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

• URL: http://arxiv.org/abs/2112.00840v3
• Date: Mon, 13 Feb 2023 14:03:22 GMT
• Title: Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Authors: Zhanna Kuznetsova and Francesco Toppan
• Abstract summary: "$10-fold way" refers to the combined classification of the $3$ associative division algebras (of real, complex and quaternionic numbers) and of the $7$, $mathbb Z$-letter-graded, superdivision algebras. 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 $mathbb Z$timesmathbb Z$-graded physics, we classify the associative $mathbb Z$times
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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).

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