Simple finite-dimensional double algebras

A double algebra is a linear space V equipped with linear map V⊗V→V⊗V. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space V is naturally described by a linear operator R on the algebra End V of linear transformations of V. Double Lie algebras correspond in this sense to skew-symmetric Rota-Baxter operators, double associative algebra structures - to (left) averaging operators.