Splitting of operations, Manin products and Rota-Baxter operators

2011/06/29 by Chengming Bai, Olivia Bellier, Li Guo +1 · 4 citations
Mathematics · #math.QA #math.CO #math.RA #msc:18D50 #msc:17A30 #msc:17A36 #msc:17B99 #msc:17C99

paper · pdf · doi:10.1093/imrn/rnr266

published as Int. Math. Res. Not. IMTN (2012) · 33 pages

arxiv created 2011/06/29 · arxiv updated 2013/02/05

Abstract

This paper provides a general operadic definition for the notion of splitting the operations of algebraic structures. This construction is proved to be equivalent to some Manin products of operads and it is shown to be closely related to Rota-Baxter operators. Hence, it gives a new effective way to compute Manin black products. The present construction is shown to have symmetry properties. Finally, this allows us to describe the algebraic structure of square matrices with coefficients in algebras of certain types. Many examples illustrate this text, including the case of Jordan algebras.

Cited by

• Homotopy theory of post-Lie algebras
• Computational Methods for Rota-type operators on 4-dimensionnal nilpotent Leibniz Algebras
• Rota-Baxter type operators on trusses and derived structures
• Hom-Poisson superalgebras with nondegenerate bilinear forms and compatible Hom-anti-pre-Poisson superalgebras

Related

• Splitting of operads and Rota-Baxter operators on operads
• On the computation of Manin products for operads
• A trick to compute certain Manin products of operads
• Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra
• Classification of monomial Rota-Baxter operators on k[x]