On the Popov–Pommerening conjecture for linear algebraic groups

Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H\subseteq G$ an observable subgroup normalised by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 1970s that the invariant algebra $k[X]^{H}$ is finitely generated. We prove the conjecture for: (1) subgroups of $\operatorname{SL}_{n}(k)$ closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of $\operatorname{SL}_{n}(k)$.

Differential Invariants of the (2+1)-Dimensional Breaking Soliton Equation

We construct the differential invariants of Lie symmetry pseudogroups of the (2+1)-dimensional breaking soliton equation and analyze the structure of the induced differential invariant algebra. Their syzygies and recurrence relations are classified. In addition, a moving frame and the invariantization of the breaking soliton equation are also presented. The algorithms are based on the method of equivariant moving frames.