Adv. appl. Clifford alg. 99 (9999), 1–14 DOI 10.1007/s00006-003-0000 Advances in c 2013 Birkhäuser Verlag Basel/Switzerland Applied Clifford Algebras Multiplicative representation of a hyperbolic non distributive algebra Manuel Fernández-Guasti and Felipe Zaldívar Abstract. The multiplicative or polar representation of hyperbolic sca- tor algebra in 1 + n dimensions is introduced. The transformations be- tween additive and multiplicative representations are presented. The addition and product operations are consistently defined in either rep- resentation using additive or multiplicative variables. The product is shown to produce a rotation and scaling for equal director components and solely a scaling in the orthogonal components. Mathematics Subject Classification (2010). MSC2010, 30G35, 20M14. Keywords. Polar representation, non-distributive algebras, hypercom- plex numbers. 1. Introduction The polar representation of complex numbers has been successfully extended to hyperbolic numbers [1] and also to complex quaternions [2]. The purpose of this communication is to present a polar representation of hyperbolic scators in 1 + n dimensions and the geometrical interpretation of the addition and product operations. In [3], we introduced a non distributive algebra given by elements of the form ϕ = (f0; f1, ··· , fj, ··· , fn), with f0, fj ∈ R. (1.1) The first component is called the scalar component, whereas subsequent com- ponents are named director components. Addition is defined component- wise. The hyperbolic product of two scators α = (a0; a1, ··· , an) and β = (b0; b1, ··· , bn) is defined by γ = αβ = (g0; g1, ··· , gn), where the scalar component is n Y akbk g0 = a0b0 (1 + ) (1.2) a0b0 k=1 2 M. Fernández-Guasti and F. Zaldívar AACA and the jth director component, for j = 1 to n, is n Y akbk aj bj gj = a0b0 (1 + )( + ). (1.3) a0b0 a0 b0 k6=j The condition for the definition of the product in 1 + n dimensions is the 1+n 1+n subset E ⊂ R where a0b0 6= 0 if ajbj 6= 0 for two or more director components. A slightly different elliptic product definition is also possible, leading to a different topology [4]. Every scator ϕ ∈ E1+n can be represented as n X ϕ = f0 + fjeˆj, (1.4) j=1 th where eˆj is the standard basis with 1 in the j position and zero everywhere else. We identify the scalar f0 with the scator f0 = (f0; 0, ··· , 0). These co- efficients will be referred to as additive variables. Hyperbolic scators form a group under the product operation provided that non invertible elements and zero divisors are excluded. However, the product is, in general, not distribu- tive over addition. The restricted space conditions 2 2 2 2 a0 > aj and b0 > bj for all j, (1.5) are sufficient conditions to satisfy group properties under the product oper- ation provided that a0, b0 6= 0. The restricted space subset is defined by the scators whose director components are smaller than the scalar component n 1+n n X 1+n o Esl = ϕ = f0 + fjeˆj ∈ E : |fj| < |f0| ∀j = 1, . , n . (1.6) j=1 Pn Pn Given scators α = a0 + j=1 ajeˆj and β = b0 + j=1 bjeˆj, addition becomes n X α + β = (a0 + b0) + (aj + bj)eˆj. (1.7) j=1 and their product is n n n ( aj + bj ) Y akbk X Y akbk a0 b0 αβ = a0b0 1 + + a0b0 1 + a b eˆj. (1.8) a0b0 a0b0 (1 + j j ) k=1 j=1 k=1 a0b0 Remark 1.1. From (1.8), if the two factors have only one non vanishing di- Pn rector component, αβ = a0b0 + ajblδjl + j=1 (b0aj + a0bj) eˆj. Whence, if the scalar components are then zero and aj = bl = 1, we obtain eˆjeˆl = δjl, a Kronecker’s delta. It follows that if α, β are two scators such that the non zero director components of one are different from the non zero director com- ponents of the other, then X X X X a0 + ajeˆj b0 + bleˆl = a0b0 + b0ajeˆj + a0bleˆl for all j 6= l. j l j l Vol. 99 (9999) Multiplicative representation of hyperbolic scators 3 Pn The conjugate of ϕ = f0 + j=1 fjeˆj is n ∗ X ϕ = f0 − fjeˆj, (1.9) j=1 and its norm or modulus squared is n 2 Y fj kϕk2 = ϕ · ϕ∗ = f 2 1 − . (1.10) 0 f 2 j=1 0 Pn A non-zero scator ϕ = f0 + j=1 fjeˆj is invertible provided that f0 6= ±fj for all j. The inverse of ϕ is 1 ϕ−1 = ϕ∗. (1.11) f 2 2 Qn j f0 j=1(1 − 2 ) f0 2. Multiplicative representation Given a scator ϕjeˆj with a single director component, by Remark 1.1, eˆjeˆj = 2m 2m+1 1 and thus its even and odd powers are (eˆj) = 1 and (eˆj) = eˆj for m ≥ 0 an integer. Heuristically, we may consider the exponential function of a scator ϕjeˆj with a single director component in the argument as a formal series of the form ∞ m X (ϕjeˆj) eϕj eˆj = . m! m=0 Again, heuristically, we may write ∞ 2m ∞ 2m+1 X ϕj X ϕj eϕj eˆj = + eˆ , 2m! (2m + 1)! j m=0 m=0 notice that on the right-hand side we have the series for the hyperbolic cosine and hyperbolic sine. The exponential function of a single director component scator in the argument is then defined by ∞ m X (ϕjeˆj) eϕj eˆj = = cosh(ϕ ) + sinh(ϕ )eˆ . (2.1) m! j j j m=0 Now, consider n + 1 single director component scators of the form ϕ0, ϕ1eˆ1, ..., ϕneˆn with ϕ0 > 0. In the multiplicative representation, the scator is given by the product n Y ϕ = ϕ0 exp(ϕjeˆj), ϕ0, ϕj ∈ R. (2.2) j=1 4 M. Fernández-Guasti and F. Zaldívar AACA The variables ϕj are called the multiplicative director components of the sca- tor ϕ and we shall see that ϕ0 is its norm. Sometimes we list the multiplicative components of a scator as an ordered sequence ϕ = (ϕ0; ϕ1, ··· , ϕn), with ϕj ∈ R. (2.3) Remark 2.1. Given two scators α = (α0; 0, ··· , αj, ··· 0), β = (β0; 0, ··· , βj, ··· 0) having a single non-vanishing multiplicative component in the same place j, their product is αβ = (α0 exp(αjeˆj))(β0 exp(βjeˆj)) = α0β0(cosh(αj) + sinh(αj)eˆj)(cosh(βj) + sinh(βj)eˆj) = α0β0(cosh(αj) cosh(βj) + sinh(αj) cosh(βj)eˆj + sinh(βj) cosh(αj)eˆj + sinh(αj) sinh(βj)eˆjeˆj) = α0β0(cosh(αj + βj) + sinh(αj + βj)eˆj) = α0β0 exp((αj + βj)eˆj); The third equality is because the product is distributive over addition for single-component scators, the fourth equality is due to eˆjeˆj = 1 and the hyperbolic function sum of angles identities, and the last equality follows from (2.1). Therefore, we have shown that the general addition theorem holds for equal j director components α0β0 exp((αj + βj)eˆj) = (α0 exp(αjeˆj))(β0 exp(βjeˆj)). (2.4) The product of two arbitrary scators in the multiplicative representation is then n n n Y Y Y αβ = α0 exp(αjeˆj)β0 exp(βjeˆj) = α0β0 exp((αj + βj)eˆj). (2.5) j=1 j=1 j=1 i.e., the scalar components are multiplied while the multiplicative director components are added component-wise. 1+n For a scator ϕ ∈ Esl there are two representations: 1. An additive representation of the form n X ϕ = f0 + fjeˆj. (2.6) j=1 2. A multiplicative representation of the form n Y ϕ = ϕ0 exp(ϕjeˆj). (2.7) j=1 Addition is particularly simple in the additive representation whereas the product is specially simple in the multiplicative representation. Vol. 99 (9999) Multiplicative representation of hyperbolic scators 5 The relationships between these two representations are as follows: Given a scator in multiplicative form (2.7), invoking (2.1) and performing the cor- responding products using the orthogonal scator’s distributivity stated in Remark (1.1), we obtain n n Y ϕj eˆj Y ϕ = ϕ0 e = ϕ0 (cosh(ϕj) + sinh(ϕj)eˆj) j=1 j=1 n n n Y X Y = ϕ0 cosh(ϕk) + ϕ0 cosh(ϕk) tanh(ϕj)eˆj. (2.8) k=1 j=1 k=1 This last expression is the additive representation but with multiplicative variables. Comparison with the additive representation (1.4), gives the fol- lowing multiplicative to additive transformations n Y f0 = ϕ0 cosh(ϕk), (2.9a) k=1 n Y fj = ϕ0 cosh(ϕk) tanh(ϕj). (2.9b) k=1 Conversely, given a non zero scator in additive form (1.4), its (additive) com- ponents are given by (2.9a) and (2.9b), from where we obtain the multiplica- tive director components −1 fj ϕj = tanh ( ). (2.10a) f0 The multiplicative scalar component is obtained from (2.9a) and the above q 1 2 expression, recalling that = 1 − tanh ϕj, hence cosh ϕj n s Y f 2 ϕ = f 1 − k . (2.10b) 0 0 f 2 k=1 0 The multiplicative scalar ϕ0 is the scator magnitude or norm as can be seen by comparing equation (2.10b) with equation with (1.10). Since the multi- plicative scalar components are multiplied in the product operation (2.5), the norm of a scator product is equal to the product of the factor’s norms. A complexified version of this identity has been used to derive Lagrange’s identity for complex numbers and some other higher order identities [5]. The multiplicative representation with additive variables is then s n n 2 Y Y fj fj ϕ = ϕ eϕj eˆj = f 1 − exp tanh−1( ) . (2.11) 0 0 f 2 f j=1 j=1 0 0 6 M.