It has long been known that the category of 2-dimensional cobordisms is freely generated by a commutative Frobenius algebra, the circle. This result allows for a complete classification of TQFTs (Topological Quantum Field Theories) in dimension 2. In this talk I will discuss similar algebraic presentations in dimension 3 and 4 due to Bobtcheva and Piergallini. In both cases, the fundamental algebraic structures are provided by certain Hopf algebras called BPH algebras. I will also present examples of such algebras and the TQFTs they induce. This is a joint work with A. Beliakova, I. Bobtcheva, and R. Piergallini.