Modular representations of the Jordan superalgebras \(D(t)\) and \(K_3\) (Q948710)

The irreducible Jordan representations of the simple superalgebra \(D(t)\) over an algebraically closed field of characteristic \(p\neq 2\) are classified. The right multiplication by the identity of the superalgebra \(D(t)\) on any irreducible superbimodule is either the zero map or \({1\over 2}\text{Id}\) or the identity \(\text{Id}\). In the first case the module is trivial, the second case for the superalgebra \(D(t)\) in zero characteristic was considered by \textit{C. Martínez} and \textit{E. Zelmanov} in Can. Math. Bull. 45, No. 4, 653--671 (2002; Zbl 1029.17024). The third case was studied independently by the same authors [Trans. Am. Math. Soc. 358, No. 8, 3637--3649 (2006; Zbl 1141.17024)] and by the author of the present paper [J. Algebra Appl. 4, No. 1, 1--14 (2004; Zbl 1062.17015)]. The author uses a Jordan superalgebra \(D(\lambda,\mu)\) such that \(D(1,t)=D(t)\). The irreducible representations of this more symmetric superalgebra \(D(\lambda,\mu)\) can be obtained from those of simple \(3\)-dimensional Lie algebra \(\text{sl}_2\). The \(F\)-superalgebra \(D(\lambda,\mu)\) can be described as the one whose even part is \(Fe_1\oplus F e_2\) where \(e_i^2=0\) (\(i=1,2\)), \(e_1e_2=0\), and odd part \(F x\oplus F y\) such that \(e_i x={1\over 2}x\), \(e_i y={1\over 2}y\) and \(x y=\lambda e_1+\mu e_2\). For any module over \(D(\lambda,\mu)\) the operators \({2\over{\lambda+\mu}}X\circ Y\), \({2\over{\lambda+\mu}}X^2\) and \({2\over{\lambda+\mu}}Y^2\) generate \(\text{sl}_2\) (here \(A\) stands for the right multiplication operator \(R_a\) of the module). Thus the irreducible representations of this algebra can be given in terms of those of \(\text{sl}_2\). The author finds six cases in the classification theorem. As a corollary she obtains a classification of the finite-dimensional irreducible representations of the Kaplansky superalgebra \(K_3\) in the case of characteristic \(p\neq 2\).