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

TL;DR

This paper classifies 13 new associative ${f Z}_2 imes {f Z}_2$-graded superdivision algebras, extending the well-known 10-fold way, with implications for topological insulators, superconductors, and graded physics models.

Contribution

It introduces a classification of 13 new superdivision algebras with ${f Z}_2 imes {f Z}_2$ grading, expanding the existing 10-fold way framework.

Findings

Identified 13 inequivalent superdivision algebras beyond the 10-fold way.

Classified these algebras into real, complex, and quaternionic series.

Used an extended alphabetic presentation of Clifford algebras to derive results.

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 $7$, ${\mathbb Z}_2$-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 ${\mathbb Z}_2\times{\mathbb Z}_2$-graded physics (classical and quantum invariant models, parastatistics) we classify the associative ${\mathbb Z}_2\times {\mathbb Z}_2$-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 $2\times 2$ real matrices and in each word the symbol of tensor product is skipped). The $13$ inequivalent ${\mathbb Z}_2\times {\mathbb Z}_2$-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).