Simple finite-dimensional double algebras

TL;DR

This paper classifies simple finite-dimensional double algebras, showing their non-existence or triviality in certain cases, and relates their structures to specific linear operators like Rota-Baxter and averaging operators.

Contribution

It provides a complete classification of simple finite-dimensional double algebras and links their structures to well-known linear operators.

Findings

Simple finite-dimensional Lie double algebras do not exist over any field.

All simple finite-dimensional associative double algebras over algebraically closed fields are trivial.

Every simple finite-dimensional associative double algebra over an arbitrary field is commutative.

Abstract

A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. 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 $\End V$ 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.