Commutators and powers of infinite unitriangular matrices

In the paper we consider some commutator-type and power-type matrix equations in the group UT(∞,K) of infinite dimensional unitriangular matrices over a fieldK. We introduce a notion of a power outer commutator ω1k k (x1, ..., xk) and a power Engel commutator e1k k (x, y) as outer (respectively Engel) commutators modified by allowing powers of letters instead of letters alone. For a given infinite unitriangular matrix A we discuss the matrix equations xk = A, ω1k k (x1, ..., xk) = A and e l,m1,...,mk k (x, y) = A in variables x, x1, ..., xk, y. As a main result, we provide the necessary and sufficient conditions for solvability of these equations.