On the Drinfeld doubles of Laistrygonian algebras in positive characteristic

In this work, we study restricted Laistrygonian algebras over a field of positive characteristic $p$. We compute their Drinfeld double ${\mathtt{D}}_{\mathtt{p}}(\mathtt{q},\mathtt{a})$ and give explicit presentations by generators and relations. We prove that ${\mathtt{D}}_{\mathtt{p}}(\mathtt{q},\mathtt{a})$ is a pointed Hopf algebra. For ${\mathtt{D}}_{\mathtt{p}}(\mathtt{1},\mathtt{a})$, we describe an abelian exact sequence of Hopf algebras and identify one of its quotients as a restricted enveloping Lie algebra. This exact sequence enables us to classify the finite-dimensional simple ${\mathtt{D}}_{\mathtt{p}}(\mathtt{1},\mathtt{a})$-modules. We also introduce an infinite-dimensional Hopf algebra covering the restricted Drinfeld double and study its relation with the corresponding finite-dimensional case. Using this covering together with a Frobenius map, we construct a $3\times 3$ commutative diagram whose rows and columns are exact sequences of Hopf algebras. In particular, our approach provides a unified treatment of the Drinfeld doubles associated with restricted Laistrygonian algebras in arbitrary positive characteristic.