### Abstract

Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex D.A; A/, called the Tate–Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex D.A; A/computes the singular Hochschild cohomology of A. We construct a cyclic (or Calabi–Yau) A-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an L-infinity algebra structure extending the classical Gerstenhaber bracket, on D.A; A/. Moreover, we prove that the cohomology algebra H .D.A; A// is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes’ boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.

Original language | English (US) |
---|---|

Pages (from-to) | 297-361 |

Number of pages | 65 |

Journal | Journal of Noncommutative Geometry |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- A-infinity algebras
- Frobenius algebras
- L-infinity algebras
- String topology
- Tate–Hochschild complex

### ASJC Scopus subject areas

- Algebra and Number Theory
- Mathematical Physics
- Geometry and Topology

### Cite this

*Journal of Noncommutative Geometry*,

*13*(1), 297-361. https://doi.org/10.4171/JNCG/325

**Singular hochschild cohomology and algebraic string operations.** / Rivera, Manuel; Wang, Zhengfang.

Research output: Contribution to journal › Article

*Journal of Noncommutative Geometry*, vol. 13, no. 1, pp. 297-361. https://doi.org/10.4171/JNCG/325

}

TY - JOUR

T1 - Singular hochschild cohomology and algebraic string operations

AU - Rivera, Manuel

AU - Wang, Zhengfang

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex D.A; A/, called the Tate–Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex D.A; A/computes the singular Hochschild cohomology of A. We construct a cyclic (or Calabi–Yau) A-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an L-infinity algebra structure extending the classical Gerstenhaber bracket, on D.A; A/. Moreover, we prove that the cohomology algebra H .D.A; A// is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes’ boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.

AB - Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex D.A; A/, called the Tate–Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex D.A; A/computes the singular Hochschild cohomology of A. We construct a cyclic (or Calabi–Yau) A-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an L-infinity algebra structure extending the classical Gerstenhaber bracket, on D.A; A/. Moreover, we prove that the cohomology algebra H .D.A; A// is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes’ boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.

KW - A-infinity algebras

KW - Frobenius algebras

KW - L-infinity algebras

KW - String topology

KW - Tate–Hochschild complex

UR - http://www.scopus.com/inward/record.url?scp=85064348393&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064348393&partnerID=8YFLogxK

U2 - 10.4171/JNCG/325

DO - 10.4171/JNCG/325

M3 - Article

AN - SCOPUS:85064348393

VL - 13

SP - 297

EP - 361

JO - Journal of Noncommutative Geometry

JF - Journal of Noncommutative Geometry

SN - 1661-6952

IS - 1

ER -