\(\mathbb{Z}_{2}-\)Graded Lie Algebra of Quaternions and Superconformal Algebra in \(D=4\) Dimensions

Abstract

In the present discussion, we have studied the \(\mathbb{Z}_{2}-\)\(grading\) of the quaternion algebra \((\mathbb{H})\). We have made an attempt to extend the quaternion Lie algebra to the graded Lie algebra by using the matrix representations of quaternion units. The generalized Jacobi identities of \(\mathbb{Z}_{2}-graded\) algebra then result in symmetric graded partners \((N_{1},N_{2},N_{3})\). The graded partner algebra \((\mathcal{F})\) of quaternions \((\mathbb{H})\) thus has been constructed from this complete set of graded partner units \((N_{1},N_{2},N_{3})\), and \(N_{0}=C\). Keeping in view the algebraic properties of the graded partner algebra \((\mathcal{F})\), the \(\mathbb{Z}_{2}-graded\) superspace \((S^{l,m})\) of quaternion algebra \((\mathbb{H})\) has been constructed. It has been shown that the antiunitary quaternionic supergroup \(UU_{a}(l;m;\mathbb{H})\) describes the isometries of \(\mathbb{Z}_{2}-graded\) superspace \((S^{l,m})\). The Superconformal algebra in \(D=4\) dimensions is then established, where the bosonic sector of the Superconformal algebra has been constructed from the quaternion algebra \((\mathbb{H})\) and the fermionic sector from the graded partner algebra \((\mathcal{F})\): asymmetric space, convex set, \(\delta\)-sun, \(\gamma\)-sun.