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Simple finite-dimensional double algebras. (English) Zbl 1437.16014

Summary: A double algebra is a linear space \(V\) equipped with linear map \(V \otimes V \rightarrow V \otimes V\). 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 \(\mathrm{End}V\) 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.

MSC:

16K20 Finite-dimensional division rings
16T25 Yang-Baxter equations
16W99 Associative rings and algebras with additional structure
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