Algebras with one operation including Poisson and other Lie-admissible algebras

By Martin Markl, Elisabeth Remm. 2004

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Abstract 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 dihedral cohomology. The main trick, which we call the polarization, will be used to represent an algebra with one operation without any specific symmetry as an algebra with one commutative and one anticommutative operations. 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-parameter 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 (∗-product) in a neat and elegant way.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/543cbcc85272f7d65ebccd4f5f003e891d50eb4d

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