On finite representations of conformal algebras

TL;DR

This paper proves that finite torsion-free associative conformal algebras have finite faithful representations, explores the possibility of joining conformal units, and establishes the existence of finite faithful representations for certain Lie conformal algebras.

Contribution

It demonstrates that all finite torsion-free associative conformal algebras have finite faithful representations and explores conditions for Lie conformal algebras.

Findings

Finite torsion-free associative conformal algebras have finite faithful representations.

Not all conformal algebras can have a conformal unit joined.

Every torsion-free finite solvable Lie conformal algebra has a finite faithful representation.

Abstract

We prove a finite torsion-free associative conformal algebra to have a finite faithful conformal representation. As a corollary, it is shown that one may join a conformal unit to such an algebra. Some examples are stated to demonstrate that a conformal unit can not be joined to any torsion-free associative conformal algebra. In particular, there exist associative conformal algebras of linear growth and even locally finite ones that have no finite faithful representation. We also consider the problem of existence of a finite faithful representation for a torsion-free finite Lie conformal algebra (the analogue of Ado's Theorem). It turns out that the conformal analogue of the Poincare-Birkhoff-Witt Theorem would imply the Ado Theorem for finite Lie conformal algebras. We prove that every torsion-free finite solvable Lie conformal algebra has a finite faithful representation.