F. Deformed gauge structure

If we denote by
and
two independent sets of gauge parameters,

then the concrete form of the commutators among the deformed gauge transformations of the fields associated with (285) and (286) (and generically written as in (162)) read as

The gauge parameters from the right-hand side of the above formulas are defined through

where

In addition, we made the notations

Related to the first-order reducibility, the transformations (163)
are given by

while the first-order reducibility relations (164) read as

Regarding the second-order reducibility, the transformations (165) take the concrete form

such that the second-order reducibility relations (166) become

Finally, we investigate the third-order reducibility, for which the transformations (167) can be written as

while that the third-order reducibility relations (168) are listed below

2018-03-26