Dual unitary matrices and unit dual quaternions

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1 Dual unitary matrices and unit dual quaternions Erhan Ata and Yusuf Yayli Abstract. In this study, the dual complex numbers defined as the dual quaternions have been considered as a generalization of complex numbers. In addition, the dual unitary matrices that are more general form than unitary matrices were obtained. Finally, the group of the dual symplectic matrices was attained by using symplectic structure upon dual quaternions. In particular, the group of symplectic matrices that are isomorphic to dual unitary matrices was studied. M.S.C. 2000: 15A33. Key words: dual quaternion, dual unitary matrix, dual complex number, symplectic matrix. 1 Introduction The algebra H = {q = a a 1 i + a 2 j + a 3 k : a 0, a 1, a 2, a 3 R} of quaternions is defined as the 4-dimensional vector space over R having a basis {1, i, j, k} with the following properties i 2 = j 2 = k 2 = 1, ij = ji = k. It is clear that H is an associative and not commutative algebra and 1 is the identity element of H. A quaternion q = a a 1 i + a 2 j + a 3 k is pieced into two parts with scalar piece S q = a 0 and vectorial piece V q = a 1 i+a 2 j+a 3 k. We also write q = S q + V q. The conjugate of q = S q + V q is then defined as q = S q V q. We call a quaternion pure if its scalar part vanishes. The pure quaternions form the three-dimensional linear subspace ImH = {a 1 i + a 2 j + a 3 k : a 1, a 2, a 3 R} = {q H : q = q} of H spanned by {i, j, k}. Summation of two quaternions q = S q + V q and p = S p + ( V p is defined Vq as q + p = (S q + S p ) + + ) V p.multiplication of a quaternion q = S q + V q with a scalar λ R is defined as λq = λs q + λ V q. In addition,quaternionic multiplication of two quaternions q = a a 1 i + a 2 j + a 3 k and p = b 0 + b 1 i + b 2 j + b 3 k is defined Vq qp = S q S p, V p + S qvp + S pvq + V q Λ V p, Vq where, V p = a 1 b 1 + a 2 b 2 + a 3 b 3, V q Λ V p = (a 2 b 3 a 3 b 2 ) i (a 1 b 3 a 3 b 1 ) j + (a 1 b 2 a 2 b 1 ) k. Thus, with this multiplication operation, H is called real quaternion Differential Geometry - Dynamical Systems, Vol.10, 2008, pp c Balkan Society of Geometers, Geometry Balkan Press 2008.

2 2 Erhan Ata and Yusuf Yayli algebra[5. Quaternionic multiplication satisfies the following properties as in [6: For any two quaternions q and p we have qp = pq and the formula for the inner product q, p = qp+pq 2. In particular, if q = p, we obtain q 2 = q, q = qq, the usual Lengith- Identity. With this, the quaternionic inverse of a nonzero quaternion q H can be written as q 1 = q. The 3-sphere S 3 H in quaternionic calculus is like the unit q 2 circle S 1 C in complex calculus. In fact, S 3 = {q H: q = 1} constitutes a group under quaternionic multiplication; this is an immediate consequence of the fact that quaternionic multiplication is normed as qp = q p. Definition 1. Each element of the set D = { A = a + εa : a, a R and ε 0, ε 2 = 0 } = {A = (a, a ) : a, a R} is called a dual number. A dual number A = a + εa can be expressed in the form A = ReA + εdua, where ReA = a and DuA = a.the conjugate of A = a + εa is defined as A = a εa. Summation and multiplication of two dual numbers are defined as similar to the complex numbers but it is must be forgotten that ε 2 = 0.Thus, D is a commutative ring with a unit element [3. Definition 2. The ring H D = {Q = A 0 + A 1 i + A 2 j + A 3 k : A 0, A 1, A 2, A 3 D} of is defined as the 4-dimensional vector space over D having a basis {1, i, j, k} with the same multiplication property of basis elements in real quaternions. Each element of H D is called a dual quaternion,dual numbers A 0, A 1, A 2, A 3 are called components of dual quaternion Q. A quaternion Q = A 0 + A 1 i + A 2 j + A 3 k is pieced into two parts with scalar piece S Q = A 0 and vectorial piece V Q = A 1 i + A 2 j + A 3 k. We also write Q = S Q + V Q. Moreover, any dual quaternion Q can be written in the form Q = q + εq,where q, q H. The conjugate of Q = S Q + V Q is then defined as Q = S Q V Q. We call a dual quaternion pure if its scalar part vanishes. The pure dual quaternions form the three-dimensional linear subspace ImH D = {A 1 i + A 2 j + A 3 k : A 1, A 2, A 3 D} = { Q H D : Q = Q } of H D spanned by {i, j, k} [4. Summation of two dual quaternions Q = S Q + V Q and P = S P + ( V P is defined VQ as Q + P = (S Q + S P ) + + ) V P. Multiplication with a scalar λ R of a dual quaternion Q = S Q + V Q is defined as λq = (λa 0 ) + (λa 1 )i + (λa 2 )j + (λa 3 )k.in addition,dual quaternionic multiplication of two dual quaternions Q = S Q + V Q and P = S P + V P is defined QP = S Q S P VQ, V P + S Q VP + S P VQ + V Q Λ V P.

3 Dual unitary matrices and unit dual quaternions 3 Thus, with this multiplication operator, H D is called dual quaternion algebra[4. Dual quaternionic multiplication satisfies the following properties: For any two quaternions QP +P Q Q and P we have QP = P Q and the formula for the inner product Q, P = 2. In particular, if Q = P, we obtain Q 2 = Q, Q = QQ, the usual Lengith-Identity. With this, the dual quaternionic inverse of a dual quaternion Q H D that its scalar part is nonzero can be written as Q 1 = Q. The 3-dual sphere Q 2 S 3 D = {Q H D : Q = 1} H D constitutes a group under dual quaternionic multiplication; this is an immediate consequence of the fact that dual quaternionic multiplication is normed as QP = Q P. The dual quaternion operator. Multiplication of two unit dual quaternions A 0 and B 0 is as follows; A 0 A0 B 0 =, B 0 + A 0 Λ B 0 = cos Θ + S sin Θ where Θ = θ + θ is the dual angle between the unit dual quaternions A 0 and B 0, and S = s0 + εs 0 = A 0 Λ B 0 A 0 Λ is a unit vector which is ortogonal to both A 0 and B 0. Also B 0 each unit dual quaternion corresponds to a directed line. In adition, the conjugate K( A 0 B 0 ) of A 0 Λ B 0 is K( A 0 B 0 ) = B 0 A 0 = (cos Θ + S sin Θ) = Q 0 ( A0 ) 1 ( B0 ) 1 and the inverses and respectively of A0 and B 0 are ( A0 ) 1 = K A0 N A0 = A 0, ( B0 ) 1 = K B0 N B0 = B 0. Thus, we can write Q 0 = ( B 0 A 0 ) = ( B0 ) 1 A0 = B 0 ( A0 ) 1, where Q 0 = cos Θ + S sin Θ is unit dual quaternion.thus, the following operator Θ Q 0 = cos Θ + S sin Θ is called dual quaternion operator. Hence, we can say that the expression Q 0 = B 0 ( A 0 ) 1 than B 0 = Q 0 A 0 which is found by left side multiplication of A 0 by Q 0 rotates A 0 around the axes S with a dual angle Θ. Since Θ = θ + εθ, a rotation of angle θ, a slide of θ occurs and θ θ is the step. This statement, we can show in the following figure. 2 Dual unitary matrices and unit dual quaternions In this section, we will firstly define the dual complex numbers similar to the complex numbers and express the dual quaternions as the dual complex numbers.

4 4 Erhan Ata and Yusuf Yayli Figure 1: Screw motion Definition 3. We know that the complex numbers set C = { z = a + bi : a, b R and i 2 = 1 } = {z = (a, b) : a, b R} has field structure. Now define the set C D as C D = { Z = A + Bi : A, B D and i 2 = 1 } = {Z = (A, B) : A, B D}. Each element of C D is called a dual complex number. A dual complex number Z = A + Bi can be expressed in the form Z = DuZ + εimz, where DuZ = A and ImZ = B. The conjugate of Z = A + Bi is defined as Z = A Bi. Summation and multiplication of any two dual complex numbers Z = A + Bi and W = C + Di are defined in the following ways, and Z + W = (A + C) + (B + D)i = (A + C, B + D) Z.W = (A + Bi)(C + Di) = (AC BD) + (AD + BC)i = (AC BD, AD + BC). We can give matrix representation of multiplying the dual complex numbers Z and W as [ [ A B C. B A D Thus, C D is a commutative ring with a unit element. Since for any dual complex number Z = A+Bi, where A, B D we can write A = a+εa and B = b+εb, where a, a, b, b R. In particular it a = b = 0, then Z = A + Bi becomes Z = a + bi since A = a and B = b, where a, b R and i 2 = 1. Namely, Z is a complex number, i.e. Z C. Hence we obtain C C D. We can give the polar form of dual complex numbers as complex numbers. Let Z = A+Bi be a unit dual complex number. Then, we get A 2 +B 2 = 1, since Z = 1. Writing A = a + εa and B = b + εb, we obtain a 2 + b 2 = 1, ab + ba = 0. Thus, the unit dual complex number Z = A + Bi can also be written as Z = cos Θ + i sin Θ =

5 Dual unitary matrices and unit dual quaternions 5 e iθ.let Z = A 0 +A 1 i and W = B 0 +B 1 i be unit dual complex numbers. Thus, we can write W = e iθ Z such that hence e iθ is called the dual complex number operator, Now, we show this. The unit dual complex numbers Z = A 0 + A 1 i and W = B 0 + B 1 i can also be writen as Z = cos Θ Z + i sin Θ Z = e iθ Z and W = cos Θ W + i sin Θ W = e iθw,where cos Θ Z = A 0, sin Θ Z = A 1 and cos Θ W = B 0, sin Θ W = B 1. Then W Z = cos Θ W + i sin Θ W cos Θ Z + i sin Θ Z = cos(θ W Θ Z ) + i sin(θ W Θ Z ) = e i(θ W Θ Z ). For Θ = Θ W Θ Z, one can obtain W Z = eiθ W = e iθ Z. Thus, the operator Θ e iθ = cos Θ + i sin Θ is called dual complex number operator. Multiplication a dual complex number by e iθ means a rotation by the dual angle Θ around the origin of this dual complex number in dual complex plane. If the dual pieces of unit dual complex numbers Z = A 0 + A 1 i and W = B 0 + B 1 i are taken zero, we obtain z = a 0 + a 1 i and w = b 0 + b 1 i, where a 0, a 1, b 0, b 1 R. Hence, it can be founded that Z = cos θ Z + i sin θ Z = e iθ Z and W = cos θ W + i sin θ W = e iθ W. For θ = θ W θ Z it can be written W Z = ei(θ W θ Z ) = e iθ W = e iθ Z. This latter statement is usualy known to be the rotation by an real angle θ around the origin on complex plane. Matrix representation of the dual quaternions We define the set H D1 = { Q = A 0 + A 1 i : A 0, A 1 D and i 2 = 1 } as a subset of dual quaternions H D. H D1 is a subalgebra with the same operations of H D. Furthermore, there is a one-to-one correspondence between every dual quaternion (A 0 + A 1 i) D and complex number ( ) A 0 + A 1 1 CD. Hence, H D1 and C D are isomorphic, i.e. H D1 = CD. Thus, the algebra H D contains a field which is isomorphic to C D. Therefore, with the operation H D C D (Q, B+C 1) H D Q(B+C 1) H D is a module over C D. However, we will consider this structure as a vector space. Since any dual quaternion Q = A 0 + A 1 i + A 2 j + A 3 k over dual complex numbers can be written in the form Q = (A 0 + A 1 i) + j(a 2 A 3 i). Hence, the vector space H D onto C D is 2-dimensional. From here it is clear that H D = Sp {1, j}. If q is

6 6 Erhan Ata and Yusuf Yayli taken to be 0, i.e q = 0,since Q = q in any dual quaternion Q = q + εq ;q, q H then H H D. Thus, ıf H D and H correspond to C D and C, respectively, we obtain H = C since the real quaternion q = a a 1 i + a 2 j + a 3 k can be written in the form q = (a 0 + a 1 i) + j(a 2 a 3 i). The following operator T : H D Hom(H D, H D ) is used to obtain matrix representation of dual quaternions.for every Q H D, we describe the transformation as, T Q : H D P H D T Q (P )=QP If dual quaternion product is considered, it can be easily seen that the operators linear. Thus, the set Hom(H D, H D ) = {T Q : Q H D } becomes a module over C D that will be considered as a vector space. If Q = A 0 + A 1 i + A 2 j + A 3 k for any Q H D, take A = A 0 + A 1 i and B = A 2 A 3 i, since for any Q H D we can write Q = 1A + jb H D, then T Q (1) = Q = 1A + jb T Q (j) = Qj = (1A + jb)j = 1( B) + ja where T Q is the corresponding matrix over C D obtained by the transformation. Hence, [ A B T Q = B A Thus, the transformation Q T Q is the 2 2 matrix representation of the algebra H D over the dual complex numbers module C D If the dual pieces of the dual numbers A 0, A 1, A 2, A 3 in dual quaternion Q = A 0 + A 1 i + A 2 j + A 3 k are taken zero, a real quaternion in the form Q = q = a a 1 i + a 2 j + a 3 k is obtained. It can be written as Q = q = 1(a 0 + a 1 i) + j(a 2 a 3 i) or Q = q = 1a + jb since H = C,where a, b C and a = a 0 + a 1 i, b = a 2 a 3 i.the matrix corresponding to the real quaternion Q = q can be found as [ a b T Q = T q = b a Thus, the 2 2 matrix representation of algebra H over complex numbers field C is obtained. Consider the set of matrices, define an transformation as, {[ } A B M 2 (C D ) = : A, B C B A D f : H D M 2 (C D ). Q f(q)=t Q

7 Dual unitary matrices and unit dual quaternions 7 Thus, let us show that f (QP ) = f(q)f(p ) or T QP = T Q T P. For Q = A 0 + A 1 i + A 2 j + A 3 k and P = B 0 + B 1 i + B 2 j + B 3 k VQ QP = S Q S P, V P + S QVP + S P VQ + V Q Λ V P = A 0 B 0 (A 1 B 1 + A 2 B 2 + A 3 B 3 ) + (A 0 B 1 + A 1 B 0 + A 2 B 3 A 3 B 2 ) i + (A 0 B 2 + A 2 B 0 + A 3 B 1 A 1 B 3 ) j + (A 0 B 3 + A 1 B 2 A 2 B 1 ) k is obtained. Then, we can write Q = (A 0 + A 1 i) + j (A 2 A 3 i), P = (B 0 + B 1 i) + j (B 2 B 3 i) QP = [(A 0 B 0 (A 1 B 1 + A 2 B 2 + A 3 B 3 )) + (A 0 B 1 + A 1 B 0 + A 2 B 3 A 3 B 2 ) i +j [(A 0 B 2 + A 2 B 0 + A 3 B 1 A 1 B 3 ) (A 0 B 3 + A 3 B 0 + A 1 B 2 A 2 B 1 ) i. Hence, [ A0 + A T Q = 1 i A 2 A 3 i A 2 A 3 i A 0 A 1 i [ B0 + B, T P = 1 i B 2 B 3 i B 2 B 3 i B 0 B 1 i T QP = [ A 0 B 0 (A 1 B 1 + A 2 B 2 + A 3 B 3 ) + (A 0 B 1 + A 1 B 0 + A 2 B 3 A 3 B 2 ) i (A 0 B 2 + AA 2 B 0 + A 3 B 1 A 1 B 3 ) (A 0 B 3 + A 3 B 0 + A 1 B 2 A 2 B 1 ) i (A 0 B 2 + A 2 B 0 + A 3 B 1 A 1 B 3 ) (A 0 B 3 + A 3 B 0 + A 1 B 2 A 2 B 1 ) i A 0 B 0 (A 1 B 1 + A 2 B 2 + A 3 B 3 ) (A 0 B 1 + A 1 B 0 + A 2 B 3 A 3 B 2 ) i is found. Since [ [ A0 + A T Q T P = 1 i A 2 A 3 i B0 + B 1 i B 2 B 3 i A 2 A 3 i A 0 A 1 i B 2 B 3 i B 0 B 1 i = [ (A 0 + A 1 i) (B 0 + B 1 i) + ( A 2 A 3 i) (B 2 B 3 i) (A 2 A 3 i) (B 0 + B 1 i) + (A 0 A 1 i) (B 2 B 3 i) (A 0 + A 1 i) ( B 2 B 3 i) + ( A 2 A 3 i) (B 0 B 1 i) (A 2 A 3 i) ( B 2 B 3 i) + (A 0 A 1 i) (B 0 B 1 i) = [ A 0B 0 (A 1 B 1 + A 2 B 2 + A 3 B 3 ) + (A 0 B 1 + A 1 B 0 + A 2 B 3 A 3 B 2 ) i (A 0 B 2 + A 2 B 0 + A 3 B 1 A 1 B 3 ) (A 0 B 3 + A 3 B 0 + A 1 B 2 A 2 B 1 ) i (A 0 B 2 + A 2 B 0 + A 3 B 1 A 1 B 3 ) (A 0 B 3 + A 3 B 0 + A 1 B 2 A 2 B 1 ) i A 0 B 0 (A 1 B 1 + A 2 B 2 + A 3 B 3 ) (A 0 B 1 + A 1 B 0 + A 2 B 3 A 3 B 2 ) i, T QP = T Q T P or f(qp ) = f(q)f(p ). Thus, the isomorphism H D {[ } = M2 (C D ) is a b found. If for M D the set of matrices M 2 (C) = : a, b C and for H b a D the algebra of real quaternions H are substituted, since the dual pieces of Q and P are equal to zero Q = q = a 0 + a 1 i + a 2 j + a 3 k, P = p = b 0 + b 1 i + b 2 j + b 3 k will be obtained. Hence, the special condition T qp = T q T p is obtained that gives the isomorphism H = M 2 (C).

8 8 Erhan Ata and Yusuf Yayli Matrix representation of the unit dual sphere The set S 3 D = {Q = (A 0 + A 1 i) + j (A 2 A 3 i) : Q H D } = { Q = A 0 + A 1 i + A 2 j + A 3 k : Q H D, A A A A 2 3 = 1 } is named as unit dual sphere in H D. The set S 3 D is a group with quaternion product. Also taken the special dual uniter group { [ } A B SU 2 (D) = K = : K M B A D, (K T )K = I 2, det K = 1 instead of M D. Theorem 4. The groups S 3 D and SU D (2) are isomorphic. Proof. We define the transformation g : S 3 D Q SU D (2) g(q)=t Q Firstly; this transformation is well defined. Because, for Q = A 0 + A 1 i + A 2 j + A 3 k SD 3, A2 0 + A A A 2 3 = 1. Hence, [ A0 + A T Q = 1 i A 2 A 3 i ( ) [ T A 2 A 3 i A 0 A 1 i Q T A0 A = 1 i A 2 + A 3 i. A 2 + A 3 i A 0 + A 1 i Thus, it is found that T Q SU 2 (D) since ( ) [ [ TQ T A0 A T Q = 1 i A 2 + A 3 i A0 + A 1 i A 2 A 3 i A 2 + A 3 i A 0 + A 1 i A 2 A 3 i A 0 A 1 i = [ = = (A 0 A 1 i) (A 0 + A 1 i) + (A 2 + A 3 i) (A 2 A 3 i) ( A 2 + A 3 i) (A 0 + A 1 i) + (A 0 + A 1 i) (A 2 A 3 i) (A 0 A 1 i) ( A 2 A 3 i) + (A 2 + A 3 i) (A 0 A 1 i) ( A 2 + A 3 i) ( A 2 A 3 i) + (A 0 + A 1 i) (A 0 A 1 i) [ A A A A [ A A A A 2 3 = I 2 and [ A0 + A det (T Q ) = det 1 i A 2 A 3 i A 2 A 3 i A 0 A 1 i = (A 0 + A 1 i) (A 0 A 1 i) ( A 2 A 3 i) (A 2 A 3 i) = A A A A 2 3 = 1. Now, we show that g (QP ) = g (Q) g (P ) or T QP = T Q T P for the function g : S 3 D Q SU D (2) g(q)=t Q,

9 Dual unitary matrices and unit dual quaternions 9 where any Q, P S 3 D. We know that for any Q, P S 3 D; Q, P H D then f (QP ) = f (Q) f (P ) or T QP = T Q T P.From here g (QP ) = g (Q) g (P ) is obtained. Thus, the isomorphism S 3 D = SU D (2) is found. If the dual pieces of dual quaternion Q S 3 D are taken zero, Q = q = a a 1 i + a 2 j + a 3 k becomes a real quaternion. Hence, Q = q S 3. So, for the matrix [ a0 + a T Q = T q = 1 i a 2 a 3 i, a 2 a 3 i a 0 a 1 i (T q T )T q = I 2 and det (T q ) = 1. This give us the isomorphism S 3 = SU (2). If the imaginary pieces of real quaternion q = a a 1 i + a 2 j + a 3 k = (a 0 + a 1 i) + j (a 2 a 3 i) are taken zero, the complex number q = a 0 + ja 2 C is obtained, where a 0, a 2 R and a a 2 2 = 1. Then, [ a0 a T q = 2 a 2 a 0,which gives (T q T ) = T T q, ( T q T ) (T q ) = I 2 and det (T q ) = 1. Thus, the isomorphism S 1 = SO (2) is found. Definition 5. Let H D be the algebra of dual quaternions; n being an integer greater than zero, we denote by H n D the product of n sets identical to H D. Elements of the set H n D = H D H D... H D { U = = (u 1, u 2,..., u n ) : u i H D, 1 i n} will be called a (dual quaternionic) vector; u 1, u 2,..., u n will be called the coordinates of this vector[2. and The addition of vectors and the scalar product in H D is defined by + : H n D H n ( D V U, ) H n D U + V =(u1,u 2,...,u n )+(v 1,v 2,...,v n ) : H n D H D H D. ( U, Q) U Q=(u1 q,u 2 q,...,u n q) Thus, H n D becomes a modul over H D. But with this structure we will call a vector space for H n D. The transformation, : H n D H n ( D V U, ) H D U, V = n K u i v i i=1 is symplectic product onto H n D, where U = (u 1, u 2,..., u n ) and V = (v 1, v 2,..., v n ) are dual quaternionic vectors. Symplectic product over H n D has similar properties

10 10 Erhan Ata and Yusuf Yayli with Hermitian product. That is to say; The symplectic product have the following properties U U V + V, W =, W +, W U U U, V + W =, V +, W U U U U, V Q =, V Q, Q, V = K Q, V, where any vectors U, V, W H n D and any quaternion Q H D. Hence U, U = n K u iu i = U, where U is a dual number and this number is called the norm of U. i=1 Definition 6. The vector space H D on which a symplectic product is defined is called a dual symplectic vector space. A linear endomorphism σ : H D H D of the vector space H D is a mapping that have the following properties ( U ) ( U ) ( V ) ( U ) ( U ) σ + V = σ + σ, σ Q = σ Q, where any vectors U, V H n D and any quaternion Q H D. Hence, the linear endomorphism of H n D are in a one-to-one correspondence with the matrices (Q ij ) which components are in H D. We shall denote by the same symbol σ the endomorphism itself and the corresponding matrix. If σ = (Q ij ) and τ = (P ij ) are two of these matrices, we denote (as usual) by στ the matrix (s ij ) with ( n ) (s ij ) = Q ij P kj k=1 ( U ) and then we have στ = σ τ, i.e. (στ) = σ ( τ ) U for every vector U H D. the set of all matrices of degree n with coefficients in H D will be denoted by M n (H D )[2. Definition 7. Let H n D be a dual symplectic vector space. If the linear endomorphism σ : H n D H n D has the following property ( U ) ( V ) U σ, σ =, V for every vectors U, V H n D, σ is called a symplectic mapping over H D [1. Proposition 8. If σ is a symplectic mapping, σ has the invers σ 1 and σ 1 is also a symplectic mapping[1.

11 Dual unitary matrices and unit dual quaternions 11 Theorem 9. Let σ be a symplectic mapping over H n D. If Q = (q ij ) is the corresponding matrix of σ, Q satisfies the following equality ( K t Q ) Q = In, where I n is the unit matrix with n type. Proof. Let the vector space H n D be spanned by the n linearly independent vectors e1, e 2,..., e n ; where e i is the vector whose j th coordinate is δ ij. We may write σ e 1, σ e j = e i, e j to be the linear endomorphism σ : H n D H n D for σ e i = n j=1 ej q ji and σ e k = n s=1 es q sk. Hence,we obtain n n ej q ji, es q sk = (δ ik ) j=1 s=1 n K qji e j, e s q sk = j,s=1 n j=1 n K qji q jk = (δ ik ) j=1 ( ) t Kqji qjk = (K Q ) t Q = I. Definition 10. The group of symplectic mappings onto H n D which are isomorphic to the group of matrices are called group of symplectic matrices and denoted in the following form as in[1 } S p (n, H D ) = {σ M n (H D ) : (K σ ) t σ = I n. If we choose n = 1 as a special case, we obtain { } S p (1, H D ) = σ M 1 (H D ) : (K σ ) t σ = 1 = {σ M 1 (H D ) : N σ = 1} = S 3 D. Then, S p (1, H D ) = S 3 D is found. If σ S p (1, H D ), it is the form [ A0 + A σ = 1 i A 2 A 3 i and (K A 2 A 3 i A 0 A 1 i σ ) t σ = 1. If the dual pieces of σ are taken zero, σ S p (1, H) since [ [ A0 A σ = 2 a0 + a = 1 i a 2 a 3 i A 2 A 0 a 2 a 3 i a 0 a 1 i. Furthermore, it is found σ S 3 since (K σ ) t σ = 1 or N σ = 1. Hence, the isomorphism S [ p (1, H) = S 3 is obtained. Now, let the imaginary pieces of σ be zero. For σ = a0 a 2, we find K a 2 a σ = σ. Since (K σ ) t σ = 1 than σ t σ = 1, we find σ S 1. Thus, 0

12 12 Erhan Ata and Yusuf Yayli the isomorphism S p (1, C) = S 1 is obtained. Finally, we may write the followings ın general: SU D (2) = S 3 D = S p (1, H D ). As a special case of this statement we find SU (2) = S 3 = Sp (1, H). It also obtained as a special case of this last statement that References SO (2) = S 1 = Sp (1, C). [1 E. Ata, Symplectic geometry on dual quaternions, D.Ü. Fen Bil. Derg., 6 (2004), [2 C. Chevalley, Theory of Lie Groups, Princeton Universty Press, [3 H.H. Hacısalihoğlu, Acceleration Axes in Spatial Kinamatics, Communications, 20A, (1971), [4 H.H. Hacısalihoğlu, Hareket Geometrisi ve Kuaternionlar Teorisi, Gazi Ünv. Publishing, [5 K. Yano and M. Kon, Structures on Manifolds, World Scientific Publishing, [6 G. Toth, Glimpses of Algebra and Geometry, Springer-Verlag, Authors addresses: Erhan Ata Dumlupınar Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Kütahya, Turkey. eata@dumlupinar.edu.tr Yusuf Yayli Ankara Üniversitesi, Fen Fakültesi, Matematik Bölümü, Ankara, Turkey. yyayli@science.ankara.edu.tr