The Even Parts For Hamiltonian Lie Superalgebras Of The Small Characteristic

Hamiltonian Lie superalgebras are very important in the finite-dimensionalLie superalgebras. Their structure and repersentation theories are very active partsin Lie theories. Determining the derivation algebras are very important subjectsin the algebras. In the paper, we aims to study the even parts of HamiltonianLie superalgebras in characteristic3. In particular, we completely determine theirderivation algebras. Firstly, we give the even part h, the suitable generating set,and the proper torus T in the finite-dimensional Hamiltonian Lie superalgebrasH. Secondly, we give the special subalgebra g of Witt modular Lie superalgebras.Then we give the weight space decompositions of g with respect to the torus T andcompute the weight vectors that have the same weights with the generators of h.We characterize the derivations vanishing on the top of h. Finally, the derivationalgebras from the even parts of Hamiltonian Lie superalgebras to the even partsof Witt modular Lie superalgebras are determined. In particular, we determine thederivation algebras of the even parts of Hamiltonian Lie superalgebras.We characterize the derivations vanishing on the top of h by the three kinds ofouter derivations and the weight space deconpositions. So the derivation algebrasfrom the even parts of Hamiltonian Lie superalgebras to the even parts of Witt mod-ular Lie superalgebras and the derivation algebras of the even parts of Hamiltoniansuperalgebras both are determined.