Beyond the $10$-fold way


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, Z2-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 Z2 × Z2-graded physics
(classical and quantum invariant models, parastatistics) we classify the associative Z2 × Z2-
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 2 × 2 real matrices and
in each word the symbol of tensor product is skipped). The 13 inequivalent Z2 × Z2-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).