Quaternionioffrono

Quaternionioffrono is a hypothetical one-parameter deformation of the quaternion algebra over the real numbers. It is defined on the same 4-dimensional real vector space as the standard quaternions but uses a modified multiplication that depends on a real parameter t. In its simplest form, elements are written as a + b i + c j + d k, and the product is the ordinary quaternion product perturbed by a bilinear term that scales with t. When t = 0, one recovers the ordinary quaternions.

The modification is guided by a fixed bilinear form Φ on the quaternion space. Different choices of Φ yield different deformation rules, and in general the resulting algebra H_t is non-associative for nonzero t. The construction is studied in the context of deformation theory as a way to illustrate how small changes to the product can alter algebraic properties while preserving the underlying vector space.

Conjugation and norm in quaternionioffrono are typically extended from the real-quaternion conjugation, but their multiplicative compatibility may fail for t ≠ 0. Representations of the algebra are built by left or right multiplication operators, with the usual 4-dimensional real representation restricted when t ≠ 0 due to loss of associativity in some cases.

In usage, quaternionioffrono appears mainly in speculative or pedagogical discussions of non-associative deformations of classical algebras. It is not part of standard mathematics and has no widely accepted canonical form; the term is encountered in thought experiments or fictional expositions designed to illustrate deformation concepts.

See also: Quaternions; Deformation theory; Non-associative algebras.