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Cauchy-Riemann Operators in Octonionic Analysis 7 1 Janne Kauhanen and Heikki Orelma ∗ † 0 2 January 31, 2017 n a J 0 3 Abstract ] V In this paper we first recall the definition of an octonion algebra C and its algebraic properties. We derive the so called e4-calculus and . using it we obtain the list of generalized Cauchy-Riemann systems in h t octonionic monogenic functions. We define some bilinear forms and a m derive the corresponding symmetry groups. [ Mathematics Subject Classification (2010). 30G35, 15A63 1 v 8 Keywords. Octonions, Cauchy-Riemann operators, Monogenic functions 9 6 8 1 Introduction 0 . 1 0 The algebra of octonions is a well known non-associative division algebra. 7 The second not so well known feature is, that we may define a function the- 1 : ory, in spirit of classical theory of complex holomorphic functions, and study v i its properties. This theory has its limitations, since the multiplication is X neither commutative nor associative. The first part of this paper is a sur- r a vey of known results, where we give a detailed definition for the octonions. We derive the so called ”e4-calculus” on it, to make our practical calcula- tions easier. Then we recall the notion of the Cauchy-Riemann operator. A function in its kernel is called monogenic. To find explicit monogenic functions directly from the definitions is too complicated, because the algebraic properties give too many limitations. To give an explicit characterization of ∗LaboratoryofMathematics,FacultyofNaturalSciences,TampereUniversityofTech- nology, Finland. Electronic address: [email protected] †Electronic address: [email protected] 1 monogenic functions, we separate variables, or represent the target space as a direct sum of subalgebras. Using this trick we obtaina list of real, complex, and quaternionic partial differential equation systems, which are all general- izations of the complex Cauchy-Riemann system. These systems allow us to study explicit monogenic functions. We compute an example, assuming that the functions are biaxially symmetric. Authors like to emphasize, that this work is the starting point for our future works onthis fascinating field of mathematics. A reader should notice, thatalthoughthealgebraiccalculationruleslookreallycomplicated, onemay still derive a practical formulas to analyze the properties of the quantities of the theory. It seems that there are two possible ways to study the octonionic analysis in our sense. In the first one, one just takes results from classical complex or quaternionic analysis and tries to prove them. The second one is to concentrate to algebraic properties and features of the theory, and try to find something totally new, in the framework of the algebra. We believe that the latter gives us deeper intuition of the theory, albeit the steps forward are not always so big. 2 On Octonion Algebra Inthissectionwerecallthedefinitionfortheoctonionsandstudyitsalgebraic properties. We develop the so called e4-calculus, which we will use during the rest of the paper to simplify practical computations. Also classical groups related to the octonions are studied. 2.1 Definition of Octonions Let us denote the field of complex numbers by C and the skew field of quaternions by H. We assume that the complex numbers are generated by the basis elements 1,i and the quaternions by 1,i,j,k with the well known defin- { } { } ing relations 2 2 2 i = j = k = ijk = 1. − We expect that the reader is familiar with the complex numbers and the quaternions. We give [2, 7, 10] as a basic reference. The so called octonions or Cayley numbers were first defined defined in 1843 by John T. Graves. Nowadays the systematic way to define octonions is the so called Cayley- Dickson construction, which we will use also in this paper. See historical remarks on ways to define the octonions in [1]. TheCayley-Dickson constructionproducesasequenceofalgebrasoverthe field of real numbers, each with twice the dimension of the previous one. The 2 previous algebra of a Cayley-Dickson step is assumed to be an algebra with a conjugation. Starting from the algebra of real numbers R with the trivial conjugation x x, the Cayley-Dickson construction produces the algebra of 7→ complex numbers C with the conjugation x + iy x iy. Then applying 7→ − Cayley-Dickson construction to the complex numbers produces quaternions H with the conjugation. The quaternion conjugation is given as follows. An arbitrary x H is of the form ∈ x = x0 +x where x0 R is the real part and x = x1i + x2j + x3k is the vector part ∈ of the quaternion x. Vector parts are isomorphic to the three dimensional Euclidean vector space R3. Then the conjugation of x obtained from the Cayley-Dickson construction is denoted by x and defined by x = x0 x. − Now the Cayley-Dickson construction proceeds as follows. Consider pairs of quaternions, i.e., the space H H. We define the multiplication for the pairs ⊕ as (a,b)(c,d) = (ac db,da+bc) − where a,b,c,d H. With this multiplication the pairs of quaternions H H ∈ ⊕ is an eight dimensional algebra generated by the elements e0 := (1,0), e1 := (i,0), e2 := (j,0), e3 := (k,0), e4 := (0,1), e5 := (0,i), e6 := (0,j), e7 := (0,k). Denoting 1 := e0 and using the definition of the product, we may write the following table. 1 e1 e2 e3 e4 e5 e6 e7 1 1 e1 e2 e3 e4 e5 e6 e7 e1 e1 1 e3 e2 e5 e4 e7 e6 − − − − e2 e2 e3 1 e1 e6 e7 e4 e5 − − − − e3 e3 e2 e1 1 e7 e6 e5 e4 − − − − e4 e4 e5 e6 e7 1 e1 e2 e3 − − − − e5 e5 e4 e7 e6 e1 1 e3 e2 − − − − e6 e6 e7 e4 e5 e2 e3 1 e1 − − − − e7 e7 e6 e5 e4 e3 e2 e1 1 − − − − We see that e0 = 1 is the unit element of the algebra. Using the table, it is an easy task to see that the algebra is not associative nor commutative. Also 3 we see that elements 1,e1,e2,e3 generates a quaternion algebra, i.e., H is { } a subalgebra. We will consider more subalgebras later. The preceding algebra is called the algebra of octonions and it is denoted by O. An arbitrary x O may be represented in the form ∈ x = x0 +x where x0 R is the real part of the octonion x and ∈ x = x1e1 +x2e2 +x3e3 +x4e4 +x5e5 +x6e6 +x7e7, where x1,...,x7 R is the vector part. Vector parts are isomorphic to the seven dimension∈al Euclidean vector space R7. The whole algebra of octonions is naturally identified as a vector space with R8. The Cayley-Dickson construction produces also naturally in O a conjugation (a,b)∗ := (a, b), − where a is the quaternion conjugation. Because there is no risk of confusion, we will denote the conjugation of x O by x. Using the definition, we have ∈ x = x0 x. − We refer [1, 2, 4] for more detailed description to the preceding construction. 2.2 Algebraic Properties In this subsection we collect some algebraic properties and results of the octonions to better understand its algebraic structure. Proposition 2.1 (O is an alternative division algebra, [10]). If x,y O ∈ then 2 2 x(xy) = x y, (xy)y = xy , (xy)x = x(yx), and each non-zero x O has an inverse. ∈ We see that the associativity holds in the case (xy)x = x(yx). Unfortu- nately, this is almost the only non-trivial case when the associativity holds: Proposition 2.2 ([2]). If x(ry) = (xr)y for all x,y O, then r is real. ∈ So we see, that use of parentheses is something what we need to keep in mind, when we compute using the octonions. The alternative properties given in Proposition 2.1 implies the following identities. 4 Proposition 2.3 (Moufang Laws, [2, 9]). For each x,y,z O ∈ (xy)(zx) = (x(yz))x = x((yz)x). The inverse element x−1 of non-zero x O may be computed as follows. ∈ We define the norm by x = √xx = √xx. A straightforward computation | | shows that the norm is well defined and 7 2 2 x = x . | | j j=0 X In addition, x x−1 = . x 2 | | An important property of the norm is the following. Proposition 2.4 (O is a composition algebra, [2, 4, 10]). The norm of O satisfies the composition law xy = x y | | | || | for all x,y O. ∈ We will say that octonions has a multiplicative norm. The composition law has algebraic implications for conjugation, since the conjugation may be written using the norm in the form x = x+1 2 x 2 1 x. | | −| | − − Proposition 2.5 ([2]). If x,y O, then ∈ x = x and xy = yx. These formulas are easy to prove by brute force computations. But the reader should notice, that actually they are consequences of the composition laws, not directly related only tooctonions. In generalwe say thatanalgebra A is a composition algebra, if it has a norm N: A R such that N(ab) = → N(a)N(b) for all a,b A. We know that R, C, H and O are composition ∈ algebras. It is an interesting algebraic task to prove that actually this list is complete. Theorem 2.6 (Hurwitz, [2]). R, C, H and O are the only composition algebras. 5 2.3 e –Calculus 4 In this subsection we study how to compute with the octonions in practise. In principle all of the computations are possible to carry out using the multiplication table. In practise, this often leads to chaos of indices, so it is better to develop another kind of calculation. Our starting point is the observation that every octonion x O may be written in the form ∈ x = a+be4 where a,b H. This form is called the quaternionic form of an octonion. If ∈ x = x0 +x1e1 + +x7e7, ··· then a = x0 +x1e1 +x2e2 +x3e3 and b = x4 +x5e1 +x6e2 +x7e3. Using the multiplication table, it is easy to prove the following. Lemma 2.7. Let i,j 1,2,3 . Then ∈ { } (a) ei(eje4) = (ejei)e4, (b) (eie4)ej = (eiej)e4, − (c) (eie4)(eje4) = ejei. Using these, we have Lemma 2.8. If a = a1e1 +a2e2 +a3e3 and b = b1e1 +b2e2 +b3e3, then (a) e4a = ae4 − (b) e4(ae4) = a (c) (ae4)e4 = a − (d) a(be4) = (ba)e4 (e) (ae4)b = (ab)e4 − (f) (ae4)(be4) = ba Using preceding formulae, it is easy to obtain similar formulas for quaternions. Lemma 2.9. Let a,b H. Then ∈ 6 (a) e4a = ae4 (b) e4(ae4) = a − (c) (ae4)e4 = a − (d) a(be4) = (ba)e4 (e) (ae4)b = (ab)e4 (f) (ae4)(be4) = ba − The preceding list is called the rules of e4-calculus for the octonions. When we compute using octonions, we drop our computations to quaternionic level and use the preceding formulas and associativity. The situation is similar to computing with complex numbers, where we usually compute by real numbers with the relation i2 = 1. − Lemma 2.10. Let x = a1 + b1e4 and y = a2 + b2e4 be octonions in the quaternionic form. Then their product in quaternionic form is xy = (a1a2 b2b1)+(b1a2 +b2a1)e4. − Proof. Apply Lemma 2.9: xy = (a1 +b1e4)(a2 +b2e4) = a1a2 +(b1e4)a2 +a1(b2e4)+(b1e4)(b2e4) = a1a2 +(b1a2)e4 +(b2a1)e4 b2b1. − Lemma 2.11. For an octonion a+be4 in the quaternionic form we have a+be4 = a be4, − 2 2 2 a+be4 = a + b . | | | | | | 2.4 Bilinear Forms and Their Invariance Groups In this section we consider some octonion valued bilinear forms and study their invariance groups. Some of the results are not new, but not well known. For the convenience of the reader we give the proofs here. 7 2.4.1 On Involutions and Subalgebras Our aim here is to study how involutions and subalgebras of octonions are related to each other. We begin with the easiest case. Therealnumbers Rmaybeidentified withtherealpartsoftheoctonions. Since the conjugation of an octonion x = x0 + x is defined by x = x0 x, − the real part Re(x) = x0 of the octonion x may be computed as 1 Re(x) = (x+x). 2 2 Because Re = Re, it is a projection of the octonion algebra onto the real numbers. We make the following conclusion. Theorem 2.12. The real numbers R is a subalgebra of the octonions O generated by the identity element 1 . The mapping { } Re: O R → is a projection. The complex numbers is a subalgebra of the octonions, and may be generated by any pair 1,e where j = 1,...,7. To find a canonical one, we j { } define an involution x∗ := a+be4 where x = a+be4 O is in quaternionic form. Then we define the complex ∈ part of an octonion by 1 Co(x) := (x+x∗) = Re(a)+Re(b)e4. 2 2 Since Co = Co, it is a projection of the octonion algebra onto the complex numbers generated by 1,e4 . We make the following conclusion. { } Theorem 2.13. The complex numbers C is a subalgebra of the octonions O generated by the elements 1,e4 . The mapping { } Co: O C → is a projection. The quaternions H is a subalgebra of the octonions O, generated by any 1,ei,ej,eiej , where i,j 1,...,7 , i = j. If x = a+ be4 O we define { } ∈ { } 6 ∈ an involution x := a be4. − b 8 Using hat, we define the quaternion part of an octonion x as 1 Qu(x) := (x+x), 2 2 that is, Qu(a+be4) = a. Since Qu = Qu webmake the following conclusion. Theorem 2.14. The quaternions H is a subalgebra of the octonions O generated by the elements 1,e1,e2,e3 . The mapping { } Qu: O H → is a projection. Next we study algebraic properties of the preceding involutions. Proposition 2.15. If x,y O, then ∈ (a) x∗∗ = x, (b) x = x, (c) xy = xy. b b Proof. Parts (a) and (b) are obvious. We prove (c). Let x = a1 +b1e4 and c bb y = a2 +b2e4. Then using the rules of e4-calculus and Lemma 2.10, we have xy = (a1 b1e4)(a2 b2e4) − − = a1a2 (b1e4)a2 a1(b2e4)+(b1e4)(b2e4) − − bb = a1a2 b2b1 (b1a2 +b2a1)e4 = xy. − − 2.4.2 Linear Mappings and their Invariance Gcroups A real linear mapping T: O O from the octonions into itself is acting on → 7 x = x e j j j=0 X as 7 Tx = T x e ij j i i,j=0 X where T R. We define the matrix representation of T by [T] := [T ] ij ij R8×8. It is∈easy to see that [TS] = [T][S] and [T−1] = [T]−1. Using matri∈x 9 representation we may define the determinant of T as det(T) := det([T]). The set of invertible linear mappings on O is denoted by GL(O). We will consider real bilinear functions B: O O O. × → For each function, we may associate a symmetry group (B) := T GL(O): B(Tx,Ty) = B(x,y) x,y O . G { ∈ ∀ ∈ } Let us next study bilinear functions generated by preceding projection mappings and the product xy. 2.4.3 SO(8) We define a bilinear form BR: O O R by × → BR(x,y) = Re(xy). One may compute 7 BR(x,y) = xjyj, j=0 X and it is well known (see e.g. [4, 10]), that this form is invariant under orthogonal transformations, i.e., (BR) = SO(8). G 2.4.4 U(4) This part is a modification of a similar result in [3], but a different definition of the octonions was used and none of the proofs were given. We define a bilinear form BC: O O C by × → BC(x,y) = Co(xy). If z = u + ve4 C, we denote Re(z) = u and Im(z) = v. Then BC = ∈ Re(BC)+Im(BC)e4 and (BC) = (Re(BC)) (Im(BC)). G G ∩G Lemma 2.16. If BC is the preceding bilinear function, then (a) (Re(BC)) = SO(8), G 10