| Autor: |
Goncharov, M. E., Kolesnikov, P. S. |
| Rok vydání: |
2016 |
| Předmět: |
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| Zdroj: |
Journal of Algebra 500 (2018), 425--438 |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1016/j.jalgebra.2017.04.020 |
| Popis: |
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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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