This project investigates a series of interrelated problems in noncommutative algebras and quantum groups. These include (1) determining commutation relations among quantum minors in the cominuscule quantum Schubert cell algebras of quantum symmetric and anti-symmetric matrices, (2) presenting defining relations for the universal bialgebras of Faddeev-Reshetikhin-Takhtajian associated to non-standard solutions of the quantum Yang-Baxter equation, (3) identifying conditions under which universal bialgebras are quantum nilpotent algebras, and (4) determining minimal generating sets for the prime ideals in the cominuscule quantum Schubert cells and their relation to total positivity.