Weakly associative algebras, Poisson algebras and deformation quantization

Abstract

Weakly associative algebras are a class of Lie-admissible algebras that generalize associative algebras and useful to extend the class of formal deformations of associative commutative algebras. Thus we obtain a new method of determining quantization deformations of Poisson algebras which generalizes the classical process. The algebraic study of weakly associative algebras is also interesting, in particular the homology of free algebras with one generator that coincides with the free flexible algebra with one generator.

Keywords:

• Deformation quantization
• flexible algebras
• nonassociative algebras
• Poisson algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A30
• 53D55
• 17B63
• 17D25

Acknowledgment

The author is grateful to Kévin Morand for his orientation on this subject and for many fruitful discussions.