• DOI:10.1007/s11005-021-01383-4
• Corpus ID: 232417130

On a Batalin–Vilkovisky operator generating higher Koszul brackets on differential forms

@article{Shemyakova2021OnAB,
  title={On a Batalin–Vilkovisky operator generating higher Koszul brackets on differential forms},
  author={Ekaterina Shemyakova},
  journal={Letters in Mathematical Physics},
  year={2021},
  volume={111},
  pages={1-30}
}

• E. Shemyakova
• Published 30 March 2021
• Mathematics
• Letters in Mathematical Physics

We introduce a formal $$\hbar $$
-differential operator $$\Delta $$
that generates higher Koszul brackets on the algebra of (pseudo)differential forms on a $$P_{\infty }$$
-manifold. Such an operator was first mentioned by Khudaverdian and Voronov in arXiv:1808.10049. (This operator is an analogue of the Koszul–Brylinski boundary operator $$\partial _P$$
which defines Poisson homology for an ordinary Poisson structure.) Here, we introduce $$\Delta =\Delta _P$$
by a different method and establish its properties. We show that this BV type operator generating higher Koszul brackets can be included in a one-parameter family of BV type formal $$\hbar $$
-differential operators, which can be understood as a quantization of the cotangent $$L_{\infty }$$
-bialgebroid. We obtain symmetric description on both $$\Pi TM$$
and $$\Pi T^*M$$
. For the purpose of the above, we develop in detail a theory of formal $$\hbar $$
-differential operators and also of operators acting on densities on dual vector bundles. In particular, we have a statement about operators that can be seen as a quantization of the Mackenzie–Xu canonical diffeomorphism. Another interesting feature is that we are able to introduce a grading, not a filtration, on our algebras of operators. When operators act on objects on vector bundles, we obtain a bi-grading.