The Structure Of A Special Class Of 3-Lie Algebras

On a commutative associative algebra A generated by a class of special functions f?C?(R3),a Nambu 3-Lie algebra T over the real field F with the multiplication[f1,f2,f3](?)=(?)(f1,f2,f3)/(?)(x,y,z),(?)f1,f2,f3 ? A is constructed,and the structure of it is studied.1)First,the nilpoten-cy,solvability,structure of derived algebra and the structure of inner derivation algebra adT are discussed.It is proved that the inner derivation algebra adT is an indecomposable Lie algebra without minimum ideals,but it can be decomposed into the semi-direct product of subalgebra P and ideal Q.2)A class of 3-Lie algebra-modules(W,??,?)with ??F,?=0 or 1 is constructed,which has a maximal indecomposable submodule P(?)=?m?ZFw?+m,where? ? F,and it is proved that(W,??,?)does not exist irreducible submodules.3)The structure of homogeneous Rota-Baxter operators of 3-Lie algebra T is studied.We conclude that ? is a nonzero homogeneous Rota-Baxter operator of weight 1 if and only if ? is one of the ten cases given in theorem 5.6,? is a nonzero homogeneous Rota-Baxter operator of weight 0 if and only if ? is one of the four possibilities described in theorem 5.8,and the concrete expressions of them are provided.At last of the paper,Rota-B axter 3-Lie Rinehart algebras are introduced,and the structure of it is discussed.