Twisted tensor products

Many noncommutative algebras in representation theory and combinatorics arise naturally as twisted tensor products: they decompose as a vector space into the tensor product of two subalgebras.  Examples of twisted tensor products and their deformations include skew polynomial rings, quantum planes, Weyl algebras, Ore extensions, universal enveloping algebras, quantum Schubert cell algebras, smash and crossed products (for group/Hopf actions), braided tensor products defined by R-matrices, some Sklyanin algebras, some Sridharan enveloping algebras, graded affine Hecke algebras, and symplectic reflection algebras.  Often a Poincare-Birkhoff-Witt property, triangular decomposition, or particularly fruitful application of the Diamond Lemma points to the underlying structure as a twisted tensor product.  We discuss resolutions for twisted tensor products and give Alexander-Whitney and Eilenberg-Zilber maps for converting between resolutions.  These maps allow one to transfer abstract homological information into concrete conditions for determining deformations of algebras.