Beyond the 10-fold Way: 13 Associative $$ {\mathbb Z}_2\times {\mathbb Z}_2$$-Graded Superdivision Algebras

The “10-fold way” refers to the combined classification of 3 associative division algebras (of real, complex and quaternionic numbers) 7, $${\mathbb Z}_2$$ -graded, superdivision (in a algebra each homogeneous element is invertible). connection 10-fold way with periodic table topological insulators superconductors well known. Motivated by recent interest in Z}_2\times {\mathbb -graded physics (classical quantum invariant models, parastatistics) we classify show that 13 inequivalent cases have be added way. Our scheme based on “alphabetic presentation Clifford algebras”, here extended graded algebras. generators are expressed as equal-length words 4-letter alphabet (the letters encode basis invertible $$2\times 2$$ real matrices word symbol tensor product skipped). split into series (4 subcases 4 each), (5 8 generators) 16 generators). As an application, parafermionic Hamiltonian possessing time-reversal particle-hole symmetries presented.