38 problems

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 $m_1,m_2,m_3,m_4\in A_n$,…

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…

Let $\mathbb{k}\langle x_1,x_2,y_1,y_2\rangle$ be the free associative algebra. Yagzhev's wildness conjecture. The automorphism … is wild, meaning that it does not belong to the su…

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 $A_n$ be the free associative algebra and let $\Omega(d,D)^{even}_*$ be the even differential-form algebra with two anticommuting differentials $d$ and $D$, equipped with the p…

Let $A_n$ be the free associative algebra on $n$ generators, regarded as a Lie algebra, and consider the surjective map … The relations in its kernel include the Jacobi relations,…

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 $A_n$ be the free associative algebra on generators $x_1,\dots,x_n$, regarded as a Lie algebra with bracket $[a,b]=a\cdot b-b\cdot a$. Let $C^{\bullet}(A_n,A_n)$ be its Lie coc…

Let $A_n$ be the free associative algebra on $n$ generators, regarded as a Lie algebra, and let $A_{n,k}$ denote its $k$-th consecutive quotient of the lower central series. Write…

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…

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 $\frak m$ be the Monster Lie algebra, and let $\scr V$ and $\scr V'$ be the $(\frak g\frak l_2,M)$-modules identified inside $\frak m$. For $v\in\scr V$ or $v\in\scr V'$, write…

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 $\mathbf{k}[x,y,z]$ be a polynomial algebra and $\mathbf{k}\langle x,y,z\rangle$ the free associative algebra. A wild $z$-automorphism is an automorphism fixing $z$ that is not…

Let $\mathbf{k}[x_1,\ldots,x_n]$ be a polynomial algebra and $\mathbf{k}\langle x_1,\ldots,x_n\rangle$ a free associative algebra. An automorphism is stably tame if, after adjoinin…

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 $F_{n,k}$ be the $k$-th tensor power of the free algebra $F_n$, and let $\mathcal{M}$ be a finitely generated free $F_{n,k}$--$F_{n,k}$ bimodule. Finite Gröbner basis conjectur…

Let $F_n=\mathbb{K}\langle x_1,\ldots,x_n\rangle$ be the free algebra, let $\Omega_{F_n/\mathbb{K}}$ be its module of noncommutative differentials, and let $\phi:F_n\to F_n$ be an…

Let ${\bf k}[x,y,z]$ be a polynomial algebra and ${\bf k}\langle x,y,z\rangle$ the free associative algebra over an arbitrary field ${\bf k}$. A $z$-automorphism fixes $z$, and a w…

Let ${\bf k}[x_1,\dots,x_n]$ be a polynomial algebra and ${\bf k}\langle x_1,\dots,x_n\rangle$ the free associative algebra. An automorphism is stably tame if, after adjoining fini…

Let $A_n$ be the free associative algebra on $n$ generators, and let $B_i(A_n)$ be its lower-central-series quotient. Over $\mathbb Z[1/2]$, let $W_n$ denote the Lie algebra of vec…

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…