Double Complexes and Cohomological Hierarchy in a Space of Weakly Invariant Lagrangians of Mechanics

For a given configuration space M and Lie algebra G whose action is defined on M the space V 0.0 of weakly G-invariant Lagrangians (i.e. Lagrangians whose motion equations left hand sides are G-invariant) is studied. The problem is reformulated in terms of the double complex of Lie algebra cochains with values in the complex of Lagrangians. Calculating the cohomology of this complex using the method of spectral sequences we arrive at the hierarchy in the space V 0.0 : The double filtration {V s.σ } (s = 0, 1, 2, 3, 4, σ = 0, 1) and the homomorphisms on every space V s.σ are constructed. These homomorphisms take values in the cohomologies of the algebra G and configuration space M. On one hand every space V s,σ is the kernel of the corresponding homomorphism, on the other hand this space is defined by its physical properties. The cohomology of the symmetries algebra has important consequences for properties of corresponding theory [1,2] and cohomological methods play essential role in many problems of modern fields theory. For example their application made more clear the understanding of algebraic origin of gauge anomalies. As it was shown in [1] one can consider axial anomalies of four-dimensional gauge theory in terms of infinitesimal cocycles in a representation of gauge group. Another example is BRST formalism which at beginning was formulated in terms of symplectic geometry of phase space expanded by the ghosts and antighosts, then it was understood [3,4,5,6,7] that the language of homological algebra is more deeply related with physical meaning of this formalism: Inclusion of ghosts and antighosts corresponds to the construction of the chain of free modules (free resolvent) on phase space of constrained system where the constrains cannot be resolved in a direct way. The operator corresponding to BRST charge becomes the differential of the complex of these resolvents. Further the investigation of local BRST cohomology was performed with use of developed homological methods. (See [8,9,10] with citations there.)