Associativity and Operator Hamiltonian Quantization of Gauge Theories

We show that the associative algebra structure can be incorporated in the BRST quantization formalism for gauge theories such that extension from the corresponding Lie algebra to the associative algebra is achieved using operator quantization of reducible gauge theories. The BRST differential that encodes the associativity of the algebra multiplication is constructed as a second-order quadratic differential operator on the bar resolution.