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 $7$, ${\mathbb{Z}}_2$-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).

20 pages; final version accepted in Advances in Applied Clifford Algebras; extra section added with application to a parafermionic oscillators hamiltonian

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Cited by in corpus (4)

• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
• On the detectability of paraparticles beyond bosons and fermions
• Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1|2)$ and Virasoro algebra