so(D−1,1)⊕so(D−1,2)so(D−1,1)⊕so(D−1,2) algebras and gravity

It is shown that using a specific semigroup, the S-expansion of the AdS   Lie algebra leads to a generalization of the so(D−1,1)⊕so(D−1,2)so(D−1,1)⊕so(D−1,2) algebras, which are called generalized AdS-Lorentz algebras.The generalized Inönü–Wigner contraction of the generalized AdS-Lorentz   algebras provides the so-called BmBm algebras. The B4B4 algebra corresponds to the so-called Maxwell algebra. The BmBm algebras can be also obtained by S-expansion of the AdS Lie algebra when we use a semigroup endowed with a multiplication law different from the law of multiplication of semigroup above.Chern–Simons as well as Born–Infeld gravities actions, which in a certain limit contain the Einstein–Hilbert Lagrangian with a cosmological term, are also constructed.