Presentations for small reflection equation algebras of type A

We give presentations, in terms of the generators and relations, for the reflection equation algebras of type $G{L}_n$ and $S{L}_n$, i.e., the covariantized algebras of the dual Hopf algebras of the small quantum groups of ${\mathfrak{g}\mathfrak{l}}_n$ and ${\mathfrak{s}\mathfrak{l}}_n$. Our presentations display these algebras as quotients of the infinite-dimensional reflection equation algebras of types $G{L}_n$ and $S{L}_n$ by identifying additional relations that correspond to twisting the nilpotency and unipotency relations of the finite-dimensional quantum function algebras. The presentations are valid for appropriately defined integral forms of these algebras.