remark 4.2 The property (4.4) implies that the actionsJmJlandJlJmcan be represented by matricesJ˜and
−J˜respectively, such thatJ˜2=−U,J˜T =−J,˜ J˜−1=−J, and thus, the property (4.3) holds also for˜ J.˜
5 Hamiltonian formalism
The geometry of the octonionH-type group is induced by the sub-Laplacian Δ0=
8 l=1
Xl2= Δx+1
4|x|2Δt+ 7 m=1
(xJm· ∇x)∂tm,
where∇x = (∂x1, . . . , ∂x8),Δx =8
l=1∂x2
l,Δt =7
m=1∂t2
m. In the calculation we used Remark 4.2. We introduce the matrix
M= 7 m=1
θmJm=
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
0 θ1 θ2 θ3 θ4 θ5 θ6 θ7
−θ1 0 θ3 −θ2 θ5 −θ4 −θ7 θ6
−θ2 −θ3 0 θ1 θ6 θ7 −θ4 −θ5
−θ3 θ2 −θ1 0 θ7 −θ6 θ5 −θ4
−θ4 −θ5 −θ6 −θ7 0 θ1 θ2 θ3
−θ5 θ4 −θ7 θ6 −θ1 0 −θ3 θ2
−θ6 θ7 θ4 −θ5 −θ2 θ3 0 −θ1
−θ7 −θ6 θ5 θ4 −θ3 −θ2 θ1 0
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦ .
Comparing the matrixMwith Table 3 we see that matrixMreflects the commutative relation betweenXl. We notice the following property.
Lemma 5.1 For anyX ∈V1the actionM Xcorresponding to the matrixMsatisfies the following rules M2X =−|θ|2UX, M3X =−|θ|2XM, M4X =|θ|4XU, M5X =|θ|4XM, . . . , (5.1) whereUis the8×8identity matrix.
P r o o f. The proof is a straightforward application of Remark 4.2.
Introducing the dual variablesξl=∂xl,θm=∂tm,l= 1, . . . ,8,m= 1, . . . ,7, we get the Hamilton function H(x, t, ξ, θ) =
8 l=1
ξl+ (xM)l·ξl 2=|ξ|2+1
4|x|2|θ|2+xM·ξ, (5.2) where “·” denotes the usual scalar product inR8.
The corresponding Hamiltonian system is
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
˙
x =∂H
∂ξ = 2ξ+xM, t˙m = ∂H
∂θm = θm
2 |x|2+xJm·ξ, m= 1, . . . ,7, ξ˙ =−∂H
∂x =−1
2|θ|2x+ξM, θ˙m =−∂H
∂zm = 0.
(5.3)
The solutionsγ(s) = (x(s), t(s), ξ(s), θ(s))of the system (5.3) are calledbicharacteristics.
Definition 5.2 LetP1(x0, t0),P2(x, t)∈G17. Ageodesic starting atP1and ending atP2is the projection of a bicharacteristicγ(s),s∈[0,1], onto the(x, t)-space, that satisfies the boundary conditions
x(0), t(0) = (x0, t0),
x(1), t(1) = (x, t).
Lemma 5.3 Any geodesic is a horizontal curve.
P r o o f. Letc(s) = (x(s), t(s))be a geodesic. The system (5.3) implies t˙m=θm
2 |x|2+1
2xJm·2ξ=θm
2 |x|2+1
2xJm·x˙+1
2xJm·(2ξ−x).˙ (5.4)
Making use of the first line of the system (5.3), we write the last term of (5.4) as 1
2xJm·(2ξ−x) =˙ 1
2xJmM·x=−θm
2 |x|2. (5.5)
Here we used (3.8) and Remark 4.2. Combining (5.4) and (5.5) we deduce t˙m=1
2xJm·x,˙ m= 1, . . . ,7. (5.6)
Therefore,c(s)is a horizontal curve by Proposition 4.1.
Lemma 5.3 shows that the second equation of the system (5.3) is nothing more than the horizontality condi- tion (4.1).
We need also the following lemma. Here and furtherUdenotes the8×8identity matrix.
Lemma 5.4 For anyX ∈V1the actionexp(2sM)Xcorresponding to the matrixexp(2sM)can be written as
exp(2sM)X = cos(2s|θ|)XU+sin(2s|θ|)
|θ| XM. (5.7)
P r o o f. We observe that exp
2sM)X = ∞ n=0
(2s)n n! MnX
=XU∞
k=0
(2s|θ|)4k
(4k)! +XM
|θ| ∞ k=0
(2s|θ|)4k+1 (4k+ 1)!
−XU∞
k=0
(2s|θ|)4k+2
(4k+ 2)! −XM
|θ| ∞ k=0
(2s|θ|)4k+3 (4k+ 3)!
by (5.1). Note that ∞ k=0
(2s|θ|)4k (4k)! −∞
k=0
(2s|θ|)4k+2
(4k+ 2)! = cos(2s|θ|) and
∞ k=0
(2s|θ|)4k+1 (4k+ 1)! −∞
k=0
(2s|θ|)4k+3
(4k+ 3)! = sin(2s|θ|).
Thus, we get (5.7).
The last equation in the Hamiltonian system (5.3) shows that the functionH(ξ, θ, x, t)does not depend on t. We obtain thatθm are constants which will be considered as Lagrangian multipliers in the next section.
Simplifying the system (5.3) we get
¨
x= 2 ˙xM. (5.8)