GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS

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1 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA Abstract. The present article applies the method of Geometric Analysis to the study H-type groups satisfying the J 2 condition and finishes the series of works describing the Heisenberg group and the quaternion H-type group. The latter class of H-type groups satisfying the J 2 condition is related to the octonions. The relations between the group structure and the boundary 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 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 noncommutative, 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 the H-type homogeneous groups, the class satisfying the J 2 Clifford algebras condition [7]. Three associative 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 [12, 16]. The homogeneous groups satisfying the J 2 condition 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 [10, 18]. In the present article we describe the construction of H-type homogeneous groups associated with the four division algebras. The Heisenberg group, the quaternion and octonion H-type groups, admitting the maximal dimension of their center, satisfy the J 2 condition. Applying the 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 # 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 # and of the University of Bergen. 1

2 2 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA arbitrary points of the groups. We discuss the cardinality of geodesics depending on the location of final points. The complex action is presented. Part of this article is based on a lecture presented by the first author during the Symposium on 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 for his invitation. The final version of this paper was in part written while the authors visited the National Center for Theoretical Sciences and National Tsing Hua University during January 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 [7]. Let G be a real Lie algebra, equipped with the Lie bracket [, ], which can be written as an orthogonal direct sum, G = V 1 V 2, [V 1, V 1 ] V 2, [V 1, V 2 ] = [V 2, V 2 ] = 0. Suppose that G is endowed with a scalar product,. Define the linear mapping J : V 2 End(V 1 ) by the formula (2.1) J Z X, X = Z, [X, X ], X, X V 1, Z V 2, whence (2.2) J T Z = J Z, Z V 2. We say that G is H-type if (2.3) J 2 Z = Z 2 U for all Z in V 2, where U denotes the identity mapping. The H(eisenberg)-type groups were introduced 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) J Z J Z + J Z J Z = 2 Z, Z U, Z, Z V 2, see [7]. When there exists a linear mapping J : V 2 End(V 1 ) satisfying (2.2) and (2.3), V 1 is called the Clifford module over V 2. The algebra G (or the Clifford module associated with G) satisfies the J 2 condition if, whenever X V 1 and Z, Z V 2 with Z, Z = 0, then there exists Z in V 2 such that (2.5) J Z J Z X = J Z X. We present here a result from [7] giving the classification of H-type algebras satisfying the J 2 condition. Denote by G0 n the Euclidean n-dimensional space, by Gn 1 the n-dimensional Heisenberg algebra, by G3 n the n-dimensional quaternion H-type algebra, and by G1 7 the octonion H-type algebra. The lower index corresponds to the topological dimension of V 2 and the upper index reflects the real, complex, quaternion and octonion topological dimensions of V 1. Theorem 2.1 ([7]). Suppose that G is an H-type algebra satisfying the J 2 condition. Then G is isometrically isomorphic to G n 0, Gn 1, Gn 3 or to G1 7. Before we describe the general construction of groups G n 0, Gn 1, Gn 3, and G1 7, we would like to remind the Cayley-Dickson construction of division algebras R (real numbers), C (complex numbers), H (quaternion numbers), and O (octonion numbers). The Cayley-Dickson construction explains why each one of the algebras fits neatly inside the next. Recall that the division

3 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 3 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 numbers a, 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 defined as in (2.6) and the product as in (2.7). We obtains the quaternion numbers H that form a noncommutative 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 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. 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 J 2 condition We present a general construction of the H-types algebras, satisfying J 2 condition. Using the Cayley-Dickson product, we first describe the following groups: Euclidean n-dimensional space G n 0 = Rn, the Heisenberg group G n 1, the quaternion H-type group Gn 3, and the octonion H- type group G 1 7. Then we obtain the corresponding algebras Gn 0, Gn 1, Gn 3, and G1 7 as infinitesimal representations of the groups The Euclidean space. The space G n 0 = Rn is a trivial example of an H-type group since all commutative relations vanish. The underlying space V 1 is identified with R n via the exponential map which is identity in this case. The center V 2 is the empty set The Heisenberg group G n 1. 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 1 2 = 1, i 2 = 1. Take a complex number z = (x 1, x 2 ), x 1, x 2 R, and a real number t. Define a new noncommutative law between elements h = [z, t] C R and p = [z, t ] C R by (3.1) hp = [z, t][z, t ] = [z + z, t + t (zi) z ], where firstly we take the Cayley-Dickson product zi = (x 1, x 2 )(0, 1) and then the scalar product of vectors z, z R 2. This multiplication law can be deduced from the matrix product of upper triangular 3 3-matrices [6]. If we use the representation of i as the (2 2) matrix then the group low can be written as [ i = ], hp = [z, t][z, t ] = [z + z, t + t (zi) z ].

4 4 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA Using the algebraic form of a complex number z = x 1 + ix 2 = Rez + i Imz, we can write (3.1) in the form hp = [z, t][z, t ] = [z + z, t + t Im(z z )], where z z is a Cayley-Dickson product of z by z. The non-commutativity of new multiplication law in C R is seen for the last variable t from the one dimensional real space, that corresponds to the existence of only one imaginary unit. It can be shown that C R with new non-commutative multiplication forms a Lie group with the identity element [0, 0], the left translation L h (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-dimensional vectors of complex numbers w = (z 1,..., z n ), 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 law between the elements h = [w, t] and p = [w, t ] C n R is transformed into the following one hp = [w, t][w, t ] = [w + w, t + t + 1 n (z l i) z l 2 ] = [w + w, t + t (wj) w ] = [w + w, t + t Im(w w )], where w w = n z l z l. The Heisenberg algebra G1 n, n = 2k, k N, is identified with the left invariant vector fields on the tangent bundle at the identity element of the group. It splits into the direct sum V 1 V 2, where V 1 = span(x 11, X 21, X 21, X 22,..., X 1n, X 2n ) with a basis given by X 1l = x1l 1 2 x 2l t, and X 2l = x2l x 1l t, l = 1,..., n. The vector field X = (X 11,..., X 2 n ) = ( x (xj) t) with x = (x 11,..., x 2 n ), x = ( x11,..., x2n ), is a natural analogue of the Euclidean gradient. The subspace V 2 is one dimensional and generated by Z = t. Since [X 1l, X 2l ] = Z and other commutators vanish, we verify the condition (2.1). The endomorphism J Z is represented by the matrix J which possesses properties (2.2), (2.3). The J 2 condition holds trivially, since only one J Z is different from Quaternion group G n 3. 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 unity 1 = (1, 0), 1 2 = 1 and three imaginary unities i 1 = (i, 0), i 2 = (0, 1), i 3 = (0, i), such that i 2 1 = i 2 2 = i 2 3 = i 1 i 2 i 3 = 1. The Cayley-Dickson product is no longer commutative, for example, (3.2) i 1 i 2 = i 2 i 1 = i 3, i 2 i 3 = i 3 i 2 = i 1, i 3 i 1 = i 1 i 3 = i 2 In order to design the quaternion H-type group G 1 3, we take a quaternion q = (z 1, z 2 ), z 1, z 2 C, and three real numbers t 1, t 2, t 3 that reflects the three dimensional setting of the space of the imaginary quaternions. Define a new non-commutative law between elements h = [q, t 1, t 2, t 3 ] H R 3 and p = [q, t 1, t 2, t 3 ] H R3 by hp = [q, t 1, t 2, t 3 ][q, t 1, t 2, t 3] (3.3) = [q + q, t 1 + t (qi 1) q, t 2 + t (qi 2) q, t 3 + t (qi 3) q ],

5 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 5 where qi k, k = 1, 2, 3 is the Cayley-Dickson product for the quaternions and is the scalar product in R 4. As in the case of the Heisenberg group we can use the matrix representation of imaginary units i 1 = , i 2 = and rewrite the group law (3.3) in the form , i 3 = hp = [q + q, t 1 + t (qi 1) q, t 2 + t (qi 2) q, t 3 + t (qi 3) q ]. Using the imaginary unities we can represent a quaternion in the algebraic form as q = α + i 1 β + i 2 γ + i 3 δ = α + i 1 Im 1 q + i 2 Im 3 q + i 3 Im 3 q. Then the multiplication law (3.3) takes the form hp = [q, t 1, t 2, t 3 ][q, t 1, t 2, t 3], (3.4) = [q + q, t 1 + t Im 1(q q ), t 2 + t Im 2(q q ), t 3 + t Im 3(q q )], 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 = (q 1,..., q n ), w = (q 1,..., q n). Each of the matrices i m, m = 1, 2, 3, is changed by the block diagonal matrix M m = diag i m with n (4 4)- dimensional matrices i m on the main diagonal. The multiplication law between the elements h = [w, t 1, t 2, t 3 ], p = [w, t 1, t 2, t 3 ] Hn R 3 is the following hp = [w, t 1, t 2, t 3 ][w, t 1, t 2, t 3] = [w + w, t 1 + t n (q l i 1 ) q l 2, t 2 + t n (q l i 2 ) q l, t 3 + t n (q l i 3 ) q l ] = [w + w, t 1 + t (wm 1) w, t 2 + t (wm 2) w, t 3 + t (wm 3) w ] = [w + w, t 1 + t Im 1(w w ), t 2 + t Im 2(w w ), t 3 + t Im 3(w w )], where w w = n q l q l. The quaternion algebra G n 3, n = 4k, k N, is the direct sum of V 1 V 2, where with V 1 = span(x 11, X 21, X 31, X 41,..., X 1n, X 2n, X 3n, X 4n ) X 1l (w, t) = x1l + 1 2( x2l t1 x 3l t2 x 4l t3 ), (3.5) X 2l (w, t) = x2l + 1 2( x1l t1 + x 4l t2 x 3l t3 ), X 3l (w, t) = x3l + 1 2( x4l t1 + x 1l t2 + x 2l t3 ), X 4l (w, t) = x4l + 1 2( x3l t1 x 2l t2 + x 1l t3 ), l = 1,... n

6 6 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA and w = (q 1,..., q n ) = (x 11, x 21, x 31, x 41,..., x 1 n, x 2 n, x 3 n, x 4 n ). The latter system of vector fields can be written as X = (X 11,..., X 4n ) = ( x (xm m ) tm ), 2 with x = (x 11,..., x 4 n ), x = ( x11,..., x4 n ). The subspace V 2 is spanned by {Z 1, Z 2, Z 3 } with Z k = tk. The following commutator relations m=1 [X 1l, X 2l ] = Z 1, [X 1l, X 3l ] = Z 2, [X 1l, X 4l ] = Z 3, [X 2l, X 3l ] = Z 3, [X 2l, X 4l ] = Z 2, [X 3l, X 4L ] = Z 1, hold for l = 1,..., n and others vanish. Thus, the condition (2.1) is verified. The endomorphisms J Zm are represented by matrices M m, m = 1, 2, 3. The J 2 condition holds by the relation (3.2) and it is independent of elements X V 1. Remark 3.1. If we involve into the construction only two imaginary units, then we obtain the quaternion H-type group with two dimensional center V 2. Taking into consideration one of the i k, k = 1, 2, 3, we get a group isomorphic to the Heisenberg group G n Octonion H-type group O 1 7. Octonion numbers, that we think of as a pair of quaternion numbers, has one real unity 1 = (1, 0), 1 2 = 1 and 7 imaginary unities j 1 = (i 1, 0), j 2 = (i 2, 0), j 3 = (i 3, 0), j 4 = (0, 1), j 5 = (0, i 1 ) j 6 = (0, i 2 ), j 7 = (0, i 3 ), whose squares equal 1. The rule of multiplication is presented in Table 1. The product of Table 1. Rules of multiplication of j m j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 1 1 j 3 j 2 j 5 j 4 j 7 j 6 j 2 j 3 1 j 1 j 6 j 7 j 4 j 5 j 3 j 2 j 1 1 j 7 j 6 j 5 j 4 j 4 j 5 j 6 j 7 1 j 1 j 2 j 3 j 5 j 4 j 7 j 6 j 1 1 j 7 j 6 j 6 j 7 j 4 j 5 j 2 j 7 1 j 5 j 7 j 6 j 5 j 4 j 3 j 6 j 5 1 octonions is not associative, for example, j 1 (j 2 j 4 ) = j 7, (j 1 j 2 )j 4 = j 7. We take an octonion w = (q 1, q 2 ), q 1, q 2 H and 7 real numbers t k, k = 1,..., 7, that correspond to 7-dimensional space of imaginary octonions. Define a new non-commutative law for elements h = [w, t] = [w, t 1,..., t 7 ], p = [w, t ] = [w, t 1,..., t 7 ] O R7 by hp = [w, t][w, t ] = [w, t 1,..., t 7 ][w, t 1,..., t 7] (3.6) = [w + w, t 1 + t (wj 1) w,..., t 7 + t (wj 7) w ] = [w + w, t 1 + t Im 1(w w ),..., t 7 + t Im 7(w w )], where wj m, m = 1,..., 7, and w w are the Cayley-Dickson product and is the scalar product in R 8.

7 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 7 There is no matrix representation of j k since the multiplication between j k is not associative, but the matrix multiplication is so. Nevertheless, it is possible to associate a matrix J m with any imaginary unit j m that will represent the corresponding endomorphism J Zm, m = 1,..., 7. The matrices J m are given in the Appendix. Using J m we write the multiplication law (3.6) as follows (3.7) hp = [w, t][w, t ] = [w + w, t 1 + t (wj 1) w,..., t 7 + t (wj 7) w ]. Notice some properties of the matrices J m : (3.8) J 2 m = U, J T m = J m, J 1 m = J m, m = 1,..., 7, where U is the (7 7) identity matrix. The product of the matrices J m does not correspond to the product of the corresponding imaginary unities j m, for example, j 1 j 2 = j 3, but J 1 J 2 J 3. The matrices J m do 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 octonion H-type algebra G7 1 is the direct sum V 1 V 2, where V 1 = span(x 1,..., X 8 ) with (3.9) X l (w, t) = xl (xj m ) l tm, l = 1,..., 8, 2 m=1 where w = (x 1,..., x 8 ) and (xj m ) l is the l-th coordinate of the vector xj m. We give the coefficients (xj m ) l in the Table 2. The subspace V 2 is spanned by {Z 1,..., Z 7 } with Z m = tm. Table 2. The product xj m t1 t2 t3 t4 t5 t6 t7 X 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 X 2 x 1 x 4 x 3 x 6 x 5 x 8 x 7 X 3 x 4 x 1 x 2 x 7 x 8 x 5 x 6 X 4 x 3 x 2 x 1 x 8 x 7 x 6 x 5 X 5 x 6 x 7 x 8 x 1 x 2 x 3 x 4 X 6 x 5 x 8 x 7 x 2 x 1 x 4 x 3 X 7 x 8 x 5 x 6 x 3 x 4 x 1 x 2 X 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 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: ( 8 7 ) exp x k X k + t m Z m G 1 7. k=1 The J 2 condition says that given X = (α, β) and Z, Z V 2 with Z, Z = 0 (for instance corresponding to the multiplication by j 1 and j 2 ), there exists Z in V 2, such that m=1 J Z J Z X = J Z X. Let Z = (a, b). In order to find a and b, we have to solve the linear system of 8 equations with 8 unknown variables.

8 8 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA Table 3. Non-vanishing commutators X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 1 0 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 X 2 Z 1 0 Z 3 Z 2 Z 5 Z 4 Z 7 Z 6 X 3 Z 2 Z 3 0 Z 1 Z 6 Z 7 Z 4 Z 5 X 4 Z 3 Z 2 Z 1 0 Z 7 Z 6 Z 5 Z 4 X 5 Z 4 Z 5 Z 6 Z 7 0 Z 1 Z 2 Z 3 X 6 Z 5 Z 4 Z 7 Z 6 Z 1 0 Z 3 Z 2 X 7 Z 6 Z 7 Z 4 Z 5 Z 2 Z 3 0 Z 1 X 8 Z 7 Z 6 Z 5 Z 4 Z 3 Z 2 Z 1 0 Example. Let Z = j 1, Z = j 2, X = (α, β). We look for the element Z = (a, b) corresponding to the action J 1 J 2 in the equation ( ) (3.10) J 1 J 2 (α, β) = (i 1, 0) (i 2, 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) = j 3. If X = (α, β) = (0, 0,..., 0, 1), then a = ( (0, 0)(0, 1) ) and b = ( (0, 0)(0, 0) ) or (a, b) = j Octonion H-type group G 1 7 The Heisenberg group has been studied extensively by many mathematicians, see for instance [3, 6, 7, 14, 15]. The quaternion H-group was studied in [8, 9]. We concentrate our attention on the octonion H-group following the ideas developed in [2, 4, 5, 6]. There is an essential difference between the cases G n 1 and Gn 3. Even for Gn 3 the J 2 condition (2.5) is rather trivial, since it does not depend on X V 1. In the case of G 1 7 it essentially depends on X V 1 as it was shown in the example. The endomorphisms J m, m = 1,..., 7, are represented by matrices J m. But the composition action of two endomorphisms J l J k does not correspond to the action of the product of the corresponding matrices J l J k. The multiplication law (3.7) defines the left translation L q (p) of the element p by the element q. The Lie algebra is identified with the set of left invariant vector fields whose basis is given by (X, Z) = (X 1,..., X 8, Z 1,..., Z 7 ). A basis of one-forms dual to (X, Z) is dx = (dx 1,..., dx 8 ), and dϑ = (dϑ 1,..., dϑ 7 ) with dϑ m = dt 1 2 (xj m dx). The subspace T H q of the tangent space T q, q G 1 7 defined by the formula ker(dϑ) = 0 is called the horizontal subspace. Since dϑ(x) = 0, the horizontal subspace at q G 1 7 is V 1 (q) = span{x 1 (q),..., X 8 (q)}. We say that an absolutely continuous curve c(s) : [0, 1] G 1 7 is horizontal if the tangent vector ċ(s) satisfies ċ(s) = 8 a l(s)x l (c(s)). The definition of the 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) ṫ m = 1 2 xj m ẋ. The following properties of horizontal curves can be easily obtained (see also [8]).

9 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 9 (i) If a curve c(s) = (x(s), t(s)) is horizontal, then 8 ċ(s) = ẋ l (s)x l (c(s)). (ii) Left translation L q of a horizontal curve c(s) is a horizontal curve c = L q (c) with the velocity 8 c(c) = ẋ l (s)x l ( c(s)). (iii) The acceleration vector c(s) of a horizontal curve c(s) is a horizontal vector such that 8 c(s) = ẍ l (s)x l (c(s)). The following equalities (4.2) xj m w = x wj m for all x, w R 8, m = 1,..., 7, (particularly xj m x = 0) are used in the proof of the last assertion. Since Z l, Z m = 0, Z l, Z m V 2, m l, the action J m J l possesses the following property (4.3) J m J l + J l J m = 0 m, l = 1,..., 7, m l by (2.4). Remark 4.1. The property (4.3) implies that the actions J m J l and J l J m can be represented by matrices J and J respectively, such that J 2 = U, J T = J, J 1 = J, and thus, the property (4.2) holds also for J. 5. Hamiltonian formalism The geometry of the octonion H-type group is induced by the sub-laplacian 8 0 = Xl 2 = x x 2 t + (xj m x ) tm, where x = ( x1,..., x8 ), x = 8 2 x l, t = 7 m=1 2 t m. In the calculation we used Remark 4.1. We introduce the matrix 0 θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 θ 1 0 θ 3 θ 2 θ 5 θ 4 θ 7 θ 6 θ 2 θ 3 0 θ 1 θ 6 θ 7 θ 4 θ 5 7 M = θ m J m = θ 3 θ 2 θ 1 0 θ 7 θ 6 θ 5 θ 4 m=1 θ 4 θ 5 θ 6 θ 7 0 θ 1 θ 2 θ 3. θ 5 θ 4 θ 7 θ 6 θ 1 0 θ 3 θ 2 θ 6 θ 7 θ 4 θ 5 θ 2 θ 3 0 θ 1 θ 7 θ 6 θ 5 θ 4 θ 3 θ 2 θ 1 0 Comparing the matrix M with Table 3 we see that matrix M reflects the commutative relation between X l. We notice the following property. Lemma 5.1. For any X V 1 the action MX corresponding to the matrix M satisfies the following rules (5.1) M 2 X = θ 2 UX, M 3 X = θ 2 XM, M 4 X = θ 4 XU, M 5 X = θ 4 XM,..., where U is the 8 8 identity matrix. m=1

10 10 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA Proof. The proof is a straightforward application of Remark 4.1. Introducing the dual variables ξ l = xl, θ m = tm, l = 1,..., 8, m = 1,..., 7, we get the Hamilton function 8 ( ) 2 (5.2) H(x, t, ξ, θ) = ξ l + (xm) l ξ l = ξ x 2 θ 2 + xm ξ, where denotes the usual scalar product in R 8. The corresponding Hamiltonian system is ẋ = 2ξ + xm ṫ m θ m = θm 2 x 2 + xj m ξ, m = 1,..., 7. (5.3) ξ = H x = 1 2 θ 2 x + ξm θ m = H z m = 0. The solutions γ(s) = (x(s), t(s), ξ(s), θ(s)) of the system (5.3) are called bicharacteristics. = H ξ = H Definition 5.2. Let P 1 (x 0, t 0 ), P 2 (x, t) G 1 7. A geodesic starting at P 1 and ending at P 2 is the projection of a bicharacteristic γ(s), s [0, 1], onto the (x, t)-space, that satisfies the boundary conditions ( ) ( ) x(0), t(0) = (x0, t 0 ), 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) ṫ m = θ m 2 x xj m 2ξ = θ m 2 x xj m ẋ xj m (2ξ ẋ). Making use of the first line of the system (5.3), we write the last term of (5.4) as 1 (5.5) 2 xj m (2ξ ẋ) = 1 2 xj mm x = θ m 2 x 2. Here we used (3.8) and Remark 4.1. Combining (5.4) and (5.5) we deduce (5.6) ṫ m = 1 2 xj m ẋ, 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 V 1 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 ) XM. θ Proof. We observe that exp ( (2s) n 2sM)X = M n X = XU n! n=0 k=0 (2s θ ) 4k+2 XU XM (4k + 2)! θ k=0 (2s θ ) 4k (4k)! k=0 + XM θ (2s θ ) 4k+3 (4k + 3)! k=0 (2s θ ) 4k+1 (4k + 1)!

11 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 11 by (5.1). Note that and Thus, we get (5.7). k=0 k=0 (2s θ ) 4k (4k)! (2s θ ) 4k+1 (4k + 1)! k=0 k=0 (2s θ ) 4k+2 (4k + 2)! (2s θ ) 4k+3 (4k + 3)! = cos(2s θ ) = sin(2s θ ). The last equation in the Hamiltonian system (5.3) shows that the function H(ξ, θ, x, t) does not depend on t. We obtain that θ m are constants which can be used as Lagrangian multipliers. Simplifying the system (5.3) we get (5.8) ẍ = 2ẋM. Solving (5.8) we deduce (5.9) ẋ(s) = ẋ(0) exp(2sm), where ẋ(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) ẋ(s) = cos(2s θ )ẋ(0)u + sin(2s θ ) ẋ(0)m. θ Integrating from 0 to s we get (5.11) x(s) = 1 cos(2s θ ) 2 θ 2 ẋ(0)m + sin(2s θ ) ẋ(0)u. 2 θ Let us describe the t-components of a geodesic curve. If a curve is geodesic, then it is horizontal by Lemma 5.3, and we deduce that (5.12) ṫ m (s) = 1 2 J mx(s) ẋ(s) = θ m ẋ(0) 2 4 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) t m (s) = θ m ẋ(0) 2 4 θ 2 ( sin(2s θ ) ) s, m = 1,..., 7. 2 θ 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, k 1,..., k 6 ), where k 1,..., k 6 are constants. The curve c(s) = (x(s), t(s)) is horizontal. Indeed, ṫ 1 (s) = s2 2, 1 2 (xj 1 ẋ) = s2 2, ṫ k (s) =0 and 1 2 (xj k ẋ) = 0, k = 2,..., 7.

12 12 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA On the other hand, the curve c(s) does not satisfy the system (5.8). The system (5.8) admits the form 1 = 2( θ 1 θ 2 s θ 3 ) 0 = 2(θ 1 s θ 3 s + θ 2 ) 1 = 2(θ 2 s + θ 3 θ 1 ) 0 = 2(θ 3 s θ 2 + θ 1 s) Summing up the first and the third equation, and then, the second and the forth ones, we write the latter system as 2 = 4θ 1 0 = 4θ 1 s 1 = 2(θ 2 s + θ 3 θ 1 ) 0 = 2(θ 3 s θ 2 + θ 1 s). We see that the first and the second equations contradict each other. Lemma 5.6. A curve c is geodesic on the group G 1 7 if and only if (i) c(s) is a horizontal curve and (ii) c(s) satisfies c(s) = 2ċ(s)M. Proof. If a curve is geodesic, then it is horizontal by Lemma 5.3. Proposition 4.1 implies that the vector c is also horizontal: c(s) = 8 ẍl(s)x l (c(s)). Since ẍ(s) = 2ẋ(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) ṫ m = 1 2 xj m ẋ = θ m 2 x 2 + xj m ξ = H, m = 1,..., 7. θ m as it was proved in Lemma 5.3. We see that c(s) satisfies the equations of the second line of (5.3). The condition (ii) of Lemma 5.6 admits the form ẍ(s) = 2ẋ(s)M in the coordinate functions. Define the following curve γ(s) = (x(s), t(s), ξ(s), θ) in the cotangent space, where (5.15) ξ = ẋ(s) x(s)m and θ = (θ 1,..., θ 7 ) is constant. The relation (5.15) implies the equations of the first and the last lines of (5.3). Differentiating (5.15), we get ξ = ẍ 2 1 2ẋM = ẋm ẋm 2 = 1 2 (2ξ + xm)m = ξm 1 2 θ 2 x, by the condition (ii) of Lemma 5.6, (5.1) and (5.15). Thus, γ(s) satisfies the Hamiltonian system (5.3). Then, the projection of γ onto the (x, t)-space, that coincides with c(s), is geodesic. Let us start with an auxiliary result. 6. Connectivity by geodesics Proposition 6.1. The kinetic energy E = 1 2 ẋ 2 is preserved along geodesics. Proof. In fact, de m ds = ẋ ẍ = 2ẋ ẋm = 0 by the property (4.2) of the matrices J m. Now we can discuss the connectivity property case by case. Case 1. Connectivity between 0 = (0, 0) and P = (x, 0), x 0. The following is the first main result of this section.

13 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 13 Theorem 6.1. A smooth curve c(s) is horizontal with constant t-coordinates t 1,..., t 7 if and only if c(s) = (a 1 s,..., a 8 s, t 1,..., t 7 ) with a l R and 8 a2 l 0. In other words, there is only one geodesic joining the origin with a point (x, 0). Proof. Let c(s) be a horizontal curve with constant t-coordinates t 1,..., t 7. Then ṫ m = 0 and (5.12), (5.10), (5.11), and (4.1) imply 0 = ṫ m = θ m ẋ(0) 2 2 θ 2 sin2 (s θ ), m = 1,..., 7. Since the energy ẋ(0) 2 does not vanish we deduce, that θ m = 0 for all m = 1,..., 7. The Hamiltonian system (5.3) is reduced to the next one ẋ = 2ξ 0 = xj m ξ, m = 1,..., 7 ξ = 0 θ m = 0. We see that ξ is a constant vector. Taking into account that x(0) = 0, we get x(s) = (a 1 s,..., a 8 s) with a l = 2ξ l. This proves the statement. Now, let us assume that c(s) = (a 1 s,..., a 8 s, t 1,..., t 7 ) with constant t-components. Set as = (a 1 s,..., a 8 s). Then, ṫ m = 0 and 1 2 (as)j m (as) = s 2 aj m 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 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 ẋ(0) since we do not have enough information about the behavior of x-coordinates. Theorem 6.2. There are infinitely many geodesics c = c(s), s [0, 1], joining the origin with a point (0, t). The corresponding equations are (6.1) x (k) (s) = 4 sin2 (kπs) ẋ(0) 2 ( (6.2) t (k) (s) = t ẋ(0)t + sin(2kπs) ẋ(0)u, k N, 2kπ ), k N, s sin(2kπs) 2kπ where T is a matrix that is obtained from the matrix M replacing θ m by t m (1). The lengths l(c) of corresponding geodesics c(s) are l 2 (c) = 4πk t. Proof. Substituting s = 1 in (5.11) and using (4.2), we calculate 0 = x(1) 2 = sin2 ( θ ) θ 2 ẋ l (0) 2. Since the kinetic energy E = ẋ(0) 2 2 does not vanish we deduce that θ = ( 7 θ2 l ) 1/2 = kπ, k N. Equalities (5.13) give for s = 1 (6.3) t m = θ m ẋ(0) 2, m = 1,..., 7. 4(πn) 2

14 14 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA We find the unknown constants θ m = 4t mπ 2 n 2. Substituting θ ẋ(0) 2 m in (5.11), (5.13), we obtain the equations (6.1) and (6.2) for geodesics and the matrix T. To calculate the length of geodesics, we observe that t = θ ẋ(0) 2 4(kπ) 2 Thus the square of the length l(c) of a geodesic c(s) is ( l (c) = ẋ(s) ds) = ẋ(0) 2 = 4kπ t, k N. 0 Case 3. Connectivity between (0, 0) and (x, t), x 0, t 0. Now, we will look for a solution of equation (5.8) with the boundary conditions Further we need the function (6.4) µ( θ ) = x(0) = 0, t(0) = 0, x(1) = x, t(1) = t. θ sin 2 ( θ ) cot( θ ) = ẋ(0) 2 4kπ from (6.3). that was introduced by Gaveau in [11] and studied in detailes by Beals, Gaveau, Greiner in [2, 3, 4]. By the following lemma, one finds some basic properties of the function µ. θ Lemma 6.2. The function µ(θ) = cot θ is an increasing diffeomorphism of the interval sin 2 θ ( π, π) onto R. On each interval (kπ, (k + 1)π), k = 1, 2,..., the function µ has a unique critical point c k. On this interval the function µ strictly decreases from + to µ(c k ), and then, strictly increases from µ(c k ) to +. Moreover, Hence, µ(c k ) as k. µ(c k ) = c k, k N. 4 t x µ 10 5 π 2π 3π 4π 5π 6π θ 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 sin 4. z

15 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 15 It follows that µ (c k ) = 0 c k = tan(c k ). Plugging in z = c k into the definition of µ, we have c k µ(c k ) = sin 2 (c k ) cot(c k) The proof of the lemma is therefore complete. = tan(c k) cos(c k ) sin(c k ) sin 2 (c k ) = sin(c k) cos 2 (c k ) sin(c k ) sin 2 (c k ) cos(c k ) = sin 3 (c k ) sin 2 (c k ) cos(c k ) = tan(c k) = c k. Theorem 6.3. Given a point P = (x, t) with x 0, t 0, there are finitely many geodesics joining 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 ( 4 sin x (k) 2 ( θ k ) ( tan(s θ k ) cot( θ k ) 1 ) x (s) = sin(s θ k ) cos(s θ k ) θ k cos( θ k ) sin( θ k ) x T (6.6) + ( tan(s θ k ) + cot( θ k ) ) ) xu, k = 1, 2,..., N, (6.7) t (k) 2s θ k sin 2 (2s θ k ) (s) = t, k = 1, 2,..., N, θ k cos( θ k ) sin( θ k ) where T is the matrix that obtained from the matrix M replacing θ m by t m (1). The lengths of these geodesics are (6.8) l 2 n = ν( θ k )( x t ), where ν( θ k ) = θ 2 k sin( θ k )(sin( θ k ) cos( θ k ))+ θ k. Proof. We put s = 1 in (5.11) and get (6.9) ẋ(0) 2 = θ 2 sin 2 ( θ ) x 2. Substituting s = 1 into (5.13) and ẋ(0) 2 from (6.9) we obtain (6.10) t m = θ m x 2 ( θ ) 4 θ sin 2 ( θ ) cot( θ ), m = 1,..., 7. Then (6.11) t = x 2 4 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). Equalities (6.10) and (6.11) give θ m = t m t θ. We set s = 1 in (5.11) and express ẋ(0) as θ ( k ẋ(0) = sin( θ k ) cos( θ k ) x U + tan( θ ) k) 1. M θ k

16 16 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA Expanding Finally ( U + tan( θ k) θ k M) 1 into the series and applying (5.1), we get ( U + tan( θ k) θ k ) 1 M = ( U tan( θ ) k) M ( 1) k tan 2k ( θ k ) θ k k=0 = U cos 2 ( θ k ) M sin( θ k) cos( θ k ) θ k. (6.12) ẋ(0) = x ( θ k cot( θ k )U M ). Substituting ẋ(0) and θ m = t m 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) starting from O = (0, 0). The square of the length of a geodesic c is ( lk 2 1 ) 2 (c) = θ 2 ẋ(0) ds = k x 2 sin 2 ( θ k ). Notice that x t = x 2( 1 + µ( θ k ) ) = sin2 ( θ k )( 1 + θ 2 µ( θ k ) ) lk 2 (c) k by (6.11). From the latter equalities we obtain (6.8) Complex Hamiltonian mechanics Our aim now is to study the complex action which may be used to obtain the length of real geodesics. Definition 7.1. A complex geodesic is the projection of a solution of the Hamiltonian system (5.3) with the non-standard boundary conditions x(0) = 0, x(1) = x, t(0) = 0, t(1) = t, and θ m = iτ m, m = 1,..., 7, onto the (x, t)-space. 7 Let us introduce the notation iτ for the vector ( iτ 1,..., iτ 7 ). Then τ = m=1 τ m 2 and θ = i τ. Notice, that we should treat the missing directions apart from the directions in the underlying space. Definition 7.2. The modifying complex action is defined as (7.1) f(x, t, τ) = i m τ m t m ( (ẋ, ξ) H(x, t, ξ, τ) ) ds. We present some useful calculations following from the system (5.3). ξ ẋ =2 ξ 2 + xm ξ = 1 2 ẋ 2 1 xm ẋ, 2 (7.2) ξ 2 = ẋ xm ẋ + 1 ẋ 2 xm xm = xm ẋ θ 2 x 2, xm ξ = 1 2 xm ẋ 1 2 θ 2 x 2.

17 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 17 Making use of formulas (5.2), (7.2), and the value of the energy E = ẋ(0) 2 2 = θ 2 deduce f(x, t, τ) = i m τ m t m + ẋ(0) cosh(2s τ ) ds sin θ 2 x 2 2, we = i m τ m t m + x 2 τ coth τ. 4 (7.3) The complex action function satisfies the Hamilton-Jacobi equation Indeed, we have Then, 7 m=1 7 m=1 τ m f τ m + H(x, t, x f, t f) = f. f = it m i x 2 τ m µ(i τ ), m = 1,..., 7, τ m 4 τ H(x, t, f x, f t τ m f τ m + H(x, t, f x, f tz ) = i m = i m In the critical points τ c, where f τ m ) = H(x, t, ξ, τ) = x 2 4 τ m t m + x 2 τ coth τ = f. 4 = 0, we have τ 2 sinh 2 τ. τ m t m + x 2 τ ( iµ(i τ ) + τ ) 4 sinh 2 τ f(x, t, τ c ) = H(x, t, x f, t f) = E 2 = 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 the H-type groups with m-dimensional center, where m varies from 1 to 7. The octonion H-type group with 3-dimensional center G 1 3 is isomorphic to the quaternion H-type group G2 3, whereas the octonion H-type group with 1-dimensional center G 1 1 is isomorphic to the Heisenberg group G 4 1. If the dimension m of the center is equal to 2, 4, 5, 6, then the H-type groups do not satisfy the J 2 condition. The J 2 condition essentially expresses the multiplication law between the imaginary unities on C, H, and O. The results of Section 6 contain the equations of geodesics for H-type groups with an arbitrary dimensional center from 1 to 7, since in the calculations we used only the relation (2.4). The same is true for results of Section Action of H-type groups on the Siegel upper half space Let C n+1 be the (n + 1)-dimensional complex space. We use the notation z = (z, z n+1 ), where z = (z 1,..., z n ) C n. The set n U n = {(z 1,..., z n+1 ) C n+1 : 4Re z n+1 > z 2 = z l 2 }

18 18 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA defines the Siegel upper half space. Let B C denotes the unit ball in C n+1 : Then the Cayley transformation B C = {(w 1,..., w n+1 ) C n+1 : w n+1 = 1 z n z n+1, w l = n+1 w l 2 < 1}. z l 1 + z n+1, l = 1,..., n, and its inverse z n+1 = 1 w n+1 2w l, z l =, l = 1,..., n, 1 + w n w n+1 show that the unit ball D and the Siegel upper half space U n are biholomorphically equivalent. Let H n+1, n + 1 = 4k, k N, be an (n + 1)-dimensional quaternion vector space over the field of real numbers. The elements of H n+1 are (n + 1)-tuples of quaternions that we denote by q = (q, q n+1 ), q = (q 1,..., q n ) H n, with the norm q 2 = n+1 q2 l. The Siegel upper half space in H n+1 can be defined by analogy with the complex case as: n U n = {(q 1,..., q n+1 ) H n+1 : 4Re q n+1 > q l 2 = q 2 }. The unit ball B H in H n+1 is B H = {(h 1,..., h n+1 ) H n+1 : n+1 h l 2 < 1}. Since the multiplication of quaternion is not commutative there are two forms of Cayley transformation that give the symmetric geometry. The (left) Cayley transformation, mapping the Siegel upper half space U n onto the unit ball B H has the form q n+1 = (1+h n+1 ) 1 (1 h n+1 ) = (1 + h n+1 )(1 h n+1) 1 + h n+1 2, q l = 2h l (1+h n+1 ) 1 = 2h l(1 + h n+1 ) 1 + h n+1 2, for l = 1,..., n. The inverse transformation from B H onto U n is h l = q l (1 + q n+1 ) 1 = q l(1 + qn+1 ) 1 + q n+1 2, h n+1 = (1 q n+1 )(1 + q n+1 ) 1 = (1 q n+1)(1 + qn+1 ) 1 + q n+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 O 2. We shall consider only 2-dimensional case of octonion vector space over the field of real numbers: O 2 = {o = (o, o 2 ) : o, o 2 O} with the norm o 2 = o 2 + o 2 2. The Siegel upper half space U 1 and the unit disk B O can be defined as U 1 = {(o, o 2 ) O 2 : 4Re o 2 > o 2 }, The transformations B O = {(r, r 2 ) O 2 : r 2 + r 2 2 < 1}. o = 2r (1 + r 2 ) 1, o 2 = (1 + r 2 ) 1 (1 r 2 ), map the Siegel upper half space onto the unit ball and the map r = o (1 + o 2 ) 1, r 2 = (1 o 2 )(1 + o 2 ) 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,

19 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 19 where x is a complex number, a quaternion or an octonion and xy is the Cayley-Dickson product. Let us denote by (q, q) an element of one of the above mentioned Siegel upper half spaces and by (h, h) a point from the corresponding unit ball. Then and Since (1 + q) 1 2 = (1 + q) 2, we have h 2 = h (h ) = q 2 (1 + q) 1 2, h 2 = hh = 1 q 2 (1 + q) 1 2, h 2 + h 2 = ( q q 2 ) (1 + q) 1 2 < 1. q q 2 = q 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 C n+1, H n+1 or O 2. We denote by p = (q, q) a point from the Siegel upper half space U n of K. The boundary of the space U n is n U n = {(q, q) K : 4Re q = q 2 = q l 2 }. We mention here three automorphisms of the domain U n : dilation, rotation and translation. Dilation. Let p = (q, q) U n. For each positive number δ we define a dilation δ p by δ p = δ (q, q) = (δq, δ 2 q). The non-isotropy of the dilation comes from the definition of U n. Rotation. For each unitary linear transformation R that acts on C n, H n or O, we define the rotation R(p) on U n by R(p) = R(q, q) = (R(q ), q). Both, the dilation and the rotation extend to mappings on the boundary U n. Translation We use the notation G for the H-type groups G n 1, Gn 3, G1 7. To each element [w, t] of G we associate the following affine self mapping of U n. (Notice that it is holomorphic map for the cases C n+1, H n+1 ). (8.1) [w, t] : (q, q) (q + w, q + w w q + i t). Here i t = dim V 2 i k t k. 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 + 2Re w q 1, we obtain 4Re (q + w w q ) q + w 2 = 4Re q q 2. Hence, the transformation (8.1) maps U n onto itself and preserves the boundary U n. Let us check that the mapping (8.1) defines an action of the group G on the space U n. 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 ω (w + ω) q w 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 (w + ω) q i Im w ω + i (s + t) ).

20 20 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA Observing that w + ω i Im w ω = w ω Re w ω i Im w ω = w ω w ω, we conclude that (8.3) and (8.4) give the same result. Thus, (8.1) gives us a realization of G as a group of affine (q-holomorphic) bijections of U n. We can identify the elements of U n with the boundary via its action at the origin where h = [w, t]. Thus h(0) = [w, t] : (0, 0) (w, w 2 + i t), G [w, t] (w, w 2 + i t) U n. We may use the following coordinates (q, t, r) = (q, t 1,..., t dim V2, r) on U n : U n (q, q) = (q, t, r), where t k = Im k q, k = 1,..., dim V 2, r = r(q, q) = 4Re q q 2. If 4Re q = q 2, then we get coordinates on the boundary U n of the Siegel upper half space where t k are as above and r = r(q, q) = 0. U n (q, q) = (q, t 1,..., t dim V2 ), 9. Appendix We give here the precise forms of matrices J m J 1 = , J 2 = J 3 = J 5 = , J 4 =, J 6 = ,,,

21 GEOMETRIC ANALYSIS ON H-TYPE GROUPS RELATED TO DIVISION ALGEBRAS 21 J 7 =

22 22 OVIDIU CALIN, DER-CHEN CHANG, IRINA MARKINA References [1] Baez J. C. The octonions. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, [2] Beals R., Gaveau B., and Greiner P. C. Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians, I, II, III. Bull. Sci. Math., 21 (1997), no. 1 3, 1 36, , [3] Beals R., Gaveau B., and Greiner P. C. Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. 79 (2000), no. 7, [4] Calin O., Chang D. C., and Greiner P. C. On a step 2(k + 1) sub-riemannian manifold. J. Geom. Anal., 14 (2004), no. 1, 1 18 [5] Calin O., Chang D. C., and Greiner P. C. Real and complex Hamiltonian mechanics on some subriemannian manifolds. Asian J. Math., 18, No. 1 (2004), [6] Calin O., Chang D. C., and Greiner P. C. Geometric Analysis on the Heisenberg Group and Its Generalizations, to be published in AMS/IP series in advanced mathematics, International Press, Cambridge, Massachusetts, [7] Cowling M., Dooley A. H., Korányi A., and Ricci F. H-type groups and Iwasawa decompositions. Adv. Math. 87 (1991), no. 1, [8] Chang D. C., Markina I. Geometric analysis on quaternion H-type groups. J. Geom. Anal. 16 (2006), no. 2, [9] Chang D. C., Markina I. Anisotropic quaternion Carnot groups: geometric analysis and Green s function. (submited) [10] Chang D. C., Markina I. Quaternion H-type group and differential operator λ, to appear in Science in China, Series A: Mathematics, (2007) [11] Gaveau B. Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), no. 1-2, [12] Gürlebeck K., Sprössig W. Quaternionic and Clifford Calculus for Physicists and Engineers John Wiley and Sons, Chichester, pp. [13] Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratics forms. Trans. Amer. Math. Soc. 258 (1980), no. 1, [14] Kaplan A. On the geometry of groups of Heisenberg type. Bull. London Math. Soc. 15 (1983), no. 1, [15] Korányi A. Geometric properties of Heisenberg-type groups. Adv. in Math. 56 (1985), no. 1, [16] Porteous I. R. Clifford algebras and the classical groups. Cambridge Studies in Advanced Mathematics, 50. Cambridge University Press, Cambridge, pp. [17] Ricci F. The spherical transform on harmonic extensions of H-type groups. Differential geometry (Turin, 1992). Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), no. 4, (1993). [18] Stein E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, 1993, 695 pp. Department of Mathematics, Eastern Michigan University, Ypsilanti, MI, 48197, USA address: ocalin@emunix.emich.edu Department of Mathematics, Georgetown University, Washington D.C , USA Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30013, ROC address: chang@georgetown.edu Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway address: irina.markina@uib.no