On irreducible infinite conformal algebras

Abstract

The associative conformal algebra CendN and the corresponding general Lie conformal algebra gcN are the most important examples of simple conformal algebras which are not finite (see Sect. 2.10 in [K1]). One of the most important open problems of the theory of conformal algebras is the classification of infinite subalgebras of CendN and of gcN which act irreducibly on C[∂] N . (For a classification of such finite algebras, in the associative case see Theorem 2.6 of the present paper, and in the (more difficult) Lie case see [CK] and [DK].) The classical Burnside theorem states that any subalgebra of the matrix algebra MatN C that acts irreducibly on C N is the whole algebra MatN C. This is certainly not true for subalgebras of CendN (which is the “conformal” analogue of MatN C). There is a family of infinite subalgebras CendN,P of CendN , where P(x) ∈ MatN C[x], det P(x) 6= 0, that still act irreducibly on C[∂] N . One of the conjectures of [K2] states that there are no other infinite irreducible subalgebras of CendN . This conjecture was recently proved by Kolesnikov [Ko]. In the Lie conformal case, we have a conjecture on the classification of infinite Lie conformal subalgebras of gcN acting irreducibly on C[∂] N , see Conjecture 4.4. This conjecture agrees with recent results of E. Zelmanov [Z2] and A. De Sole - V. Kac.