On d-Dimensional Lattice (co)sine n-Algebra简

We present the(co)sine n-algebra which is indexed by the d-dimensional integer lattice.Due to the associative operators,this generalized(co)sine n-algebra is the higher order Lie algebra for the n even case.The particular cases are the d-dimensional lattice sine 3 and cosine 5-algebras with the special parameter values.We find that the corresponding d-dimensional lattice sine 3 and cosine 5-algebras are the Nambu 3-algebra and higher order Lie algebra,respectively.The limiting case of the d-dimensional lattice(co)sine n-algebra is also discussed.Moreover we construct the super sine n-algebra,which is the super higher order Lie algebra for the n even case.