# Conformally equivariant quantization and symbol maps associated with $n$-ary differential operators on weighted densities

## TL;DR

This paper constructs a unique conformally equivariant symbol map and quantization map for $n$-ary differential operators acting on weighted densities, within the framework of the orthosymplectic superalgebra $rak{osp}(1|2)$.

## Contribution

It establishes the existence and uniqueness of a canonical conformally equivariant symbol map and provides explicit formulas for the associated quantization map for these operators.

## Key findings

- Proved the existence of a unique conformally equivariant symbol map.
- Derived explicit expressions for the quantization map.
- Enhanced understanding of $n$-ary differential operators on weighted densities.

## Abstract

We are interested in the study of the space of $n$-ary differential operators denoted by $\mathfrak{D}_{\underline{\l},\mu}$ where $\underline{\l}=(\l_{1},...,\l_{n})$ acting on weighted densities from $\frak F_{\l_1}\otimes\frak F_{\l_2}\otimes...\otimes\frak F_{\l_n}$ to $\frak F_{\mu}$ as a module over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$. As a consequence, we prove the existence and the uniqueness of a canonical conformally equivariant symbol map from $\mathfrak{D}_{\underline{\lambda},\mu}^k$ to the corresponding space of symbols as well for the explicit expression of the associated quantization map.