Quasi-associativity and Cayley-Dickson algebras

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . It is shown that, if the imaginary part of a hypercomplex number in more than 2 dimensions is squared to be non-positive, the linearly independent imaginary units defining the hypercomplex number should anti-commute with each other. The simplest examples are Hamilton’s quaternions (4 dimensions) and octonions (8 dimensions). This leads to an inductive construction of Cayley–Dickson algebras A2n via “quasi-associativity” for positive integer n ≥ 3. The “quasiassociativity” guaranteeing anti-commutativity among the imaginary units for non-associative algebras is a mixture of associativity and non-associativity and can be formulated inductively. It allows us to compute Cayley–Dickson products to define A2n for n > 3 that are no longer alternative but flexible. In other words, general Cayley–Dickson algebras A2n for n ≥ 3 are defined by the “quasi-associativity” for the product of basis elements. In addition, a new 2D quaternion-valued matrix representation of quaternions, which is equivalent to the well known Pauli representation, is also proposed. The Study determinant of the quaternionic matrix, which obeys the composition law, is modified such that the composition law of quaternions is derived from the composition property of a modified Study (Study-like) determinant. It can easily be generalized to arbitrary Cayley–Dickson algebras A2n , although the associativity of the matrix product is lost for n ≥ 3. In addition, the associativity of the product of the matrix elements is to be replaced by the “quasi-associativity” for n ≥ 3. It turns out that the composition property of the Study-like determinant holds true only for quaternions and octonions in accord with the Hurwitz theorem. The same is true, of course, for real and complex numbers that are also represented by 2 × 2 matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .