  A matrix modeling formulation for translation-invariant noncommutative gauge
theories is given in the setting of differential graded algebras and quantum
groups. Translation-invariant products are discussed in the setting of
{\alpha}-cohomology and it is shown that loop calculations are entirely
determined by {\alpha}-cohomology class of star product in all orders.
Noncommutative version of geometric quantization and (anti-) BRST
transformations is worked out which leads to a noncommutative description of
consistent anomalies and Schwinger terms.
