7 problems

Let $K$ be an algebraically closed field, let $n\geq 2$ and $m\geq 1$ be integers, and let $T_n(K)$ denote the algebra of $n\times n$ upper triangular matrices over $K$. For…

Let $K$ be a field, let $n\geq 2$ be an integer, and let $p$ be a multilinear polynomial in noncommuting variables. Let $T_n(K)$ denote the algebra of all $n\times n$ upper triangu…

Let $K$ be a field and let $UT_n$ denote the algebra of all $n\times n$ upper triangular matrices over $K$. Let $f$ be a multilinear polynomial, and let $f(UT_n)$ denote its image…

Let $\eta_1$ and $\eta_2$ be gradings on $UT_n^{(-)}$. Say that $\eta_1$ and $\eta_2$ correspond to a 1-kill-coarsening when passing from one grading to the other moves exactly one…

Let $G$ be a group and let $(UT_n^{(-)},\eta)$ be an elementary $G$-grading, where $UT_n^{(-)}$ denotes the Lie algebra of upper triangular $n\times n$ matrices. A tree $\mu$ is a…

Two-summand conjecture. Then

Let $T_m(K)$ denote the algebra of upper triangular $m\times m$ matrices over a field $K$, and let $p$ be a multilinear polynomial in noncommutative variables. Fagundes–de Mello co…