• Corpus ID: 189928397

# Pre-Calabi-Yau algebras and double Poisson brackets

@article{Iyudu2019PreCalabiYauAA,
title={Pre-Calabi-Yau algebras and double Poisson brackets},
author={Natalia Iyudu and Maxim Kontsevich and Yannis Vlassopoulos},
journal={arXiv: Rings and Algebras},
year={2019}
}
• Published 17 June 2019
• Mathematics
• arXiv: Rings and Algebras
We give an explicit formula showing how the double Poisson algebra introduced in \cite{VdB} appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $A\oplus A^*$. Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra $A$ and emphasizes the special role of the fourth…
3 Citations

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## References

SHOWING 1-10 OF 15 REFERENCES

### Noncommutative smooth spaces

• Mathematics
• 2000
We will work in the category Al gk of associative unital algebras over a fixed base field k. If A € Ob(Algk), we denote by 1A € A the unit in A and by m A : A ⊗ A—→A the product. For an algebra A, we

### Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry

• Mathematics
• 2008
We develop a geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. The geometric approach clarifies several

### Double Poisson brackets on free associative algebras

• Mathematics
• 2012
We discuss double Poisson structures in sense of M. Van den Bergh on free associative algebras focusing on the case of quadratic Poisson brackets. We establish their relations with an associative

### Double Poisson algebras

In this paper we develop Poisson geometry for non-commutative algebras. This generalizes the bi-symplectic geometry which was recently, and independently, introduced by Crawley-Boevey, Etingof and

### Relative Calabi–Yau structures

• Mathematics
Compositio Mathematica
• 2019
We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and

### Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the

### FORMAL (NON)-COMMUTATIVE SYMPLECTIC GEOMETRY

Some time ago B. Feigin, V. Retakh and I had tried to understand a remark of J. Stasheff [S1] on open string theory and higher associative algebras [S2]. Then I found a strange construction of

### On the Deformation of Rings and Algebras

CHAPTER I. The deformation theory for algebras 1. Infinitesimal deformations of an algebra 2. Obstructions 3. Trivial deformations 4. Obstructions to derivations and the squaring operation 5.