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Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

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Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology. / Iyudu, Natalia; Kontsevich, Maxim.
In: arXiv.org, 24.11.2020.

Research output: Contribution to Journal/Magazine › Journal article

Harvard

Iyudu, N & Kontsevich, M 2020, 'Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology', arXiv.org.

APA

Iyudu, N., & Kontsevich, M. (2020). Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology. arXiv.org.

Vancouver

Iyudu N, Kontsevich M. Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology. arXiv.org. 2020 Nov 24.

Author

Iyudu, Natalia ; Kontsevich, Maxim. / Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology. In: arXiv.org. 2020.

Bibtex

@article{26f50c676acc414b912cde82a324c11d,
title = "Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology",
abstract = " We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $\xi\delta$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $\xi\delta$-monomials. This subcomplex and a basis of $\xi\delta$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $\xi\delta$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure. ",
keywords = "math.RA, math-ph, math.AG, math.CO, math.MP, 16A22, 16S37",
author = "Natalia Iyudu and Maxim Kontsevich",
note = "33 pages",
year = "2020",
month = nov,
day = "24",
language = "Undefined/Unknown",
journal = "arXiv.org",

}

RIS

TY - JOUR

T1 - Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

AU - Iyudu, Natalia

AU - Kontsevich, Maxim

N1 - 33 pages

PY - 2020/11/24

Y1 - 2020/11/24

N2 - We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $\xi\delta$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $\xi\delta$-monomials. This subcomplex and a basis of $\xi\delta$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $\xi\delta$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.

AB - We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $\xi\delta$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $\xi\delta$-monomials. This subcomplex and a basis of $\xi\delta$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $\xi\delta$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.

KW - math.RA

KW - math-ph

KW - math.AG

KW - math.CO

KW - math.MP

KW - 16A22, 16S37

M3 - Journal article

JO - arXiv.org

JF - arXiv.org

ER -