Let A be a Koszul Calabi-Yau algebra. We show that there exists an isomorphism of Batalin-Vilkovisky algebras between the Hochschild cohomology ring of A and that of its Koszul dual algebra A ! . This confirms (a generalization of) a conjecture of R. Rouquier.2010 Mathematics Subject Classification. 14A22, 16E40, 16S38, 55U30.where the Batalin-Vilkovisky operator on each side is the pull-back of the Connes operator via the Poincaré duality. − The isomorphisms (2) and (3) are not true with respect to the usual grading. They are in fact isomorphism with respect to a certain bigrading; see §5 for details.Quite recently, the above mentioned results of Ginzburg and Tradler are generalised to their twisted versions. More precisely, when a twisted Calabi-Yau algebra or a Frobenius algebra has semisimple Nakayama automorphism, then its Hochschild cohomology ring is still a Batalin-Vilkovisky algebra; see Kowalzig and Krähmer [22, Theorem 1.7] and Lambre, the third author and Zimmermann [25, Theorem 0.1]. We shall consider the twisted version of our main result and related applications in a future work.This paper is organized as follows: §2 recalls some basic facts about bar/cobar constructions and twisting morphisms. Our basic reference of this section is the recent book of Loday and Vallette ([28]). §3 collects the definitions of Hochschild and cyclic (co)homology of algebras and coalgebras;§4 reviews some basic facts about Koszul algebras; §5 computes the Hochschild (co)homology of Koszul algebras and their Koszul dual; §6 studies Koszul Calabi-Yau algebras; §7 proves Theorem A; and the last §8, gives an application of the previous results to the cyclic homology of Calabi-Yau algebras. Throughout this paper, k denotes a field of arbitrary characteristic. It is supposed to be of zero characteristic when talking about cyclic (co)homology.Acknowledgements. The authors would like to thank Farkhod Eshmatov, Ji-Wei He and Dong Yang for many helpful conversations. Nearly at the same time when the first version of paper was put in arXiv, a paper of Estanislao Herscovich also appeared in arXiv ([17]), which also deals with Koszul duality and Hochschild (co)homology. There is certain overlap between our paper and his, mostly in Sections 2-5. We recommend his paper as a companion to ours. Estanislao proves that the Hochschild (co)homology of a Koszul algebra and that of its Koszul dual form dual differential calculi in a certain sense, he, however, does not consider Calabi-Yau algebras and Batalin-Vilkovisky structures. Bar/cobar construction and twisting morphismsThe goal of this section is to recall the bar/cobar construction and twisting morphisms; for details, we refer the reader to Loday-Vallette [28].In this paper, we shall use chain complexes and homological grading everywhere, that is, a complex is of the form/ / · · · and the differential decreases the index. Let V be a k-vector space, denote V * = Hom k (V, k) be its k-dual. For an abelian group G, written additively, a G-graded vector space V = ⊕ g∈G V g is a direct su...
A new type of mesoionic insecticide triflumezopyrim is mainly used to control rice planthoppers, leafhoppers, etc. In order to study the uptake and translocation characteristics of this new insecticide in rice (Oryza sativa), a method for the detection of triflumezopyrim in rice, soil, and water was established using liquid–liquid extraction and QuEChERS sample pretreatment combined with liquid chromatography–triple quadrupole tandem mass spectrometry. The distribution of triflumezopyrim in rice was investigated after hydroponic treatment and foliar treatment at the concentrations of 2.5 and 5 mg·L–1 within the ranges of 24, 48, and 72 h. The results showed that triflumezopyrim could be absorbed by roots and form a systematic distribution in rice by hydroponic treatment; meanwhile, it could also be absorbed by leaves and transported to the bottom leaves under foliar treatment, but no triflumezopyrim was detected in the roots. Thus, triflumezopyrim exhibited high acropetal translocation within the rice plant. This study provides an important scientific basis for the development of an application strategy of triflumezopyrim to control planthoppers and leafhoppers as well as for the residue detection method and safety evaluation.
Let M be a smooth, simply-connected, closed oriented manifold, and L M the free loop space of M. Using a Poincaré duality model for M, we show that the reduced equivariant homology of L M has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.
Spirotetramat is a pesticide with bidirectional systemicity and can effectively control pests by inhibiting the biosynthesis of fatty acids. In this study, adsorption and desorption behaviors of spirotetramat in six soils and its interaction mechanism were studied using the batch equilibrium method and infrared radiation. The results showed that the adsorption and desorption behaviors of spirotetramat conformed to the Freundlich isotherm model. The values of adsorption capacities K F‑ads ranged from 2.11 to 12.40, and the values of desorption capacities K F‑des varied from 2.97 to 32.90. From the hysteresis coefficient, spirotetramat was easily desorbed from the test soils. The adsorption capacity of the soil to spirotetramat enhanced with an increasing temperature. Moreover, the changes in pH values and exogenous addition of humic acid and surfactant could also affect soil adsorption capacity, but for desorption, there was no correlation.
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