as explained there, this derived L-∞ algebroid has a Chevalley-Eilenberg algebra – a graded-commutative dg-algebra – which is equipped with the structure of a BV-algebra, hence with a Poisson n-algebra bracket and a BV-operator , such that there is a “Hamiltonian” for the differential on
so that the condition on the differential is equivalently
This is an extension of the classical action: the BV-action . This equation is called the classical master equation.
Under quantization the graded-commutative dg-algebra becomes a non-commutative dg-algebra. For instance in deformation quantization it becomes a non-commutative algebra over the power series ring in a formal parameter .
In general such a deformation breaks the condition by terms of order . For a consistent BV-quantization (…) the condition is that after quantization the relation is
This equation is called the quantum master equation in the context of BV-quantization.
If after quantization this condition does not hold one says that the system has a gauge quantum anomaly . See there for more references on this point.