Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras

Abstract

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, 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 Z 2 × Z 2 -graded physics (classical and quantum invariant models, parastatistics) we classify the associative Z 2 × Z 2graded 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 Z 2 × 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).