On the Drinfeld double of the restricted Jordan plane in characteristic 2

We consider the restricted Jordan plane in characteristic 2, a finite-dimensional Nichols algebra quotient of the Jordan plane that was introduced by Cibils, Lauve and Witherspoon. We extend results from arXiv:2002.02514 on the analogous object in odd characteristic. We show that the Drinfeld double of the restricted Jordan plane fits into an exact sequence of Hopf algebras whose kernel is a normal local commutative Hopf subalgebra and the cokernel is the restricted enveloping algebra of a restricted Lie algebra $m$ of dimension 5. We show that $u\left(m\right)$ is tame and compute explicitly the indecomposable modules. An infinite-dimensional Hopf algebra covering the Drinfeld double of the restricted Jordan plane is introduced. Various quantum Frobenius maps are described.