Locally Higher-order Differentiable Maps On Trigonometric Algebra

In this paper,using induction,we mainly study higher ξ-Lie derivable maps at reciprocal elements,higher ξ-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 ξ-Lie derivable map,higher derivation）and so on.In Chapter 2,we mainly discuss non-global higher ξ-Lie derivable maps on triangular algebras.Let u=Tri（A,M,B）be a triangular algebra.{φ_n}_n∈N:u→u be a sequence of linear maps（φ₀=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 {φ_n}_n∈N satisfies （?） for any U,V∈U with UV=VU=1,i+j=n then {φ_n}_n∈N is a higher derivation.In the second section,we prove that {φ_n}_n∈N satisfies （?） for any U,V ∈ u with U （?） V=P1,then {φ_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 {φ_n}_n∈N:u→u satisfies （?） for any U,V Eu with U（?）V ∈ Q,then {φ_n}_n∈N is a higher derivation,where φ₀=id is the identity map.