Double-graded quantum superplane

Abstract

A ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the \emph{double-graded quantum superplane}. The commutation rules between the coordinates, their differentials and partial derivatives are explicitly given. Furthermore, we show that an extended version of the double-graded quantum superplane admits a natural Hopf ${\mathbb{Z}}_2^2$-algebra structure.

arXiv comment20 pages, amslatex, romp.sty, version 3: final journal version, to appear in "Reports on Mathematical Physics"

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Cited by in corpus (6)

• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded parastatistics in multiparticle quantum Hamiltonians
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the quantization
• Inequivalent quantizations from gradings and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ parabosons
• Classification of minimal ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie (super)algebras and some applications
• A classification of lowest weight irreducible modules over ${\mathbb{Z}}_2^2$-graded extension of $osp(1|2)$
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras