Demonstratio Mathematica (Apr 2025)
Characterization generalized derivations of tensor products of nonassociative algebras
Abstract
Consider A{\mathcal{A}} and ℬ{\mathcal{ {\mathcal B} }} to be nonassociative unital algebras. Under the assumption that either one of them has finite dimensions or that both are finite dimensions, a generalized derivation is an additive map ℱ:A→A{\mathcal{ {\mathcal F} }}:{\mathcal{A}}\to {\mathcal{A}} associated with a derivation d{\mathcal{d}} of A{\mathcal{A}} if ℱ(uv)=ℱ(u)v+ud(v){\mathcal{ {\mathcal F} }}\left(uv)={\mathcal{ {\mathcal F} }}\left(u)v+u{\mathcal{d}}\left(v) for all u,v∈Au,v\in {\mathcal{A}}. The objective of this study is to characterize and elucidate the structure of a generalized derivation on the tensor product of nonassociative algebras. Specifically, we prove that if ℱ{\mathcal{ {\mathcal F} }} is a generalized derivation of A⊗ℬ{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} associated with a derivation d{\mathcal{d}} of A⊗ℬ{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}, then ℱ=ℒu+d{\mathcal{ {\mathcal F} }}={{\mathcal{ {\mathcal L} }}}_{u}+{\mathcal{d}}, where ℒu{{\mathcal{ {\mathcal L} }}}_{u} is a left multiplication by uu and uu belongs to the left nucleus of A⊗ℬ{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} (i.e., ℒur=ur{{\mathcal{ {\mathcal L} }}}_{u}r=ur for all r∈A⊗ℬr\in {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} and u∈Nl(A⊗ℬ)u\in {N}_{l}\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }})). Moreover, every generalized derivation of A⊗ℬ{\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} can be represented as the sum of the derivations of the three categories: (i) w+aduw+a{\mathcal{d}}u, where u,w∈N(A⊗ℬ)u,w\in N\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}), (ii) ℒz⊗f{{\mathcal{ {\mathcal L} }}}_{z}\otimes f, where ff is a derivation of ℬ{\mathcal{ {\mathcal B} }} and z∈Z(A)z\in Z\left({\mathcal{A}})(the center of A{\mathcal{A}}), and (iii) g⊗ℒwg\otimes {{\mathcal{ {\mathcal L} }}}_{w}, where gg is a derivation of A{\mathcal{A}} and w∈Z(ℬ)w\in Z\left({\mathcal{ {\mathcal B} }}).
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