# Lagrangian Relations and Quantum L∞ Algebras

## TL;DR

This paper explores quantum L∞ algebras and introduces a new way to define relationships between them using Lagrangian relations and shifted symplectic vector spaces.

## Contribution

A new category for quantum L∞ algebras is proposed, enabling morphisms via Lagrangian relations and distributional half-densities.

## Key findings

- Morphisms in the proposed category can be described using formal half-densities and Lagrangian relations.
- The composition of these morphisms recovers the homotopy transfer construction for quantum L∞ algebras.
- A new notion of relation between quantum L∞ algebras is introduced through this category.

## Abstract

Quantum
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\begin{document}$$L_{\infty }$$\end{document}L∞ algebras are higher loop generalizations of cyclic
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\begin{document}$$L_{\infty }$$\end{document}L∞ algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of
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\begin{document}$$(-1)$$\end{document}(-1)-shifted symplectic vector spaces and distributional half-densities, originally proposed by Ševera. Morphisms in this category can be given both by formal half-densities and Lagrangian relations; we prove that the composition of such morphisms recovers the construction of homotopy transfer of quantum
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\begin{document}$$L_{\infty }$$\end{document}L∞ algebras. Finally, using this category, we propose a new notion of a relation between quantum
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\begin{document}$$L_{\infty }$$\end{document}L∞ algebras.