On derivation deviations in an abstract pre-operad. (English) Zbl 1002.18009
The authors study pre-operads and algebraic structures on them. In particular they investigate the cup product \(\smile\), the total composition \(\bullet\), the pre-coboundary operator \(\delta\) and the tribraces \(\{.,.,.\}\). The derivation deviation of the pre-coboundary operator has been defined in terms of the pre-coboundary operator and the total composition.
The authors derive a generalization of the Gerstenhaber identity which relates the derivation deviation to the total composition (resp.commutator) and the cup product [M. Gerstenhaber, Ann. Math., II. Ser. 78, 267-288 (1963; Zbl 0131.27302)]. The Gerstenhaber formula will be obtained if the arguments in the tribraces are Hochschild cocycles, i.e. if \(\delta h=0\).
The authors derive a generalization of the Gerstenhaber identity which relates the derivation deviation to the total composition (resp.commutator) and the cup product [M. Gerstenhaber, Ann. Math., II. Ser. 78, 267-288 (1963; Zbl 0131.27302)]. The Gerstenhaber formula will be obtained if the arguments in the tribraces are Hochschild cocycles, i.e. if \(\delta h=0\).
Reviewer: Bernhard Drabant (Walldorf)
MSC:
| 18D50 | Operads (MSC2010) |
Citations:
Zbl 0131.27302References:
| [1] | DOI: 10.1016/0022-4049(93)90041-Q · Zbl 0827.17021 · doi:10.1016/0022-4049(93)90041-Q |
| [2] | DOI: 10.2307/1970343 · Zbl 0131.27302 · doi:10.2307/1970343 |
| [3] | DOI: 10.2307/1970553 · Zbl 0182.05902 · doi:10.2307/1970553 |
| [4] | Gerstenhaber M., Deformation Theory and Quantum Groups with Applications to Mathematical Physics 134 pp 51– (1990) · doi:10.1090/conm/134/1187279 |
| [5] | DOI: 10.1155/S1073792895000110 · Zbl 0827.18004 · doi:10.1155/S1073792895000110 |
| [6] | DOI: 10.1080/00927870008826902 · Zbl 0948.18004 · doi:10.1080/00927870008826902 |
| [7] | Loday J.-L., Contemp. Math. pp 202– (1997) |
| [8] | Markl M., Contemp. Math. 227 pp 235– (1999) · doi:10.1090/conm/227/03259 |
| [9] | May J. P., Lecture Notes in Math. pp 271– (1972) |
| [10] | Shnider S., Quantum Groups from Coalgebras to Drinfeld Algebras (1994) · Zbl 0845.17015 |
| [11] | Stasheff J. D., Trans. Amer. Math. Soc. 108 pp 275–292, 293– (1963) |
| [12] | Stasheff J., Contemp. Math. 202 pp 53– (1997) · doi:10.1090/conm/202/02593 |
| [13] | Voronov A. A., Preprint math. |
| [14] | DOI: 10.1007/BF01077036 · Zbl 0849.16010 · doi:10.1007/BF01077036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
