Simple finite-dimensional double algebras

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.

References in corpus (4)

• Double Poisson vertex algebras and non-commutative Hamiltonian equations
• Averaging algebras, Schröder numbers, rooted trees and operads
• Rota-Baxter operators on the polynomial algebras, integration and averaging operators
• Homogeneous conformal averaging operators on semisimple Lie algebras

Cited by in corpus (20)

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• The controlling $L_{\infty }$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples
• Cohomology and deformations of weighted Rota-Baxter operators
• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• $3$-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras
• Spectrum of Rota-Baxter operators
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Nonabelian embedding tensors
• Nonunital decompositions of the matrix algebra of order three
• Rota-Baxter operators of weight zero on upper-triangular matrices of order three
• Deformations and cohomologies of embedding tensors on 3-Lie algebras
• An example of a simple double Lie algebra
• Rota-Baxter operators of weight zero on the matrix algebra of order three without unit in kernel
• Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra
• Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures
• Rota-Baxter operators on $\mathrm{Cur}({\mathrm{sl}}_2(\mathbb{C}))$
• Formal integration of complete Rota-Baxter Lie algebras