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Division Algebra English The matrices representation of division algebras, new properties, the Cartan-Weyl basis of SO(n) and the 2 n -dimensional harmonic oscillator basis Mehdi Hage-Hassan To cite this version: Mehdi Hage-Hassan. The matrices representation of division algebras, new properties, the Cartan- Weyl basis of SO(n) and the 2 n -dimensional harmonic oscillator basis. 2015. hal-01222005 HAL Id: hal-01222005 https://hal.archives-ouvertes.fr/hal-01222005 Preprint submitted on 8 Nov 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The matrices representation of division algebras, new properties, the Cartan-Weyl basis of SO(n) and the 2n-dimensional harmonic oscillator basis M. Hage-Hassan* Université Libanaise, Faculté des Sciences Section (1) Hadath-Beyrouth Abstract We derive by a simple recurrence method the matrices representation of division algebras or Hurwitz matrices (Hn, n = 2,4,8,16). We introduce a generalization of the polar angles and coordinates and we show the connection between Hn and the polypolar matrices transformations. We find new properties of Hn matrices. We also find that the matrices Hn are generating functions of Cartan-Weyl basis of SO (n). We define the poly- cylindrical coordinates and we find the 2n-dimensional harmonic oscillator basis for n≥1. 1. Introduction The real division algebras are: real numbers, complex numbers, quaternion, and Octonion. These algebras and the orthogonal groups SO(n) are very important in mathematics and physics [1-10]. This real division algebra is originated of the famous problem solving "the sums of squares" [10-11]. So we try to determine the values of n so that there is a solution of the identity: 2 2 2 2 2 2 2 2 2 x1 x2 ... xn x1 x2 ... xn z1 z2 ... zn (1.1) With zi are bilinear forms xi, yi, i, j = 1... n. This problem was solved by Euler and Fermat for n = 2, by Hamilton for n = 4 and by Grave and Cayley for n = 8. Hurwitz shown that the equation (1.1) has solutions only for n = 1, 2, 4, and 8. And every solution corresponds to an algebra which is isomorphic to anti-symmetric matrices Hn (u1... un), representing the division algebras [11-12]. In this work we determine Hn (u) by a simple recurrence method. Then, by developing ii ii z j re , z j re ,( j 1k),k 1,2,4. and by introducing a new parameterization of{ j } polypolar angles, we find that the expansion of the polypolar coordinates are expressed in terms of matrices Mn with Mn = Hn (1... 1). By calculating the generating function of the cylindrical oscillator basis [13-14], we have observed a property of H2 (u) and after a simple calculation we found that this new property may be generalized to Hn (u) n = 4, 8, 16. The matrices representing the infinitesimal generators [15-20] of SO (n) are the anti- symmetric matrices which require the study of the connection between these matrices 1 and Hn (u). We find that Hn are generating matrices of Cartan-Weyl basis [15] of SO(2) ... SO(6) . The theory of N-dimensional harmonic oscillator is very useful in the study of Molecular vibration [9] and quantum physics [21-23]. It is natural to extend the polypolar coordinates, for that we generalize the cylindrical basis of the harmonic oscillator [18-20] by introducing a new parameterization of cylindrical coordinates [14, 23]. And thus we determine the eigenstates of the harmonic oscillator in N-dimensional, N= 2n with n> 1. Finally we give a simple application of division algebras. In this paper we expose the derivation of the representation matrices of division algebras in section two. In section three we introduce the polypolars coordinates and its relation with the representation matrix of division algebra. The new properties of Hurwitz matrices or the representation matrices of division algebras are given in section four. In section five we give the representation matrix of division algebras and the Cartan-Weyl basis of . The generalized cylindrical coordinates and the N- dimensional radial function of the oscillators will be given in section six. In section seven we expose the derivation the wave function of the N-dimensional harmonic oscillator. In the appendix we give the polycylindric coordinates for n = 4 and the new simple application of division algebras. 2. The matrices representation of division algebras We determine the matrices representation of division algebras [11], using a recurrence method and taking as a starting point trigonometry formulas. The matrix representations of the real is H1 (u) u1I . 2.1 The matrix representation of complex algebra We begin with the well-known trigonometric formulas: 2 2 cos cos (/ 2) sin (/ 2) sin 2cos(/ 2)sin(/ 2) (2.1) Multiplying these expressions by r: x u 2 u 2 , x 2u u (2.2) 1 1 2 2 1 2 With u1 r cos( / 2), u2 r sin( / 2) 2 2 2 2 2 2 2 2 And x1 x2 (u1 u2 ) or r (u ) (2.3) We write (2.1) in matrix form: x u u u 1 1 2 2 t (H 2 (u)) (U 2 ) (2.4) x2 u2 u1 u1 H 2 (u1,u2 ) is an orthogonal matrix and is anti-symmetric with: 2 t 2 2 H 2 (u) u1I Ju 2 , and J 1, H 2 (u)H 2 (u) (u1 u2 )I (2.5) 2 t And H 2 (u) is the transpose of H 2 (u) . 2.2 The matrix representation of Quaternion The generalization of the transformation (2.2) and (2.3) is obtained by setting: x1 u1 u2 u 3 2 (2.7) x 2 u2 u1 u 4 2 2 2 2 2 2 2 We add x3 (u1 u2 ) (u3 u4 ) for the relationship r (u ) . We find the Octonion quadratic transformation, [7-8], R4 R3 x u u u u u 1 3 1 2 3 4 x2 u4 u2 u1 u4 u3 t H (u) , H (u) (2.8) x 4 u 4 u u u u 3 1 3 4 1 2 0 u2 u4 u3 u2 u1 In quaternion’s notations we write: e1 I,e2 J, e3 K, e4 L . 2 2 2 Q= H 4 (u) u1e1 u2e2 u3e3 u4e4 , With J K L 1 (2.9) 2.3 The matrix representation of Octonion We obtain the orthogonal matrix H8(u) by generalization of the transformation (2.6). We find the Octonion quadratic transformation R8 R5 : x u 1 8 x2 u7 t H (u) , x 4 u 3 6 x u 4 5 We add x (u 2 u 2 u 2 u 2 ) (u 2 u 2 u 2 u 2 ) so that r 2 (u 2 )2 , we find: 5 1 2 3 4 5 6 7 8 u u u u u u u u 1 2 3 4 5 6 7 8 u2 u1 u4 u3 u6 u5 u8 u7 u u u u u u u u 3 4 1 2 7 8 5 6 u u u u u u u u H (u) 4 3 2 1 8 7 6 5 (2.10) 8 u u u u u u u u 5 6 7 8 1 2 3 4 u6 u5 u8 u7 u2 u1 u4 u3 u7 u8 u5 u6 u3 u4 u1 u2 u u u u u u u u 8 7 6 5 4 3 2 1 3 8 with H (u) u I u e , e e 1, 8 1 i2 i i i i 8 8 2 And t H (u) u I u e , t H (u) H (u) u (2.11) 8 1 i2 i i 8 8 i1 i a- From the orthogonality Hn, n = 1, 2, 4 it is simple to prove the identity (1.1) b- We also derive the non-orthogonal matrix H16 by the same method of recurrence. 2.4 The Hurwitz theorem: There are only anti-symmetric matrices and orthogonal Hn only if: n=1, 2, 4,8 or “R, C, H=Q and O” (2.12) 3. The polypolar coordinates and the matrix representation of division algebra We will define first the polypolar coordinates and then we determine the connection between these coordinates and the Hurwitz matrices of division algebras. 3.1 The polypolar coordinates i j We consider the n-couples of complex numbers z j r e , z j , ( j 1k),k 1,2,4 and {i } are functions of angular variables ,, ,...etc. For n=2, 1 , 2 1 ( ) , 2 ( ) , For n=3 3 ( ) , 4 ( ) We will show that the developments z j , z j , ( j 1k),k 1,2,4 are expressed simply with the help of Hn (1). 3.2 The polar coordinates and Hurwitz matrix of the complex variable We consider the polar transformation i z1 r e r(cos sin) x1 iy1 (3.1) i z1 r e r(cos sin) x1 iy1 This polar transformation is written in the matrix form by: z x 1 1 1 1 M 2 , With M 2 (1,i) H 2 (1) (3.2) z1 iy1 1 1 3.3 The polar coordinates and Hurwitz matrix of the quaternion We consider the case of two complex variables z1 and z2 with: i( ) z1 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 , i( ) z2 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 , 4 i( ) z1 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 , i( ) z2 re r(cos isin)(cos isin ) x1 iy1 ix 2 y2 (3.3) We write the polar transformation by: z x 1 1 1 1 1 1 z2 iy1 1 1 1 1 M , M (1,i,i,1) H (1) (3.4) z 4 ix 4 4 1 1 1 1 1 2 z2 y2 1 1 1 1 3.4 The polar coordinates of the Hurwitz matrix of Octonion and Sedenion 3.4.1 The polar coordinates of the Hurwitz matrix of the Octonion We consider the case of four complex variables (z1,..., z4 , z1,..., z4 ) whose angular variables are functions of three variables, , with: 1 ( ) , 2 ( ) , 3 ( ) , 4 ( ) (3.5) The transformation in this case becomes: i1 z1 re r(cos i sin)(cos isin )(cos isin ) x1 iy1 ..
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