Locally Higher-order Differentiable Maps On Trigonometric Algebra

In this paper,using induction,we mainly study higher ?-Lie derivable maps at reciprocal elements,higher ?-Lie derivable maps by Jordan product idempotents and higher derivable maps by Jordan product square zero on triangular algebras.The details are as follows:In Chapter 1,we give some common symbols,definitions(for example,triangular algebra,higher ?-Lie derivable map,higher derivation)and so on.In Chapter 2,we mainly discuss non-global higher ?-Lie derivable maps on triangular algebras.Let u=Tri(A,M,B)be a triangular algebra.{?_n}_n?N:u?u be a sequence of linear maps(?₀=id is the identity map).In the first section,for any A E A,B?B,there are integers k₁,k₂ respectively,making k₁1_A-A,k₂1_B-B invertible in triangular algebras.We prove that if {?_n}_n?N satisfies (?) for any U,V?U with UV=VU=1,i+j=n then {?_n}_n?N is a higher derivation.In the second section,we prove that {?_n}_n?N satisfies (?) for any U,V ? u with U (?) V=P1,then {?_n}_n?N is a higher derivation.In Chapter 3,we mainly discuss higher derivable maps on triangular algebras by Jordan product square zero elements.Let u=Tri(A,M,B)be a triangular algebra,and Q=?U?u:U²=0}.we prove that if the maps {?_n}_n?N:u?u satisfies (?) for any U,V Eu with U(?)V ? Q,then {?_n}_n?N is a higher derivation,where ?₀=id is the identity map.