$\mathcal{N}$-Extension of duble-graded supersymmetric and superconformal quantum mechanics

In the recent paper, Bruce and Duplij introduced a double-graded version of supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic nature of $\mathcal{N}=1$ SQM to a ${\mathbb{Z}}_2^2$-graded superalgebra. In this work, we propose an extension of Bruce-Duplij model to higher values of $\mathcal{N}\mathcal{.}$ Furthermore, it is shown that our construction of double-graded SQM is a special case of the method which converts a given Lie superalgebra to a ${\mathbb{Z}}_2^2$-graded superalgebra. By employing this method one may convert a model of superconformal mechanics to its double-graded version. The simplest example of $\mathcal{N}=1$ double-graded superconformal mechanics is studied in some detail.