2004/12/10 by Martin Markl, Markl, Martin, Elisabeth Remm +1 · 4 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Sphingolipid Metabolism and Signaling #math.AT #math.KT
paper · pdf · doi:10.48550/arxiv.math/0412206
19 pages
arxiv created 2004/12/10 · openalex publication_date 2004/12/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a new kind of symmetry for operads, the dihedrality, responsible for the existence of the dihedral cohomology. The main trick, which we call the polarization, will be to represent an algebra with one operation without any specific symmetry as an algebra with one commutative and one anticommutative operation. We will try to convince the reader that this change of perspective might sometimes lead to new insights and results. This point of view was used by Livernet and Loday to introduce a one-parametric family of operads whose specialization at 0 is the operad for Poisson algebras, while at a generic point it equals the operad for associative algebras. We study this family and explain how it can be used to interpret the deformation quantization in a neat and elegant way.