The L-algebra of Hurwitz primes

Conway and Smith proved that up to recombination of conjugate primes and migration of units, the only obstruction to unique factorization in the ring of Hurwitz integers in the quaternions is metacommutation of primes with distinct norm. We show that the Hurwitz primes form a discrete L⁎-algebra, a quantum structure which provides a general explanation for metacommutation. L-algebras arise in the theory of Artin-Tits groups, quantum logic, and in connection with solutions of the quantum Yang-Baxter equation. It is proved that every discrete L⁎-algebra admits a natural embedding into a right ℓ-group, which yields a new class of Garside groups.