Algebraic monoid structures on the affine 3-space
Ivan Arzhantsev, Roman Avdeev, Yulia Zaitseva
TL;DR
The paper resolves the full classification of algebraic monoid structures on the affine 3-space $\mathbb{A}^3$ by reducing noncommutative cases to the commutative setting via left and right commutative reductions and a rank-based case analysis. The main result, Theorem A3_theor, lists all isomorphism classes: toric and unipotent cases ($3M$, $3A$, $U_3$) and three parametric families of noncommutative monoids described by integers $(b,b',c,c')$ and, when present, polynomials $Q_p$; rank-1 and rank-2 cases are treated in detail and isomorphism criteria are explicit. The paper also analyzes fundamental algebraic properties (centers, idempotents, kernels, zeros) of all classified monoids and connects them to multiplicative structures on 3-dimensional algebras. An independent Appendix classifies reductive monoids on arbitrary affine spaces, showing they correspond to matrix-algebra direct products $\mathrm{R}(n_1,\dots,n_k)$, with isomorphism determined by the unordered multiset of dimensions. These results advance the understanding of affine algebraic monoids and provide a concrete, complete landscape for $\mathbb{A}^3$ and, more broadly, reductive cases.
Abstract
We complete the classification of algebraic monoid structures on the affine 3-space. The result is based on a reduction of the general case to that of commutative monoids. We also study various algebraic properties of all monoids appearing in the classification.