Regular Borel subalgebras of the Lie Algebra of Aut(${\mathbb{A}}^2$)

Abstract

In this paper, we find all regular Borel subalgebras of Lie(Aut(${\mathbb{A}}^2$)), i.e., maximal solvable subalgebras generated by homogeneous derivations with respect to the standard ${\mathbb{Z}}^2$-grading. It follows that a regular Borel subalgebra has derived length $2$ or $3$. We also describe isomorphism classes of such subalgebras. On the way, we give a combinatorial description of the regular Abelian subalgebras and the regular solvable subalgebras of Lie(Aut(${\mathbb{A}}^2$)).

28 pages