Associative conformal algebras with finite faithful representation

We describe all irreducible conformal subalgebras of Cend_N. The classification of simple and semisimple associative conformal algebras with finite faithful representation follows from this description.Comment: Final version, 29 pages, to appear in Adv. Mat

On representations of dialgebras and conformal algebras

In this note, we observe a relation between dialgebras (in particular, Leibniz algebras) and conformal algebras. The purpose is to show how the methods of conformal algebras help solving problems on dialgebras, and, conversely, how the ideas of dialgebras work for conformal algebras.Comment: 11 page

On finite representations of conformal algebras

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.Comment: 22 page