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Biquaternions Lie Algebra and Complex-Projective Spaces Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 CJMS. 4(2)(2015), 227-240 Biquaternions Lie Algebra and Complex-Projective Spaces Murat Bekar 1 and Yusuf Yayli 2 1 Department of Mathematics and Computer Sciences, Konya Necmettin Erbakan University, 42090 Konya, Turkey 2 Department of Mathematics, Ankara University, 06100 Ankara, Turkey Abstract. In this paper, Lie group and Lie algebra structures of unit complex 3-sphere S 3 are studied. In order to do this, adjoint C representation of unit biquaternions (complexified quaternions) is obtained. Also, a correspondence between the elements of S 3 and C the special bicomplex unitary matrices SU C2 (2) is given by express- 2 ing biquaternions as 2-dimensional bicomplex numbers C2. The 3 3 3 relation SO(R ) =∼ S ={±1g = RP among the special orthogonal 3 3 group SO(R ), the quotient group of unit real quaternions S ={±1g 3 and the projective space RP is known as the Euclidean-projective space [1]. This relation is generalized to the Complex-projective space and is obtained as SO( 3) ∼ S 3 ={±1g = P3. C = C C Keywords: Bicomplex numbers, real quaternions, biquaternions (complexified quaternions), lie algebra, complex-projective space. 2000 Mathematics subject classification: 20G20, 32C15, 32M05. 1. Introduction It is known that the special orthogonal group SO(3) forms the set of 3 all the rotations in 3-dimensional Euclidean space E , which preserves lenght and orientation [2]. Thus, Lie algebra of the Lie group SO(3) 3 corresponds to the 3-dimensional Euclidean space R , with the cross 1 Corresponding author: [email protected] Received: 07 August 2013 Revised: 21 November 2013 Accepted: 12 February 2014 227 228 Murat Bekar, Yusuf Yayli product operation. Killing bilinear form of the unit 3-sphere is the inner 3 product in E . Adjoint representation adO(3) of the orthogonal group O(3) is the group of all isometries that preserves the inner product on the Lie algebra of O(3) [1]. Similar to SO(3), the special orthogonal 3 3 group SO(C ) forms the set of all the rotations in C , which preserves lenght and orientation. In this paper, Lie group and Lie algebra structures of unit complex 3- sphere S 3 are studied. In order to do this, adjoint representation of unit C biquaternions (complexified quaternions) is obtained. From the adjoint representation of S 3 , it is obtained that the group adS 3 is the group of C C all the complex isometries that preserves the complex metric tensor on the Lie algebra of S 3 . Also, Killing bilinear form on the Lie algebra of C the group S 3 is obtained. It is shown that the Killing bilinear form of C S 3 can be given with the metric tensor in 3. It is obtained that S 3 C E C is not compact, because the value of the Killing bilinear form of S 3 is C obtained complex. 3 ∼ 3 3 The relation SO(R ) = S ={±1g = RP among the special orthog- 3 onal group SO(R ), the quotient group of the unit real quaternions 3 3 S ={±1g and the projective space RP is known as the Euclidean- projective space [1]. Finally, this relation is generalized to the Complex- projective space and is obtained as SO( 3) ∼ S 3 ={±1g = P3. C = C C 2. Preliminaries In this section, initially we will present basics of lie group and lie al- gebra structures. Afterwards, we will present basics of real quaternions, bicomplex numbers and biquaternions (complexified quaternions). 2.1. Basics of Lie Group and Lie Algebra Structures. Definition 2.1. Left and right translation on the group G, determined by an element a 2 G, are the mappings La : G ! G defined by x 7! La(x) = ax and Ra : G ! G defined by x 7! Ra(x) = xa for all x 2 G, respectively [3]. Definition 2.2. Let M be a topological manifold with an atlas S. Then, M is called a differentiable manifold if transformations of coordinates have all partial derivatives of all orders. M is called an analytic manifold if transformations of coordinates are real analytic functions at every point [3]. Biquaternions Lie Algebra and Complex-Projective Spaces 229 Definition 2.3. Let G be a group. G is called a Lie group if G is an analytic manifold and the mapping φ: G ×G ! G defined by φ(x; y) = xy is analytic, for all x; y 2 G [3]. Definition 2.4. An algebra V is called a Lie algebra if the operation of multiplication in V denoted by [u; v] for u; v 2 V satisfies (1)[ u; v] = −[v; u] (antisymetric); (2) u; [v; w] + v; [w; u] + w; [u; v] = 0 for all w 2 V (Jacobi identity); The operation [u; v] is called the bracket of vectors u and v. First condition implies that [u; u] = 0 [3]. Definition 2.5. Let G be a Lie group. A vector field X on G is called 0 0 left-invariant if Lg(X) = X for all g 2 G, where Lg is the differential of the mapping Lg [3]. Definition 2.6. Let e be the unit element of the Lie group G and ξ; η 2 Te(G). Denote by X and Y the left-invariant vector fields determined by ξ and η. Then we define [ξ; η] = [X; Y ]e. Thus the tangent space Te(G) at the unit element of the group becomes a Lie algebra (which is the same as XL(G), of course; we limit ourselves to the unit of the group for simplification only). This Lie algebra is called the Lie algebra of the group Te(G) and is denoted by G [3]. Definition 2.7. Let a Lie group G be given and let g be a fixed choosen element of the group G. For all h 2 G the mapping −1 Intg : G ! G defined by Intg(h) = ghg is a differentiable isomorphism of the group G and we have Intg(e) = −1 geg = e. This means that the differential of the mapping Intg at the 0 0 point e, i.e. (Intg)e, maps Te(G) into Te(G). By denoting (Intg)e with adg then adg : G ! G. The mapping ad: G ! Hom(G; G) is called the adjoint representation of the group G [3]. Definition 2.8. Let G be a Lie algebra and let us define the transfor- mation K : G × G ! R defined by K(X; Y ) = T r(AdX; AdY ) for all X; Y 2 G. Then, the form K(X; Y ) is called the Killing bilinear form on G and is a symmetric bilinear form where AdX : G ! G defined by Y 7! AdX(Y ) = [X; Y ] for all Y 2 G; and T r(AdX; AdY ) stands for the trace of the mapping AdX · AdY : G ! G defined by Z 7! AdX · AdY (Z) = X; [Y; Z]: 230 Murat Bekar, Yusuf Yayli 2.2. Basics of Real Quaternions. Real quaternion algebra H = fq = w + xi + yj + zk w; x; y; z 2 Rg is a four dimensional vector space over the field of real numbers R with a basis fi; j ; kg invented by Hamilton in 1843. Multiplication is defined by the hypercomplex operator rules i 2 = j 2 = k 2 = ij k = −1; ij = −ji = k; jk = −kj = i; ki = −ik = j : Also, H is an associative and non-commutative division ring. For any real quaternion q = w + xi + yj + zk, we define the scalar part and the vector part of q as S(q) = w and V(q) = xi + yj + zk, respectively, that is q = S(q) + V(q). The conjugate of q = S(q) + V(q) is defined as q = S(q) − V(q), also known as quaternion conjugate. If S(q) = 0, then q is called pure real quaternion. Pure real quaternions set Im(H) = fq = xi + yj + zk w; x; y; z 2 Rg is a linear subspace of H spanned by fi; j ; kg [4, 5]. 2.3. Basics of Bicomplex Numbers. The algebra of complex num- bers can be generalized to bicomplex numbers C2 = fZ = A + J B = (A; B): A; B 2 C)g where J is the complex imaginary operator (therefore J 2 = −1) distinct from the complex numbers operator I satisfying JI = IJ . For any Z = A + J B 2 C2, we define the real part and imaginary part of Z as R(Z) = A and T(Z) = B, respectively, that is Z = A + J B. The bicomplex conjugate of Z is defined as Z∗ = R(Z) − J T(Z). If R(Z) = 0, then Z is called pure bicomplex number. The set of pure bicomplex numbers ImC2 = fZ = J B : B 2 Cg is a linear subspace of C2. Let Z = A + J B and W = C + J D be any two bicomplex numbers. Then, we define the sum and multiplication of Z and W as follows: Z + W = (A + C) + J (B + D) = (A + C; B + D); ZW = (AC − BD) + J (AD + BC) = (AC − BD; AD + BC): In addition, we can write ZW in the matrix form as A −B C ZW = : BA D Thus, C2 is a commutative ring with the unit element 1 = (1; 0). Biquaternions Lie Algebra and Complex-Projective Spaces 231 For any bicomplex number Z = A + J B, where A = a0 + I a1 and 2 2 B = b0+I b1, the norm and the modulus of Z are NZ = jjZjj = A +B = 2 2 p p (a0 + b0) + J (2a0a1 + 2b0b1) and jZj = NZ = jjZjj, respectively.
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