On the Octonion-like Associative Division Algebra Juhi Khalid, Martin Bouchard School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward, Ottawa K1N6N5 Canada ABSTRACT Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over . The octonion-like algebra is the even subalgebra of Clifford algebra Cl40, () , which is isomorphic to Cl03, () and to the split-biquaternion algebra. For two seminorms described in the paper (which differ from the norm used in the original papers on the octonion-like algebra), it is shown that the octonion-like algebra is a seminormed algebra over with no zero divisors when using one of the two seminorms. Moreover, additional results related to the computation of inverse numbers in the octonion-like algebra are introduced in the paper, confirming that the octonion-like algebra is a division algebra over as long as the two seminorms are non-zero. Additional results on normalization of octonion-like numbers and some involutions are also presented. The elementary linear algebra descriptions used in the paper also allow straightforward software implementations of the octonion-like algebra. KEYWORDS: associative octonion-like algebra, octonion-like seminormed algebra, octonion-like division algebra, hypercomplex algebra. I. INTRODUCTION In [1], [2], an eight-dimensional octonion-like associative division algebra over was recently introduced and presented as a normed algebra. There has been some controversy with this octonion-like algebra, as it was noted that a new normed division algebra over would contradict Hurwitz’s theorem [3]. It was pointed out that the octonion-like algebra, which is the even subalgebra of Clifford algebra Cl40, () , is isomorphic to Cl03, () [4] which, 1 in turn, is isomorphic to the split-biquaternion algebra and to the sum of two quaternion algebras [5]. The octonion-like algebra presented in [1], [2] was also found to be the same as the 1d-up approach to conformal geometric algebra presented in [6]. Using elementary linear algebra and two scalar seminorms defined in this paper, we explain that the associative octonion-like algebra over is a seminormed algebra and it is also a division algebra as long as the two seminorms are non-zero. Therefore, as an associative seminormed division algebra over , the octonion-like algebra does not contradict Hurwitz’s theorem. It should be noted that the two scalar seminorms used in this paper differ from the norm used in the original papers on the octonion-like algebra [1], [2], where a multi-dimensional non-scalar norm was first defined and subsequently constrained to become a scalar. The associativity property found in the octonion-like algebra (like all Clifford algebras but unlike octonions which are non-associative) can be an important criterion for the practical applicability of an algebra. A proof for the preservation of the two seminorms under multiplication of octonion-like numbers (ZXY , ZXY ) is provided in the Appendix of this paper, using elementary linear algebra. Because of the preservation of the seminorms under multiplication, the octonion-like algebra contains no zero-divisors using the two seminorms. It will be shown in this paper that the computation of an inverse octonion-like number involves a matrix inversion, and that the case of a singular matrix only occurs when one of the two seminorms of an octonion-like number is zero, therefore the octonion-like algebra is also a division algebra as long as the two seminorms are non-zero. Additional results on normalization of octonion-like numbers and some involutions are also presented in the paper. The explanation for the seminormed nature of the octonion-like algebra (and lack of contradiction with Hurwitz’s theorem), the additional results related to the computation of inverse numbers in the octonion-like algebra (with a confirmation of its division algebra status using the seminorms), the additional results on normalization of octonion-like numbers and some involutions, and the developments and proof using simple linear algebra (as opposed to more abstract geometric algebra) are the contributions of this paper. The linear algebra descriptions also allow straightforward software implementations of the octonion-like algebra. 2 II. OCTONION-LIKE ALGEBRA AND MULTIPLICATION RULES Table I presents the multiplication rules for the octonion-like algebra. From the diagonal of the table, it can be seen that in an octonion-like number: X X(0) XeXeXeXeXeXeXe (1)1234567 (2) (3) (4) (5) (6) (7) (1) the imaginary dimensions correspond to coefficients X(1) to X(6) , while coefficients X(0) and X(7) correspond to “scalar” or non-imaginary dimensions. We define X* as the conjugate (or reverse) of X , changing the sign of the coefficients for the imaginary dimensions in X , i.e.,X(1) to X(6) : * X XXeXeXeXeXeXeXe(0) (1)1234567 (2) (3) (4) (5) (6) (7) (2). Table I Table of multiplication rules in the octonion-like algebra 1st factor (row) and 2nd factor (column) of octonion-like e0 1 e1 e2 e3 e4 e5 e6 e7 number multiplication e0 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 Following these multiplication rules, the octonion-like multiplication can be detailed as: 3 ZXYXXeXeXeXeXeXeXe(0) (1)1234567 (2) (3) (4) (5) (6) (7) Y(0) YeYeYeYeYeYeYe (1)1234567 (2) (3) (4) (5) (6) (7) XY(0)(0) XY (1)(1) XY (2)(2) XY (3)(3) XY (4)(4) XY (5)(5) XY (6)(6) XY (7)(7) XY(1)(0) XY (0)(1)(3)(2)(2)(3)(5)(4)(4)(5)(7)(6)(6)(7)XY XY XY XY XY XY e1 XY(2)(0) XY (3)(1) XY (0)(2) XY (1)(3) XY (6)(4) XY (7)(5) XY (4)(6) XY (5)(7) e2 XY(3)(0) XY (2)(1) XY (1)(2) XY (0)(3) XY (7)(4) XY (6)(5) XY (5)(6) XY (4)(7)e3 (3). XY(4)(0) XY (5)(1) XY (6)(2) XY (7)(3) XY (0)(4) XY (1)(5) XY (2)(6) XY (3)(7) e4 XY(5)(0) XY (4)(1) XY (7)(2) XY (6)(3) XY (1)(4) XY (0)(5) XY (3)(6) XY (2)(7) e5 XY(6)(0) XY (7)(1) XY (4)(2) XY (5)(3) XY (2)(4) X(3)(5)YXYXYe (0)(6) (1)(7) 6 XY(7)(0) XY (6)(1) XY (5)(2) XY (4)(3) XY (3)(4) XY (2)(5) XY (1)(6) XY (0)(7) e7 ZZeZeZeZeZeZeZe(0) (1)1234567 (2) (3) (4) (5) (6) (7) From (3), we see that for dimensions eeeeee123456,,,,, substituting Xi() by Yi() and Yi() by Xi() 07i does not lead to the same value, therefore in general the octonion-like number product ZXYYX is non-commutative. Moreover, we can verify that: (XY)*** Y X (4). Also from (3), we can build a matrix product involving only real-valued numbers, where the matrix MX includes coefficients from the left-side octonion-like number X in the product ZXY : ZMYrr X (5), with T Z ZZZZZZZZ(0)(1)(2)(3)(4)(5)(6)(7) (6) r XXXXXXXX(0) (1) (2) (3) (4) (5) (6) (7) XXXXXXXX(1)(0)(3)(2)(5)(4)(7)(6) XXXXXXXX(2)(3)(0)(1)(6)(7)(4)(5) XXXXXXXX(3) (2) (1) (0) (7) (6) (5) (4) M (7) X XXXXXXXX(4) (5) (6) (7) (0) (1) (2) (3) XXXXXXXX(5) (4) (7) (6) (1) (0) (3) (2) XXXXXXXX(6) (7) (4) (5) (2) (3) (0) (1) XXXXXXXX(7)(6)(5)(4)(3)(2)(1)(0) 4 T Y YYYYYYYY(0)(1)(2)(3)(4)(5)(6)(7) (8). r The subscript r in ZYrr, stands for “real-valued elements only”. Alternatively, re- organizing the elements of the octonion-like product ZXY as: ZXY YX(0)(0) YX (1)(1) YX (2)(2) YX (3)(3) YX (4)(4) YX (5)(5) YX (6)(6) YX (7)(7) YX(1)(0) YX (0)(1) YX (3)(2) YX (2)(3) YX (5)(4) YX (4)(5) YX (7)(6) YX (6)(7) e1 YX(2)(0) YX (3)(1) YX (0)(2) YX (1)(3) YX (6)(4) Y (7)XYXYXe(5) (4) (6) (5) (7) 2 (9) YX(3)(0) YX (2)(1) YX (1)(2) YX (0)(3) YX (7)(4) YX (6)(5) YX (5)(6) YX (4)(7) e3 YX(4)(0) YX (5)(1) YX (6)(2) YX (7)(3) YX (0)(4) YX (1)(5) YX (2)(6) YX (3)(7) e4 YX(5)(0) YX (4)(1) YX (7)(2)YX(6)(3) YX (1)(4) YX (0)(5) YX (3)(6) YX (2)(7) e5 YX(6)(0) YX (7)(1) YX (4)(2) YX (5)(3) YX (2)(4) YX (3)(5) YX (0)(6) YX (1)(7) e6 YX(7)(0) YX (6)(1) YX (5)(2) YX (4)(3) YX (3)(4) YX (2)(5) YX (1)(6) YX (0)(7) e7 we can build a 2nd matrix form involving only real-valued numbers, where this time the matrix PY includes coefficients from the right-side octonion-like number Y in the product ZXY : ZPXrr Y (10), with: YYYYYYYY(0) (1) (2) (3) (4) (5) (6) (7) YYYYYYYY(1)(0)(3)(2)(5)(4)(7)(6) YYYYYYYY(2)(3)(0)(1)(6)(7)(4)(5) YYYYYYYY(3)(2)(1)(0)(7)(6)(5)(4) P (11) Y YYYYYYYY(4)(5)(6)(7)(0)(1)(2)(3) YYYYYYYY(5) (4) (7) (6) (1) (0) (3) (2) YYYYYYYY(6) (7) (4) (5) (2) (3) (0) (1) YYYYYYYY(7)(6)(5)(4)(3)(2)(1)(0) T X XXXXXXXX(0)(1)(2)(3)(4)(5)(6)(7) (12).