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arXiv · 1705.10414

${\mathbb Z}_2 \times {\mathbb Z}_2 $ generalizations of ${\cal N} = 2$ super Schr\"odinger algebras and their representations

N. Aizawa·J. Segar
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Abstract

We generalize the real and chiral $ {\cal N} =2 $ super Schr\"odinger algebras to ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded Lie superalgebras. This is done by $D$-module presentation and as a consequence, the $D$-module presentations of ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded superalgebras are identical to the ones of super Schr\"odinger algebras. We then generalize the calculus over Grassmann number to ${\mathbb Z}_2 \times {\mathbb Z}_2 $ setting. Using it and the standard technique of Lie theory, we obtain a vector field realization of ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded superalgebras. A vector field realization of the ${\mathbb Z}_2 \times {\mathbb Z}_2 $ generalization of ${\cal N} = 1 $ super Schr\"odinger algebra is also presented.

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N. Aizawa, J. Segar. 2017-05-29. ${\mathbb Z}_2 \times {\mathbb Z}_2 $ generalizations of ${\cal N} = 2$ super Schr\"odinger algebras and their representations. https://doi.org/10.1063/1.4986570

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