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

DOI10.1007/S00006-023-01263-1arXiv2112.00840MaRDI QIDQ6384598

Zhanna Kuznetsova, Francesco Toppan

Publication date: 1 December 2021

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 $7$, ${mathbbZ}_2$-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 ${mathbbZ}_{2}imes{mathbbZ}_2$-graded physics (classical and quantum invariant models, parastatistics) we classify the associative ${mathbbZ}_{2}imes{mathbbZ}_2$-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 $2imes2$ real matrices and in each word the symbol of tensor product is skipped). The $13$ inequivalent ${mathbbZ}_{2}imes{mathbbZ}_2$-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).

Mathematics Subject Classification ID

Infinite-dimensional and general division rings (16K40) Clifford algebras, spinors (15A66) Graded rings and modules (associative rings and algebras) (16W50)

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