We prove simplicity for incomplete rank 2 Kac—Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).

In this paper we characterize $K$-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat $K$-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature $\kappa \,=\,\varepsilon$, where $\varepsilon \,=\,\pm 1$ denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a $K$-contact Lorentzian manifold.

Let $R$ be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on $R$, $\Gamma \left( R \right)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra\,+\,Rb\,=\,R$. In this paper, we consider a subgraph ${{\Gamma }_{2}}\left( R \right)$ of $\Gamma \left( R \right)$ that consists of non-unit elements. We investigate the behavior of ${{\Gamma }_{2}}\left( R \right)$ and ${{\Gamma }_{2}}\left( R \right)\backslash \text{J}\left( R \right)$, where $\text{J}\left( R \right)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of ${{\Gamma }_{2}}\left( R \right)$, and the topological properties of $\text{Max}\left( R \right)$. Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

In this short note we introduce a notion called quantum injectivity of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. In particular, this provides a new characterization of amenability of locally compact groups.

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus, and 3-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, 4-genus, and 3-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of 4-genus one.

We prove that for every ordinary genus-2 curve $X$ over a finite field $\kappa$ of characteristic 2 with $\text{Aut}\left( X/\kappa \right)\,=\,\mathbb{Z}/2\mathbb{Z}\,\times \,{{S}_{3}}$ there exist $\text{SL}\left( 2,\,\kappa \left[\!\left[ s \right]\!\right] \right)$-representations of ${{\pi }_{1}}\left( X \right)$ such that the image of ${{\pi }_{1}}\left( \overline{X} \right)$ is infinite. This result produces a family of examples similar to Y. Laszlo’s counterexample to A. J. de Jong’s question regarding the finiteness of the geometric monodromy of representations of the fundamental group.