7 problems

Freeness conjecture. The dendriform trialgebra ${\bf WQSym}$ is free.

Let $(f_1,\dots,f_r) \in K\langle X\rangle^r$ and let $I^{[\Sigma]}$ be the corresponding module. Assume that there exists a finite set…

Makar-Limanov's conjecture. If $F\otimes_K A$ contains a free $K$-algebra on at least two free generators, then $A$ also contains a free $K$-algebra on the same number of free gene…

Left-bracketing conjecture. The space $\mathscr{A}$ is linearly generated by these strict left-bracketings; in particular, together with the single letters, they form a linear basi…

Non-amenability conjecture. The free skew field on $r$ generators over $K$ is non-amenable for every base field $K$.

GA conjecture. If $D$ is finitely generated as a skew field over $Z$ and $D$ is infinite dimensional over $Z$, then $D$ contains a noncommutative free group $Z$-algebra.

Let $K$ be the base field and let $K\{x_1,x_2,\ldots,x_n\}$ be the free Poisson algebra on $x_1,x_2,\ldots,x_n$. For elements $f,g$ in this algebra, write $\{f,g\}$ for their Poiss…