On the double of the (restricted) super Jordan plane (Q2680134)

Let \(k\) be an algebraically closed field. The super Jordan plane is the graded algebra generated by \(x_1, x_2\) with defining relations \[ x_1^2=0, \ x_2x_{21}=x_{21}x_2+x_1x_{21}, \] and the restricted super version (for \(\mathrm{char}\,k=p>2\)) with the above relations and \[ x_{21}^p=0, \ x_2^{2p}=0, \] where \(x_{21}=x_2x_1+x_1x_2.\) The Jordan plane and the super Jordan plane play a central role in the study of pointed Hopf algebras over abelian groups with finite Gelfand-Kirillov dimension. The authors studied the Drinfeld doubles of (suitable bosonizations of) the Jordan plane and its restricted analogue in their earlier paper [J. Algebra Appl. 20, No. 1, Article ID 2140012, 26 p. (2021; Zbl 1484.16033)]. The results are related to the enveloping algebra of \({\mathfrak{sl}_2}\), the algebras of functions on some algebraic groups and their restricted analogues. In the present paper the authors carry out a similar analysis of the super Jordan plane and its restricted version. They show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in [loc. cit.]. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of \({\mathfrak{osp}}(1|2)\).