Scaled-Hyperbolic Clifford Algebras

In this paper, starting from recently known scaled hypercomplexes \({\mathbb {H}}_{t}\), we define scaled hyperbolics \({\mathbb {D}}_{t}\) for scales \(t\in {\mathbb {R}}\). In particular, the \(\left( -1\right) \)-scaled hyperbolics \({\mathbb {D}}_{-1}\) is isomorphic to the complex field \({\mathbb {C}}\), the 0-scaled hyperbolics \({\mathbb {D}}_{0}\) is isomorphic to the dual numbers \({\textbf{D}}\), and the 1-scaled hyperbolics \({\mathbb {D}}_{1}\) is isomorphic to the classical hyperbolic numbers \({\mathcal {D}}\). For any fixed \(t\in {\mathbb {R}}\), initiated from the t-scaled hyperbolics \({\mathbb {D}}_{t}\), we construct the t-scaled-hyperbolic Clifford algebra \({\mathscr {C}}_{t}=\underrightarrow{\textrm{lim}}{\mathscr {C}}_{t,n}\), where \({\mathscr {C}}_{t,n}\) are the n-th t-scaled-hyperbolic Clifford algebras for all \(n\in {\mathbb {N}}\cup \left\{ 0\right\} \), with \({\mathscr {C}}_{t,0}={\mathbb {R}}\) and \({\mathscr {C}}_{t,1}={\mathbb {D}}_{t}\), just like the classical Clifford algebra \({\mathscr {C}}={\mathscr {C}}_{-1}\). To analyze this \({\mathbb {R}}\)-algebra \({\mathscr {C}}_{t}\), we establish an operator algebra \({\mathscr {M}}_{t}\) (over \({\mathbb {C}}\), as usual), containing \({\mathscr {C}}_{t}\), and then construct a free-probabilistic structure \(\left( {\mathscr {M}}_{t},\tau _{t}\right) \). From the operator theory, operator algebra and free probability on \({\mathscr {M}}_{t}\), we apply these analysis for studying \({\mathscr {C}}_{t}.\)