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GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISIONALGEBRAS

OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA

Abstract. The present article applies the method of Geometric Analysis to the study H-typegroups satisfying the J2 condition and finishes the series of works describing the Heisenberggroup and the quaternion H-type group. The latter class of H-type groups satisfying theJ2 condition is related to the octonions. The relations between the group structure and theboundary of the corresponding Siegel upper half space are given.

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 realnumbers are the dependable breadwinner of the family, the completed ordered field we all relyon. The complex numbers are a slightly flashier but still respectable younger brother: notordered, but algebraically complete. The quaternions, being noncommutative, are the eccentriccousin who is shunned at important family gatherings. But the octonions are the crazy olduncle nobody lets out of the attic: they are nonassociative.” These division algebras generatea special class among the H-type homogeneous groups, the class satisfying the J2 Cliffordalgebras condition [7]. Three associative algebras (R, C, H) give the origin to the most excitinggeneralization incorporating the geometric concept of the direction, so-called Clifford algebras,which Clifford himself called “geometric algebras” [12, 16]. The homogeneous groups satisfyingthe J2 condition act as translations on the corresponding hyperbolic Siegel upper half spacesand this action can be extended up to the boundary. We present precise formulas of theseactions. The elements of the groups can be associated with the points of the boundary ofthe corresponding hyperbolic spaces though the action at the origin. The Lie algebras of thecorresponding groups can be associated with left invariant vector fields on the tangent bundleto the boundary of the Siegel upper half spaces. The nonvanishing commutative relationsdefine the sub-Riemannian geometry on the boundary of those spaces. The corresponding sub-Laplace operators are closely related with the boundary behavior of holomorphic functionsdefined on the corresponding Siegel upper half spaces [10, 18].

In the present article we describe the construction of H-type homogeneous groups associatedwith the four division algebras. The Heisenberg group, the quaternion and octonion H-typegroups, admitting the maximal dimension of their center, satisfy the J2 condition. Applyingthe method of geometric analysis we obtain the exact formulas of geodesics connecting two

2000 Mathematics Subject Classification. 53C17, 53C22, 35H20.Key words and phrases. Hamiltonian formalism, H-type groups, geodesics, division algebras, the Siegel upper

half space.The first author is partially supported by the NSF grant #0631541.The second author is supported by a research grant from the United States Army Research Office and a

competitive research grant at Georgetown University.The third author is supported by grants of the Norwegian Research Council # 177355 and of the University

of Bergen.

1

2 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA

arbitrary points of the groups. We discuss the cardinality of geodesics depending on thelocation of final points. The complex action is presented.

Part of this article is based on a lecture presented by the first author during the Symposiumon Hyper-Complex Analysis which was held on December 5-8, 2006 at the University of Macau,China. The first author thanks the organizing committee, especially Professor Qian Tao forhis invitation. The final version of this paper was in part written while the authors visited theNational Center for Theoretical Sciences and National Tsing Hua University during January2007. 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 theirstay in Taiwan.

2. Definitions

We start from the basic definitions that reader can find, for instance, in [7]. Let G be a realLie algebra, equipped with the Lie bracket [·, ·], which can be written as an orthogonal directsum,

G = V1 ⊕ V2, [V1, V1] ⊆ V2, [V1, V2] = [V2, V2] = 0.Suppose that G is endowed with a scalar product 〈·, ·〉. Define the linear mapping J : V2 →End(V1) by the formula

(2.1) 〈JZX, X ′〉 = 〈Z, [X,X ′]〉, ∀X, X ′ ∈ V1, ∀Z ∈ V2,

whence

(2.2) JTZ = −JZ , ∀Z ∈ V2.

We say that G is H-type if

(2.3) J2Z = −|Z|2U

for all Z in V2, where U denotes the identity mapping. The H(eisenberg)-type groups were in-troduced by Kaplan [13] in 1970-s and have been studied extensively by many mathematicians,see for instance [7, 14, 15, 17]. The conditions (2.2), (2.3) imply

(2.4) JZJZ′ + JZ′JZ = −2〈Z, Z ′〉U, ∀Z, Z ′ ∈ V2,

see [7]. When there exists a linear mapping J : V2 → End(V1) satisfying (2.2) and (2.3), V1 iscalled the Clifford module over V2. The algebra G (or the Clifford module associated with G)satisfies the J2 condition if, whenever X ∈ V1 and Z, Z ′ ∈ V2 with 〈Z, Z ′〉 = 0, then thereexists Z ′′ in V2 such that

(2.5) JZJZ′X = JZ′′X.

We present here a result from [7] giving the classification of H-type algebras satisfying the J2

condition. Denote by Gn0 the Euclidean n-dimensional space, by Gn

1 the n-dimensional Heisen-berg algebra, by Gn

3 the n-dimensional quaternion H-type algebra, and by G17 the octonion

H-type algebra. The lower index corresponds to the topological dimension of V2 and theupper index reflects the real, complex, quaternion and octonion topological dimensions of V1.

Theorem 2.1 ([7]). Suppose that G is an H-type algebra satisfying the J2 condition. Then Gis isometrically isomorphic to Gn

0 , Gn1 , Gn

3 or to G17 .

Before we describe the general construction of groups Gn0 , Gn

1 , Gn3 , and G1

7 , we would liketo remind the Cayley-Dickson construction of division algebras R (real numbers), C (complexnumbers), H (quaternion numbers), and O (octonion numbers). The Cayley-Dickson construc-tion explains why each one of the algebras fits neatly inside the next. Recall that the division

GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 3

algebra means that each nonzero element has a unique inverse element. The Cayley-Dicksonconstruction 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 as a∗ = a and the conjugate to the pair as

(2.6) (a, b)∗ = (a∗,−b).

Then the Cayley-Dickson product is defined by

(2.7) (a, b)(c, d) = (ac− db∗, a∗d + cb).

Now we can think a pair (a, b) as a quaternion, where a, b ∈ C. The conjugate is definedas in (2.6) and the product as in (2.7). We obtains the quaternion numbers H that form anoncommutative algebra with respect to (2.7). Finally, we define an octonion as a pair (a, b)with a, b ∈ H, the conjugate as in (2.6), and the product as in (2.7). The octonions with theoperation (2.7) makes up a noncommutative, nonassociative algebra. Actually, we can continuethe Cayley-Dickson construction doubling the dimension and getting a bit worse algebras.In course we 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 J2 condition

We present a general construction of the H-types algebras, satisfying J2 condition. Using theCayley-Dickson product, we first describe the following groups: Euclidean n-dimensional spaceGn

0 = Rn, the Heisenberg group Gn1 , the quaternion H-type group Gn

3 , and the octonion H-type group G1

7. Then we obtain the corresponding algebras Gn0 , Gn

1 , Gn3 , and G1

7 as infinitesimalrepresentations of the groups.

3.1. The Euclidean space. The space Gn0 = Rn is a trivial example of an H-type group since all

commutative relations vanish. The underlying space V1 is identified with Rn via the exponentialmap which is identity in this case. The center V2 is the empty set.

3.2. The Heisenberg group Gn1 . We start from n = 1 and then generalize for an arbitrary

n = 2k, k ∈ N. Complex numbers has 2 unites, whose absolute value of square equals 1:

real 1 = (1, 0), and imaginary i = (0, 1),

such that 12 = 1, i2 = −1. Take a complex number z = (x1, x2), x1, x2 ∈ R, and a realnumber t. Define a new noncommutative law between elements h = [z, t] ∈ C × R andp = [z′, t′] ∈ C× R by

(3.1) hp = [z, t][z′, t′] = [z + z′, t + t′ +12(zi) · z′],

where firstly we take the Cayley-Dickson product zi = (x1, x2)(0, 1) and then the scalar product“·” of vectors z, z′ ∈ R2. This multiplication law can be deduced from the matrix product ofupper triangular 3× 3-matrices [6]. If we use the representation of i as 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′ +12(zi) · z′].


Using the algebraic form of a complex number z = x1 + ix2 = Rez + i Imz, we can write (3.1)in the form

hp = [z, t][z′, t′] = [z + z′, t + t′ +12Im(z∗z′)],

where z∗z′ is a Cayley-Dickson product of z∗ by z′. The non-commutativity of new multipli-cation law in C × R is seen for the last variable t from the one dimensional real space, thatcorresponds to the existence of only one imaginary unit. It can be shown that C × R withnew non-commutative multiplication forms a Lie group with the identity element [0, 0], the lefttranslation Lh(p) = hp = [z, t][z′, t′], and the inverse to h = [z, t] element h−1 = [−z,−t].

in order to present an n-dimensional analogue of the Heisenberg group we take n-dimensionalvectors of complex numbers w = (z1, . . . , zn), w′ = (z′1, . . . , z

′n). The matrix i is changed by

a block diagonal matrix J = diag i with n matrices i on the diagonal. The multiplication lawbetween the elements h = [w, t] and p = [w′, t′] ∈ Cn×R is transformed into the following one

hp = [w, t][w′, t′] = [w + w′, t + t′ +12

n∑

l=1

(zli) · z′l] = [w + w′, t + t′ +12(wJ) · w′]

= [w + w′, t + t′ +12Im(w∗w′)],

where w∗w′ =∑n

l=1 z∗l z′l.The Heisenberg algebra Gn

1 , n = 2k, k ∈ N, is identified with the left invariant vector fieldson the tangent bundle at the identity element of the group. It splits into the direct sum V1⊕V2,where V1 = span(X11, X21, X21, X22, . . . , X1n, X2n) with a basis given by

X1l = ∂x1l− 1

2x2l∂t, and X2l = ∂x2l

+12x1l∂t, l = 1, . . . , n.

The vector field X = (X11, . . . , X2 n) = (∇x + 12(xJ)∂t) with x = (x11, . . . , x2 n), ∇x =

(∂x11 , . . . , ∂x2n), is a natural analogue of the Euclidean gradient. The subspace V2 is onedimensional and generated by Z = ∂t. Since [X1l, X2l] = Z and other commutators vanish,we verify the condition (2.1). The endomorphism JZ is represented by the matrix J whichpossesses properties (2.2), (2.3). The J2 condition holds trivially, since only one JZ is differentfrom 0.

3.3. Quaternion group Gn3 . 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 ofcomplex numbers, has one real unity 1 = (1, 0), 12 = 1 and 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,

(3.2) i1i2 = −i2i1 = −i3, i2i3 = −i3i2 = −i1, i3i1 = −i1i3 = −i2

In order to design the quaternion H-type group G13, we take a quaternion q = (z1, z2), z1, z2 ∈

C, and three real numbers t1, t2, t3 that reflects the three dimensional setting of the spaceof the imaginary quaternions. Define a new non-commutative law between elements h =[q, t1, t2, t3] ∈ H× R3 and p = [q′, t′1, t

′2, t

′3] ∈ H× R3 by

hp = [q, t1, t2, t3][q′, t′1, t′2, t

′3]

= [q + q′, t1 + t′1 +12(qi1) · q′, t2 + t′2 +

12(qi2) · q′, t3 + t′3 +

12(qi3) · q′],(3.3)


where qik, k = 1, 2, 3 is the Cayley-Dickson product for the quaternions and “·” is the scalarproduct in R4. As in the case of the Heisenberg group we can use the matrix representationof imaginary units

i1 =

0 1 0 0−1 0 0 0

0 0 0 10 0 −1 0

, i2 =

0 0 1 00 0 0 −1

−1 0 0 00 1 0 0

, i3 =

0 0 0 10 0 1 00 −1 0 0

−1 0 0 0

,

and rewrite the group law (3.3) in the form

hp = [q + q′, t1 + t′1 +12(qi1) · q′, t2 + t′2 +

12(qi2) · q′, t3 + t′3 +

12(qi3) · q′].

Using the imaginary unities we can represent a quaternion in the algebraic form as

q = α + i1β + i2γ + i3δ = α + i1Im1q + i2Im3q + i3Im3q.

Then the multiplication law (3.3) takes the form

hp = [q, t1, t2, t3][q′, t′1, t′2, t

′3]

= [q + q′, t1 + t′1 +12Im1(q∗ q′), t2 + t′2 +

12Im2(q∗ q′), t3 + t′3 +

12Im3(q∗ q′)],(3.4)

where q∗ q′ is the Cayley-Dickson product of q∗ by q′.To give an n-dimensional analogue of the quaternion H−type group, we take n-dimensional

vectors of quaternion numbers w = (q1, . . . , qn), w′ = (q′1, . . . , q′n). Each of the matrices

im, m = 1, 2, 3, is changed by the block diagonal matrix Mm = diag im with n (4 × 4)-dimensional matrices im on the main diagonal. The multiplication law between the elementsh = [w, t1, t2, t3], p = [w′, t′1, t

′2, t

′3] ∈ Hn × R3 is the following

hp = [w, t1, t2, t3][w′, t′1, t′2, t

′3]

= [w + w′, t1 + t′1 +12

n∑

l=1

(qli1) · q′l, t2 + t′2 +12

n∑

l=1

(qli2) · q′l, t3 + t′3 +12

n∑

l=1

(qli3) · q′l]

= [w + w′, t1 + t′1 +12(wM1) · w′, t2 + t′2 +

12(wM2) · w′, t3 + t′3 +

12(wM3) · w′]

= [w + w′, t1 + t′1 +12Im1(w∗w′), t2 + t′2 +

12Im2(w∗w′), t3 + t′3 +

12Im3(w∗w′)],

where w∗w′ =∑n

l=1 q∗l q′l.The quaternion algebra Gn

3 , n = 4k, k ∈ N, is the direct sum of V1 ⊕ V2, where

V1 = span(X11, X21, X31, X41, . . . , X1n, X2n, X3n, X4n)

with

X1l(w, t) =∂x1l+

12(− x2l∂t1 − x3l∂t2 − x4l∂t3

),

X2l(w, t) =∂x2l+

12(x1l∂t1 + x4l∂t2 − x3l∂t3

),

X3l(w, t) =∂x3l+

12(− x4l∂t1 + x1l∂t2 + x2l∂t3

),

X4l(w, t) =∂x4l+

12(x3l∂t1 − x2l∂t2 + x1l∂t3

),

l = 1, . . . n(3.5)


and w = (q1, . . . , qn) = (x11, x21, x31, x41, . . . , x1 n, x2 n, x3 n, x4 n). The latter system of vectorfields can be written as

X = (X11, . . . , X4n) = (∇x +12

3∑

m=1

(xMm)∂tm),

with x = (x11, . . . , x4 n), ∇x = (∂x11 , . . . , ∂x4 n). The subspace V2 is spanned by {Z1, Z2, Z3}with Zk = ∂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, . . . , n and others vanish. Thus, the condition (2.1) is verified. The endomor-phisms JZm are represented by matrices Mm, m = 1, 2, 3. The J2 condition holds by therelation (3.2) and it is independent of elements X ∈ V1.

Remark 3.1. If we involve into the construction only two imaginary units, then we obtain thequaternion H-type group with two dimensional center V2. Taking into consideration one ofthe ik, k = 1, 2, 3, we get a group isomorphic to the Heisenberg group Gn

1 .

3.4. Octonion H-type group O17. Octonion numbers, that we think of as a pair of quaternion

numbers, has one real unity 1 = (1, 0), 12 = 1 and 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

Table 1. Rules of multiplication of jm

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

octonions is not associative, for example,

j1(j2j4) = −j7, (j1j2)j4 = j7.

We take an octonion w = (q1, q2), q1, q2 ∈ H and 7 real numbers tk, k = 1, . . . , 7, thatcorrespond to 7-dimensional space of imaginary octonions. Define a new non-commutative lawfor elements h = [w, t] = [w, t1, . . . , t7], p = [w′, t′] = [w′, t′1, . . . , t

′7] ∈ O× R7 by

hp = [w, t][w′, t′] = [w, t1, . . . , t7][w′, t′1, . . . , t′7]

= [w + w′, t1 + t′1 +12(wj1) · w′, . . . , t7 + t′7 +

12(wj7) · w′](3.6)

= [w + w′, t1 + t′1 +12Im1(w∗w′), . . . , t7 + t′7 +

12Im7(w∗w′)],

where wjm, m = 1, . . . , 7, and w∗w′ are the Cayley-Dickson product and “·” is the scalarproduct in R8.


There is no matrix representation of jk since the multiplication between jk is not associative,but the matrix multiplication is so. Nevertheless, it is possible to associate a matrix Jm withany imaginary unit jm that will represent the corresponding endomorphism JZm , m = 1, . . . , 7.The matrices Jm are given in the Appendix. Using Jm we write the multiplication law (3.6)as follows

(3.7) hp = [w, t][w′, t′] = [w + w′, t1 + t′1 +12(wJ1) · w′, . . . , t7 + t′7 +

12(wJ7) · w′].

Notice some properties of the matrices Jm:

(3.8) J2m = −U, JT

m = −Jm, J−1m = Jm, m = 1, . . . , 7,

where U is the (7 × 7) identity matrix. The product of the matrices Jm does not correspondto the product of the corresponding imaginary unities jm, for example,

j1j2 = −j3, but J1J2 6= −J3.

The matrices Jm do not represent the unit imaginary octonions, but they can be used to writethe group law and the left invariant basis of the corresponding algebra.

The octonion H-type algebra G17 is the direct sum V1 ⊕ V2, where V1 = span(X1, . . . , X8)

with

(3.9) Xl(w, t) = ∂xl+

12

7∑

m=1

(xJm)l∂tm , l = 1, . . . , 8,

where w = (x1, . . . , x8) and (xJm)l is the l-th coordinate of the vector xJm. We give thecoefficients (xJm)l in the Table 2. The subspace V2 is spanned by {Z1, . . . , Z7} with Zm = ∂tm .

Table 2. The product xJm

∂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

The non-vanishing commutators are given in Table 3 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

)∈ G1

7.

The J2 condition says that given X = (α, β) and Z, Z ′ ∈ V2 with 〈Z,Z ′〉 = 0 (for instancecorresponding to the multiplication by j1 and j2), there exists Z ′′ in V2, such that

JZJZ′X = JZ′′X.

Let Z ′′ = (a, b). In order to find a and b, we have to solve the linear system of 8 equationswith 8 unknown variables.


Table 3. Non-vanishing commutators

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

Example. Let Z = j1, Z ′ = j2, X = (α, β). We look for the element Z ′′ = (a, b) correspondingto the action J1J2 in the equation

(3.10) J1J2(α, β) = (i1, 0)((i2, 0)(α, β)

)= (a, b)(α, β).

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 that a =

((0, 0)(0,−1)

)and b =

((0, 0)(0, 0)

)or

(a, b) = −j3. If X = (α, β) = (0, 0, . . . , 0, 1), then a =((0, 0)(0, 1)

)and b =

((0, 0)(0, 0)

)or

(a, b) = j3.

4. Octonion H-type group G17

The Heisenberg group has been studied extensively by many mathematicians, see for in-stance [3, 6, 7, 14, 15]. The quaternion H-group was studied in [8, 9]. We concentrate ourattention on the octonion H-group following the ideas developed in [2, 4, 5, 6]. There is anessential difference between the cases Gn

1 and Gn3 . Even for Gn

3 the J2 condition (2.5) is rathertrivial, since it does not depend on X ∈ V1. In the case of G1

7 it essentially depends on X ∈ V1

as it was shown in the example. The endomorphisms Jm, m = 1, . . . , 7, are represented by ma-trices Jm. But the composition action of two endomorphisms JlJk does not correspond to theaction of the product of the corresponding matrices JlJk. The multiplication law (3.7) definesthe left translation Lq(p) of the element p by the element q. The Lie algebra is identified withthe 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) is dx = (dx1, . . . , dx8), and dϑ = (dϑ1, . . . , dϑ7) with

dϑm = dt− 12(xJm · dx).

The subspace THq of the tangent space Tq, q ∈ G17 defined by the formula ker(dϑ) = 0

is called the horizontal subspace. Since dϑ(X) = 0, the horizontal subspace at q ∈ G17 is

V1(q) = span{X1(q), . . . , X8(q)}. We say that an absolutely continuous curve c(s) : [0, 1] → G17

is horizontal if the tangent vector c(s) satisfies c(s) =∑8

l=1 al(s)Xl(c(s)). The definition ofthe horizontal space gives the following horizontality conditions.

Proposition 4.1. A curve c(s) = (x(s), t(s)) is horizontal if and only if

(4.1) tm =12xJm · x.

The following properties of horizontal curves can be easily obtained (see also [8]).


(i) If a curve c(s) = (x(s), t(s)) is horizontal, then

c(s) =8∑

l=1

xl(s)Xl(c(s)).

(ii) Left translation Lq of a horizontal curve c(s) is a horizontal curve c = Lq(c) with thevelocity

˙c(c) =8∑

l=1

xl(s)Xl(c(s)).

(iii) The acceleration vector c(s) of a horizontal curve c(s) is a horizontal vector such that

c(s) =8∑

l=1

xl(s)Xl(c(s)).

The following equalities

(4.2) xJm · w = −x · wJm for all x,w ∈ R8, m = 1, . . . , 7, (particularly xJm · x = 0)

are used in the proof of the last assertion. Since 〈Zl, Zm〉 = 0, Zl, Zm ∈ V2, m 6= l, the actionJmJl possesses the following property

(4.3) JmJl + JlJm = 0 ∀m, l = 1, . . . , 7, m 6= l

by (2.4).

Remark 4.1. The property (4.3) implies that the actions JmJl and JlJm can be representedby matrices J and −J respectively, such that J2 = −U , JT = −J, J−1 = −J, and thus, theproperty (4.2) holds also for J.

5. Hamiltonian formalism

The geometry of the octonion H-type group is induced by the sub-Laplacian

∆0 =8∑

l=1

X2l = ∆x +

14|x|2∆t +

7∑

m=1

(xJm · ∇x)∂tm ,

where ∇x = (∂x1 , . . . , ∂x8), ∆x =∑8

l=1 ∂2xl

, ∆t =∑7

m=1 ∂2tm . In the calculation we used

Remark 4.1. 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 matrix M with Table 3 we see that matrix M reflects the commutative relationbetween Xl. We notice the following property.

Lemma 5.1. For any X ∈ V1 the action MX corresponding to the matrix M satisfies thefollowing rules

(5.1) M2X = −|θ|2UX, M3X = −|θ|2XM, M4X = |θ|4XU, M5X = |θ|4XM, . . . ,

where U is the 8× 8 identity matrix.


Proof. The proof is a straightforward application of Remark 4.1. ¤Introducing the dual variables ξl = ∂xl

, θm = ∂tm , l = 1, . . . , 8, m = 1, . . . , 7, we get theHamilton function

(5.2) H(x, t, ξ, θ) =8∑

l=1

(ξl + (xM)l · ξl

)2= |ξ|2 +

14|x|2|θ|2 + xM · ξ,

where “·” denotes the usual scalar product in R8.The corresponding Hamiltonian system is

x = ∂H∂ξ = 2ξ + xM

tm = ∂H∂θm

= θm2 |x|2 + xJm · ξ, m = 1, . . . , 7.

ξ = −∂H∂x = −1

2 |θ|2x + ξMθm = − ∂H

∂zm= 0.

(5.3)

The solutions γ(s) = (x(s), t(s), ξ(s), θ(s)) of the system (5.3) are called bicharacteristics.

Definition 5.2. Let P1(x0, t0), P2(x, t) ∈ G17. A geodesic starting at P1 and ending at P2

is the projection of a bicharacteristic γ(s), s ∈ [0, 1], onto the (x, t)-space, that satisfies theboundary conditions (

x(0), t(0))

= (x0, t0),(x(1), t(1)

)= (x, t).

Lemma 5.3. Any geodesic is a horizontal curve.

Proof. Let c(s) = (x(s), t(s)) be a geodesic. The system (5.3) implies

(5.4) tm =θm

2|x|2 +

12xJm · 2ξ =

θm

2|x|2 +

12xJm · x +

12xJm · (2ξ − x).

Making use of the first line of the system (5.3), we write the last term of (5.4) as

(5.5)12xJm · (2ξ − x) =

12xJmM · x = −θm

2|x|2.

Here we used (3.8) and Remark 4.1. Combining (5.4) and (5.5) we deduce

(5.6) tm =12xJm · x, m = 1, . . . , 7.

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 condition (4.1).We need also the following lemma. Here and further U denotes the 8× 8 identity matrix.

Lemma 5.4. For any X ∈ V1 the action exp(2sM)X corresponding to the matrix exp(2sM)can be written as

(5.7) exp(2sM)X = cos(2s|θ|)XU +sin(2s|θ|)

|θ| XM.

Proof. 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 function H(ξ, θ, x, t) doesnot depend on t. We obtain that θm are constants which can be used as Lagrangian multipliers.Simplifying the system (5.3) we get

(5.8) x = 2xM.

Solving (5.8) we deduce

(5.9) x(s) = x(0) exp(2sM),

where x(0) is the initial velocity.The group structure allows us to restrict our considerations to the curves starting at the

origin. Hence, x(0) = 0. Exploiting (5.7), the equation (5.9) can be written as

(5.10) x(s) = cos(2s|θ|)x(0)U +sin(2s|θ|)

|θ| x(0)M.

Integrating from 0 to s we get

(5.11) x(s) =1− cos(2s|θ|)

2|θ|2 x(0)M +sin(2s|θ|)

2|θ| x(0)U.

Let us describe the t-components of a geodesic curve. If a curve is geodesic, then it is horizontalby Lemma 5.3, and we deduce that

(5.12) tm(s) =12Jmx(s) · x(s) =

θm|x(0)|24|θ|2

(1− cos(2s|θ|)), m = 1, . . . , 7.

by (5.10), (5.11), and Remark 4.1. Integrating equations (5.12), we get

(5.13) tm(s) =θm|x(0)|2

4|θ|2(s− sin(2s|θ|)

2|θ|), m = 1, . . . , 7.

Lemma 5.5. Not all of horizontal curves are geodesics.

Proof. To prove this proposition we present an example. Let c = c(s) be a curve. Set x(s) =( s2

2 , s, s2

2 , s, 0, 0, 0, 0), t = (− s3

6 , k1, . . . , k6), where k1, . . . , k6 are constants. The curve c(s) =(x(s), t(s)) is horizontal. Indeed,

t1(s) =− s2

2,

12(xJ1 · x) =

s2

2,

tk(s) =0 and12(xJk · x) = 0, k = 2, . . . , 7.


On the other hand, the curve c(s) does not satisfy the system (5.8). The system (5.8) admitsthe form

1 = 2(−θ1 − θ2s− θ3)0 = 2(θ1s− θ3s + θ2)1 = 2(θ2s + θ3 − θ1)0 = 2(θ3s− θ2 + θ1s)

Summing up the first and the third equation, and then, the second and the forth ones, wewrite the latter system as

2 = −4θ1

0 = 4θ1s1 = 2(θ2s + θ3 − θ1)0 = 2(θ3s− θ2 + θ1s).

We see that the first and the second equations contradict each other. ¤Lemma 5.6. A curve c is geodesic on the group G1

7 if and only if(i) c(s) is a horizontal curve and(ii) c(s) satisfies c(s) = 2c(s)M.

Proof. If a curve is geodesic, then it is horizontal by Lemma 5.3. Proposition 4.1 implies thatthe vector c is also horizontal: c(s) =

∑8l=1 xl(s)Xl(c(s)). Since x(s) = 2x(s)M by (5.8), we

obtain the necessary result.Let the curve c(s) satisfy (i) and (ii) of Lemma 5.6. The horizontality condition (i) of

Lemma 5.6 can be written in the form

(5.14) tm =12xJm · x =

θm

2|x|2 + xJm · ξ =

∂H

∂θm, m = 1, . . . , 7.

as it was proved in Lemma 5.3. We see that c(s) satisfies the equations of the second lineof (5.3). The condition (ii) of Lemma 5.6 admits the form x(s) = 2x(s)M in the coordinatefunctions. Define the following curve γ(s) = (x(s), t(s), ξ(s), θ) in the cotangent space, where

(5.15) ξ =x(s)2

− 12x(s)M and θ = (θ1, . . . , θ7) is constant.

The relation (5.15) implies the equations of the first and the last lines of (5.3). Differentiat-ing (5.15), we get

ξ =x

2− 1

2xM = xM− xM

2=

12(2ξ + xM)M = ξM− 1

2|θ|2x,

by the condition (ii) of Lemma 5.6, (5.1) and (5.15). Thus, γ(s) satisfies the Hamiltoniansystem (5.3). Then, the projection of γ onto the (x, t)-space, that coincides with c(s), isgeodesic. ¤

6. Connectivity by geodesics

Let us start with an auxiliary result.

Proposition 6.1. The kinetic energy E = 12 |x|2 is preserved along geodesics.

Proof. In fact, dEmds = x · x = 2x · xM = 0 by the property (4.2) of the matrices Jm. ¤

Now we can discuss the connectivity property case by case.

Case 1. Connectivity between 0 = (0, 0) and P = (x, 0), x 6= 0.The following is the first main result of this section.


Theorem 6.1. A smooth curve c(s) is horizontal with constant t-coordinates t1, . . . , t7 if andonly if c(s) = (a1s, . . . , a8s, t1, . . . , t7) with al ∈ R and

∑8l=1 a2

l 6= 0. In other words, there isonly one geodesic joining the origin with a point (x, 0).

Proof. Let c(s) be a horizontal curve with constant t-coordinates t1, . . . , t7. Then tm = 0and (5.12), (5.10), (5.11), and (4.1) imply

0 = tm = −θm|x(0)|22|θ|2 sin2(s|θ|), m = 1, . . . , 7.

Since the energy |x(0)|2 does not vanish we deduce, that θm = 0 for all m = 1, . . . , 7. TheHamiltonian system (5.3) is reduced to the next one

x = 2ξ0 = xJm · ξ, m = 1, . . . , 7ξ = 0θm = 0.

We see that ξ is a constant vector. Taking into account that x(0) = 0, we get x(s) =(a1s, . . . , a8s) with al = 2ξl. This proves the statement.

Now, let us assume that c(s) = (a1s, . . . , a8s, t1, . . . , t7) with constant t-components. Setas = (a1s, . . . , a8s). Then,

tm = 0 and12(as)Jm · ˙(as) =

s

2aJm · a = 0, m = 1, . . . , 7,

by (4.2). The horizontal condition (4.1) holds for all seven t-components. ¤

Case 2. Connectivity between (0, 0) and (0, t), t 6= 0.We need to solve equation (5.8) with the boundary conditions

x(0) = x(1) = t(0) = 0, t(1) = t.

We also need to know the initial velocity x(0) since we do not have enough information aboutthe behavior of x-coordinates.

Theorem 6.2. There are infinitely many geodesics c = c(s), s ∈ [0, 1], joining the origin witha point (0, t). The corresponding equations are

(6.1) x(k)(s) = 4sin2(kπs)|x(0)|2 x(0)T +

sin(2kπs)2kπ

x(0)U, k ∈ N,

(6.2) t(k)(s) = t(s− sin(2kπs)

2kπ

), k ∈ N,

where T is a matrix that is obtained from the matrix M replacing θm by tm(1). The lengthsl(c) of corresponding geodesics c(s) are l2(c) = 4πk|t|.Proof. Substituting s = 1 in (5.11) and using (4.2), we calculate

0 = |x(1)|2 =sin2(|θ|)|θ|2 |xl(0)|2.

Since the kinetic energy E = |x(0)|22 does not vanish we deduce that |θ| =

(∑7l=1 θ2

l

)1/2= kπ,

k ∈ N. Equalities (5.13) give for s = 1

(6.3) tm = θm|x(0)|24(πn)2

, m = 1, . . . , 7.


We find the unknown constants θm = 4tmπ2n2

|x(0)|2 . Substituting θm in (5.11), (5.13), we obtain theequations (6.1) and (6.2) for geodesics and the matrix T .

To calculate the length of geodesics, we observe that |t| = |θ| |x(0)|24(kπ)2

= |x(0)|24kπ from (6.3).

Thus the square of the length l(c) of a geodesic c(s) is

l2(c) =(∫ 1

0|x(s)| ds

)2= |x(0)|2 = 4kπ|t|, k ∈ N.

¤

Case 3. Connectivity between (0, 0) and (x, t), x 6= 0, t 6= 0.Now, we will look for a solution of equation (5.8) with the boundary conditions

x(0) = 0, t(0) = 0, x(1) = x, t(1) = t.

Further we need the function

(6.4) µ(|θ|) =|θ|

sin2(|θ|) − cot(|θ|)

that was introduced by Gaveau in [11] and studied in detailes by Beals, Gaveau, Greinerin [2, 3, 4]. By the following lemma, one finds some basic properties of the function µ.

Lemma 6.2. The function µ(θ) = θsin2 θ

− cot θ is an increasing diffeomorphism of the interval(−π, π) onto R. On each interval (kπ, (k + 1)π), k = 1, 2, . . ., the function µ has a uniquecritical point ck. On this interval the function µ strictly decreases from +∞ to µ(ck), andthen, strictly increases from µ(ck) to +∞. Moreover,

µ(ck) = ck, k ∈ N.

Hence, µ(ck) →∞ as k →∞.

5 10 15 20

5

10

15

20

π 2π 3π 4π 5π 6π

4|t|

|x|2

θ

µ

Figure 1. Graphics of µ = µ(θ) and µ = θ

Proof. We just concentrate ourselves on the last part of the lemma. One has

µ′(z) =(1− cos z) sin2 z − 2(z − sin z) sin z cos z

sin4 z.


It follows that µ′(ck) = 0 ⇔ ck = tan(ck). Plugging in z = ck into the definition of µ, we have

µ(ck) =ck

sin2(ck)− cot(ck)

=tan(ck)− cos(ck) sin(ck)

sin2(ck)

=sin(ck)− cos2(ck) sin(ck)

sin2(ck) cos(ck)

=sin3(ck)

sin2(ck) cos(ck)= tan(ck) = ck.

The proof of the lemma is therefore complete. ¤Theorem 6.3. Given a point P = (x, t) with x 6= 0, t 6= 0, there are finitely many geodesicsjoining the point O(0, 0) with a point P . Let |θ|k, k = 1, . . . , N , be solutions of the equation

(6.5) µ(|θ|) =4|t||x|2 ,

where µ is the function given in (6.4). Then the equations of the geodesics c = c(s), s ∈ [0, 1],are

x(k)(s) = sin(s|θ|k) cos(s|θ|k)(4 sin2(|θ|k)

(tan(s|θ|k) cot(|θ|k)− 1

)

|θ|k − cos(|θ|k) sin(|θ|k)x

|x|T

+(tan(s|θ|k) + cot(|θ|k)

)xU

), k = 1, 2, . . . , N,

(6.6)

(6.7) t(k)(s) = t2s|θ|k − sin2(2s|θ|k)

|θ|k − cos(|θ|k) sin(|θ|k) , k = 1, 2, . . . , N,

where T is the matrix that obtained from the matrix M replacing θm by tm(1). The lengths ofthese geodesics are

(6.8) l2n = ν(|θ|k)(|x|2 + 4|t|),where ν(|θ|k) = |θ|2k

sin(|θ|k)(sin(|θ|k)−cos(|θ|k))+|θ|k .

Proof. We put s = 1 in (5.11) and get

(6.9) |x(0)|2 =|θ|2

sin2(|θ|) |x|2.

Substituting s = 1 into (5.13) and |x(0)|2 from (6.9) we obtain

(6.10) tm =θm|x|24|θ|

( |θ|sin2(|θ|) − cot(|θ|)

), m = 1, . . . , 7.

Then

(6.11) |t| = |x|24

µ(|θ|).Thus we deduced the equation (6.5).

Let us fix one of the solutions of equation (6.5) |θ|k for a given point P = (x, t). Equali-ties (6.10) and (6.11) give θm = tm

|t| |θ|. We set s = 1 in (5.11) and express x(0) as

x(0) =|θ|k

sin(|θ|k) cos(|θ|k)x(U +

tan(|θ|k)|θ|k M

)−1.


Expanding(U + tan(|θ|k)

|θ|k M)−1

into the series and applying (5.1), we get

(U +

tan(|θ|k)|θ|k M

)−1=

(U− tan(|θ|k)

|θ|k M) ∞∑

k=0

(−1)k tan2k(|θ|k)

= U cos2(|θ|k)−Msin(|θ|k) cos(|θ|k)

|θ|k .

Finally

(6.12) x(0) = x(|θ|k cot(|θ|k)U−M

).

Substituting x(0) and θm = tm|t| |θ| into (5.11) and (5.13), we obtain (6.6) and (6.7).

Let ‖(x, t)‖2 = |x|2 +4|t| be a homogeneous norm of the end point of a geodesic c(s) startingfrom O = (0, 0). The square of the length of a geodesic c is

l2k(c) =(∫ 1

0|x(0)| ds

)2=

|θ|2k|x|2sin2(|θ|k)

.

Notice that

|x|2 + 4|t| = |x|2(1 + µ(|θ|k))

=sin2(|θ|k)|θ|2k

(1 + µ(|θ|k)

)l2k(c)

by (6.11). From the latter equalities we obtain (6.8). ¤

7. Complex Hamiltonian mechanics

Our aim now is to study the complex action which may be used to obtain the length of realgeodesics.

Definition 7.1. A complex geodesic is the projection of a solution of the Hamiltonian sys-tem (5.3) with the non-standard boundary conditions

x(0) = 0, x(1) = x, t(0) = 0, t(1) = t, and

θm = −iτm, m = 1, . . . , 7,

onto the (x, t)-space.

Let us introduce the notation −iτ for the vector (−iτ1, . . . ,−iτ7). Then |τ | =√∑7

m=1 τ2m

and |θ| = i|τ |. Notice, that we should treat the missing directions apart from the directions inthe underlying space.

Definition 7.2. The modifying complex action is defined as

(7.1) f(x, t, τ) = −i∑m

τmtm +∫ 1

0

((x, ξ)−H(x, t, ξ, τ)

)ds.

We present some useful calculations following from the system (5.3).

ξ · x =2|ξ|2 + xM · ξ =12|x|2 − 1

2xM · x,

|ξ|2 =|x|24− 1

2xM · x +

14xM · xM =

|x|24− 1

2xM · x +

14|θ|2|x|2,

xM · ξ =12xM · x− 1

2|θ|2|x|2.

(7.2)


Making use of formulas (5.2), (7.2), and the value of the energy E = |x(0)|22 = |θ|2

sin |θ|2|x|22 , we

deduce

f(x, t, τ) =− i∑m

τmtm +|x(0)|2

4

∫ 1

0cosh(2s|τ |) ds

=− i∑m

τmtm +|x|24|τ | coth |τ |.

The complex action function satisfies the Hamilton-Jacobi equation

(7.3)7∑

m=1

τm∂f

∂τm+ H(x, t,∇xf,∇tf) = f.

Indeed, we have∂f

∂τm= −itm − i

|x|2τm

4|τ | µ(i|τ |), m = 1, . . . , 7,

H(x, t,∂f

∂x,∂f

∂t) = H(x, t, ξ, τ) =

|x|24

|τ |2sinh2 |τ | .

Then,7∑

m=1

τm∂f

∂τm+ H(x, t,

∂f

∂x,

∂f

∂tz) = −i

∑m

τmtm +|x|2|τ |

4

(− iµ(i|τ |) +

|τ |sinh2 |τ |

)

=− i∑m

τmtm +|x|24|τ | coth |τ | = f.

In the critical points τc, where ∂f∂τm

= 0, we have

f(x, t, τc) = H(x, t,∇xf,∇tf) =E2

=l2

4(γ),

where a geodesic curve γ connects the origin with (x, t).

Remark 7.1. The construction of H-type groups described in the present paper contains theH-type groups with m-dimensional center, where m varies from 1 to 7. The octonion H-typegroup with 3-dimensional center G1

3 is isomorphic to the quaternion H-type group G23, whereas

the octonion H-type group with 1-dimensional center G11 is isomorphic to the Heisenberg

group G41. If the dimension m of the center is equal to 2, 4, 5, 6, then the H-type groups do

not satisfy the J2 condition. The J2 condition essentially expresses the multiplication lawbetween the imaginary unities on C, H, and O. The results of Section 6 contain the equationsof geodesics for H-type groups with an arbitrary dimensional center from 1 to 7, since in thecalculations we used only the relation (2.4). The same is true for results of Section 7.

8. Action of H-type groups on the Siegel upper half space

Let Cn+1 be the (n + 1)-dimensional complex space. We use the notation z = (z′, zn+1),where z′ = (z1, . . . , zn) ∈ Cn. The set

Un = {(z1, . . . , zn+1) ∈ Cn+1 : 4Re zn+1 > |z′|2 =n∑

l=1

|zl|2}


defines the Siegel upper half space. Let BC denotes the unit ball in Cn+1:

BC = {(w1, . . . , wn+1) ∈ Cn+1 :n+1∑

l=1

|wl|2 < 1}.

Then the Cayley transformation

wn+1 =1− zn+1

1 + zn+1, wl =

zl

1 + zn+1, l = 1, . . . , n,

and its inversezn+1 =

1− wn+1

1 + wn+1, zl =

2wl

1 + wn+1, l = 1, . . . , n,

show that the unit ball D and the Siegel upper half space Un are biholomorphically equivalent.Let Hn+1, n + 1 = 4k, k ∈ N, be an (n + 1)-dimensional quaternion vector space over the

field of real numbers. The elements of Hn+1 are (n + 1)-tuples of quaternions that we denoteby q = (q′, qn+1), q′ = (q1, . . . , qn) ∈ Hn, with the norm ‖q‖2 =

∑n+1l=1 q2

l . The Siegel upperhalf space in Hn+1 can be defined by analogy with the complex case as:

Un = {(q1, . . . , qn+1) ∈ Hn+1 : 4Re qn+1 >

n∑

l=1

|ql|2 = |q′|2}.

The unit ball BH in Hn+1 is

BH = {(h1, . . . , hn+1) ∈ Hn+1 :n+1∑

l=1

|hl|2 < 1}.

Since the multiplication of quaternion is not commutative there are two forms of Cayley trans-formation that give the symmetric geometry. The (left) Cayley transformation, mapping theSiegel upper half space Un onto the unit ball BH has the form

qn+1 = (1+hn+1)−1(1−hn+1) =(1 + h∗n+1)(1− hn+1)

|1 + hn+1|2 , ql = 2hl(1+hn+1)−1 =2hl(1 + h∗n+1)|1 + hn+1|2 ,

for l = 1, . . . , n. The inverse transformation from BH onto Un is

hl = ql(1 + qn+1)−1 =ql(1 + q∗n+1)|1 + qn+1|2 , hn+1 = (1− qn+1)(1 + qn+1)−1 =

(1− qn+1)(1 + q∗n+1)|1 + qn+1|2

for l = 1, . . . , n. The Cayley transformation is biholomorphic in the quaternion sense.Continuing by analogy, we define the Siegel upper half space in O2. We shall consider only

2-dimensional case of octonion vector space over the field of real numbers: O2 = {o = (o′, o2) :o′, o2 ∈ O} with the norm |o|2 = |o′|2 + |o2|2. The Siegel upper half space U1 and the unit diskBO can be defined as

U1 = {(o′, o2) ∈ O2 : 4Re o2 > |o′|2},BO = {(r′, r2) ∈ O2 : |r′|2 + |r2|2 < 1}.

The transformations

o′ = 2r′(1 + r2)−1, o2 = (1 + r2)−1(1− r2),

map the Siegel upper half space onto the unit ball and the map

r′ = o′(1 + o2)−1, r2 = (1− o2)(1 + o2)−1

acts from unit ball to the Siegel upper half space. To verify this we notice that

(xy)∗ = y∗x∗, x−1 =x∗

|x|2 ,


where x is a complex number, a quaternion or an octonion and xy is the Cayley-Dicksonproduct. Let us denote by (q′, q) an element of one of the above mentioned Siegel upper halfspaces and by (h′, h) a point from the corresponding unit ball. Then

|h′|2 = h′(h′)∗ = |q′|2|(1 + q)−1|2,|h|2 = hh∗ = |1− q|2|(1 + q)−1|2,

and|h′|2 + |h|2 = (|q′|2 + |1− q|2)|(1 + q)−1|2 < 1.

Since |(1 + q)−1|2 = |(1 + q)|−2, we have

|q′|2 + |1− q|2 = |q′|2 + 1 + |q|2 − 2Re q < |(1 + q)|2 = 1 + |q|2 + 2Re q,

that yields |q′|2 < 4Re q.Let K be one of the following spaces Cn+1, Hn+1 or O2. We denote by p = (q′, q) a point

from the Siegel upper half space Un of K. The boundary of the space Un is

∂Un = {(q′, q) ∈ K : 4Re q = |q′|2 =n∑

l=1

|ql|2}.

We mention here three automorphisms of the domain Un: dilation, rotation and translation.Dilation. Let p = (q′, q) ∈ Un. For each positive number δ we define a dilation δ ◦ p by

δ ◦ p = δ ◦ (q′, q) = (δq′, δ2q).

The non-isotropy of the dilation comes from the definition of Un.Rotation. For each unitary linear transformation R that acts on Cn, Hn or O, we define

the rotation R(p) on Un byR(p) = R(q′, q) = (R(q′), q).

Both, the dilation and the rotation extend to mappings on the boundary ∂Un.Translation We use the notation G for the H-type groups Gn

1 , Gn3 , G1

7. To each element[w, t] of G we associate the following affine self mapping of Un. (Notice that it is holomorphicmap for the cases Cn+1, Hn+1).

(8.1) [w, t] : (q′, q) 7→ (q′ + w, q +|w|24

+12w∗q′ + i · t).

Here i · t =∑dim V2

l=1 iktk. This mapping preserves the level sets, given by the function

(8.2) r(p) = 4Re q − |q′|2.In fact, since |q′ + w|2 = |q′|2 + |w|2 + 2Rew∗q1, we obtain

4Re (q +|w|24

+12w∗q′)− |q′ + w|2 = 4Re q − |q′|2.

Hence, the transformation (8.1) maps Un onto itself and preserves the boundary ∂Un.Let us check that the mapping (8.1) defines an action of the group G on the space Un. If

we compose the mappings (8.1), corresponding to elements [w, t] and [ω, s] ∈ G, we get

(8.3) [w, t] :([ω, s] : (q′, q)

)=

(w +ω + q′, q +

|w|24

+|ω|24

+12(w + ω)∗q′+

12w∗ω + i · (s + t)

).

On the other hand the transformation corresponding to the element [w, t][ω, s] is

(8.4) [w, t][ω, s] : (q′, q) =(w + ω + q′, q +

|w + ω|24

+12(w + ω)∗q′ +

12i · Im w∗ω + i · (s + t)

).


Observing that

|w + ω|24

+12i · Im w∗ω =

|w|24

+|ω|24

+12Re w∗ω +

12i · Imw∗ω =

|w|24

+|ω|24

+12w∗ω,

we conclude that (8.3) and (8.4) give the same result. Thus, (8.1) gives us a realization of Gas a group of affine (q-holomorphic) bijections of Un. We can identify the elements of Un withthe boundary via its action at the origin

h(0) = [w, t] : (0, 0) 7→ (w, |w|2 + i · t),where h = [w, t]. Thus

G 3 [w, t] 7→ (w, |w|2 + i · t) ∈ ∂Un.

We may use the following coordinates (q′, t, r) = (q′, t1, . . . , tdim V2 , r) on Un:

Un 3 (q′, q) = (q′, t, r), where tk = Imkq, k = 1, . . . , dimV2, r = r(q′, q) = 4Re q − |q′|2.If 4Re q = |q′|2, then we get coordinates on the boundary ∂Un of the Siegel upper half space

∂Un 3 (q′, q) = (q′, t1, . . . , tdim V2),

where tk are as above and r = r(q′, q) = 0.

9. Appendix

We give here the precise forms of matrices Jm.

J1 =

0 1 0 0 0 0 0 0−1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 00 0 −1 0 0 0 0 00 0 0 0 0 1 0 00 0 0 0 −1 0 0 00 0 0 0 0 0 0 −10 0 0 0 0 0 1 0

, J2 =

0 0 1 0 0 0 0 00 0 0 −1 0 0 0 0

−1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 −1 0 0 00 0 0 0 0 −1 0 0

,

J3 =

0 0 0 1 0 0 0 00 0 1 0 0 0 0 00 −1 0 0 0 0 0 0

−1 0 0 0 0 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 −1 00 0 0 0 0 1 0 00 0 0 0 −1 0 0 0

, J4 =

0 0 0 0 1 0 0 00 0 0 0 0 −1 0 00 0 0 0 0 0 −1 00 0 0 0 0 0 0 −1

−1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 0

,

J5 =

0 0 0 0 0 1 0 00 0 0 0 1 0 0 00 0 0 0 0 0 0 −10 0 0 0 0 0 1 00 −1 0 0 0 0 0 0

−1 0 0 0 0 0 0 00 0 0 −1 0 0 0 00 0 1 0 0 0 0 0

, J6 =

0 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 −1 0 00 0 −1 0 0 0 0 00 0 0 1 0 0 0 0

−1 0 0 0 0 0 0 00 −1 0 0 0 0 0 0

,


J7 =

0 0 0 0 0 0 0 10 0 0 0 0 0 −1 00 0 0 0 0 1 0 00 0 0 0 1 0 0 00 0 0 −1 0 0 0 00 0 −1 0 0 0 0 00 1 0 0 0 0 0 0

−1 0 0 0 0 0 0 0

.


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Department of Mathematics, Eastern Michigan University, Ypsilanti, MI, 48197, USAE-mail address: [email protected]

Department of Mathematics, Georgetown University, Washington D.C. 20057, USA

Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30013, ROCE-mail address: [email protected]

Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Nor-way

E-mail address: [email protected]

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