Theorem. The hyperbolic matrix recurrence [lawson2016spin, I.4, Theorem 4.1, (4.3), pp. 25--26] [fcap-0013]
Theorem. The hyperbolic matrix recurrence [lawson2016spin, I.4, Theorem 4.1, (4.3), pp. 25--26] [fcap-0013]
For finite-dimensional quadratic spaces, Lawson--Michelsohn state the signature recurrence \[\mathcal {C}\kern -2pt\ell _{p+1,q+1}\simeq _{\mathbb {R}\text {-alg}} \mathcal {C}\kern -2pt\ell _{p,q}\otimes _{\mathbb {R}}M_2(\mathbb {R}).\] TauCeti extends the recurrence to any real quadratic module \((M,Q)\) and supplies an explicit formula on generators: \[\mathcal {C}\kern -2pt\ell (Q\perp Q_{1,1})\simeq \mathcal {C}\kern -2pt\ell (Q)\otimes _{\mathbb {R}}M_2(\mathbb {R}).\] With \[\sigma _x=\begin {pmatrix}0&1\\1&0\end {pmatrix},\] the equivalence sends a generator \((m,(s,t))\) to \[\iota _Q(m)\otimes \sigma _x +1\otimes \begin {pmatrix}s&t\\-t&-s\end {pmatrix}.\] The first summand squares to \(Q(m)\), the second to \(s^2-t^2\), and the two anticommute. The universal property therefore gives the forward algebra map. In the other direction, the original Clifford algebra and the matrix algebra act through commuting algebra maps on the hyperbolic Clifford algebra; their tensor lift is inverse to the forward map. The two composites are checked on Clifford generators and pure tensors, which is why TauCeti's extension needs no finite-dimensional hypothesis. Chevalley's split-matrix theorem and orthogonal-sum calculation supply the classical structural mechanism [chevalley1954algebraic, II.2.1 and II.2.5, pp. 42--46].
Writing the displayed generator formula as \(f\), its universal extension is summarized by the commuting diagram
