One-parameter deformations of the diassociative and dendriform operads

Livernet and Loday constructed a polarization of the nonsymmetric associative operad with one operation into a symmetric operad with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of , which includes Poisson algebras. We combine this with the dendriform...

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Bibliographic Details
Published in:Linear & multilinear algebra 2017-02, Vol.65 (2), p.341-350
Main Author: Bremner, Murray R.
Format: Article
Language:English
Subjects:
Algebra
algebraic operads
Associative
Categories
Deformation
dendriform algebras
Diassociative algebras
Hermite normal form
Koszul duality
LLL algorithm
Poisson algebras
Polarization
Primary: 17A30
Secondary: 15A21
Splitting
Symmetry
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Summary:Livernet and Loday constructed a polarization of the nonsymmetric associative operad with one operation into a symmetric operad with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of , which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2016.1183563