The Hamiltonian dynamics and the canonical covariant formalism for an exotic action in three dimensions are performed. By working with the complete phase space, we report a complete Hamiltonian description of the theory such as the extended action, the extended Hamiltonian, the algebra among the constraints, the Dirac's brackets and the correct gauge transformations. In addition, we show that in spite of exotic action and tetrad gravity with a cosmological constant give rise to the same equations of motion, they are not equivalent, in fact, we show that their corresponding Dirac's brackets are quite different. Finally, we construct a gauge invariant symplectic form which in turn represents a complete Hamiltonian description of the covariant phase space.PACS numbers: 98.80.-k,98.80.Cq

A detailed Dirac and Faddeev-Jackiw formulation of Bonzom-Livine model describing gravity in three dimensions is performed. The full structure of the constraints, the gauge transformations and the generalized FaddeevJackiw brackets are found. In addition, we show that the Faddeev-Jackiw and Dirac brackets coincide.

A detailed Faddeev-Jackiw quantization of an Abelian and non-Abelian exotic action for gravity in three dimensions is performed. We obtain for the theories under study the constraints, the gauge transformations, the generalized Faddeev-Jackiw brackets and we perform the counting of physical degrees of freedom. In addition, we compare our results with those found in the literature where the canonical analysis is developed, in particular, we show that both the generalized Faddeev-Jackiw brackets and Dirac's brackets coincide to each other. Finally we discuss some remarks and prospects.

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