19 problems

Let $K$ be a field of characteristic $0$, let $K[x,y,z]$ be the commutative polynomial algebra, and let $K\langle x,y,z\rangle$ be the free associative algebra. An automorphism is…

Let $A_n$ be the free associative algebra and let $\varphi(d,D):A_n\to\Omega(d,D)^{even}_*$ be the algebra map sending each generator $x_i$ to $x_i$. For $m_1,m_2,m_3,m_4\in A_n$,…

Wn-module conjecture. Every quotient $A_{n,k}$ for $k>1$ is a $W_n$-module in the class $\mathcal C$; the Lie bracket on $gr^+(A_n)$ is $W_n$-equivariant.

Let $A_n$ be the free associative algebra on generators $x_1,\dots,x_n$, regarded as a Lie algebra with bracket $[a,b]=a\cdot b-b\cdot a$. For a monomial…

Let $R$ be an algebra over a field $K$, let $z$ be a free generator, and let $x_1,\dots,x_n$ be noncommuting free generators. Write $K\langle x_1,\dots,x_n\rangle$ for the free ass…

Let $K\langle x,y,z\rangle$ be the free associative algebra over a field $K$. A coordinate is called wild if it is the image of a generator under a wild automorphism and no tame au…

The Anick wildness conjecture. The automorphism $\nu_1$ of $K\langle x,y,z\rangle$ is wild. The conjecture was later proved by Umirbaev, who established that the Anick automorphism…

Let $K$ be a field of characteristic $0$, and let … be the Anick automorphism of $K\langle x,y,z\rangle$. An automorphism is wild if it does not belong to the group generated by el…

Let $K\langle X_d\rangle$ be the free associative algebra on $X_d$, equipped with the operation $\circ$, and let $(F,\circ)$ be an $S$-subalgebra. A minimal generating system is a…

Let $F_n=\mathbb{K}\langle x_1,\ldots,x_n\rangle$ be the free associative algebra, and let $\mathbb{T}_{n-1}$ be the $(n-1)$-dimensional algebraic torus. Free-associative torus-act…

Let $F_n=\mathbb{K}\langle x_1,\ldots,x_n\rangle$ be the free associative algebra, and let $\mathbb{T}_{n-1}$ act effectively on $F_n$. Free associative Białynicki-Birula conjectur…

Let $R$ be a $\mathbb{K}$-algebra, and let $\mathbb{K}\langle x_1,\ldots,x_n\rangle$ and $\mathbb{K}\langle y\rangle$ denote free associative algebras. Free associative cancellatio…

Let $A_n(\mathbb Z)$ be the free associative algebra on $n$ generators, and let $B_\ell(A_n(\mathbb Z))[m_1,\ldots,m_n]$ denote the multihomogeneous component of multidegree…

Let $A_n(\mathbb F_p)$ be the free associative algebra on $n$ generators over the finite field $\mathbb F_p$, and let $B_2(A_n(\mathbb F_p))[m_1,\ldots,m_n]$ denote its multihomoge…

Composition-factor conjecture. The vectors $v_{k,0}$ and $z_{k,0}$ are non-zero in $B_3(A'_{2n})$, and the $H_{2n}$-module composition factors of $B_3(A'_{2n})$ are

Arbitrary-field fractional-power conjecture. Then $g^{m/n}\in K((X))$ has a monomial of positive degree containing a negative power of an indeterminate in $X$.

Makar-Limanov and J.-T. Yu's conjecture. The series $g^{m/n}$ has a monomial of positive degree containing a negative power of an indeterminate in $X$.

Makar-Limanov and Jie-Tai Yu's conjecture. For every $m>n$ not divisible by $n$, the formal power series

Jie-Tai Yu's conjecture. If neither $\operatorname{deg}(f)$ nor $\operatorname{deg}(g)$ divides the other, then