Beyond the 10-fold way: 13 associative Z_2times Z₂-graded superdivision algebras

The "10-fold way" refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, {mathbb Z}₂-graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in {mathbb Z}_2times{mathbb Z}₂-graded physics (classical and quantum invariant models, parastatistics) we classify the associative {mathbb Z}_2times {mathbb Z}₂-graded superdivision algebras and show that 13 inequivalent cases have to be added to the 10-fold way. Our scheme is based on the "alphabetic presentation of Clifford algebras", here extended to graded superdivision algebras. The generators are expressed as equal-length words in a 4-letter alphabet the letters encode a basis of invertible $2times 2$ real matrices and in each word the symbol of tensor product is skipped. The 13 inequivalent {mathbb Z}_2times {mathbb Z}₂-graded superdivision algebras are split into real series (4 subcases with 4 generators each), complex series (5 subcases with 8 generators) and quaternionic series (4 subcases with 16 generators).