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
Singular hochschild cohomology and algebraic string operations. / Rivera, Manuel; Wang, Zhengfang.
In: Journal of Noncommutative Geometry, Vol. 13, No. 1, 01.01.2019, p. 297-361.Research output: Contribution to journal › Article
}
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 -