The problem of classification of infinite subalgebras of Cend N and of gc N that acts irreducibly on C[∂] N is discussed in this paper. IntroductionSince the pioneering papers [BPZ] and [Bo], there has been a great deal of work towards understanding of the algebraic structure underlying the notion of the operator product expansion (OPE) of chiral fields of a conformal field theory. The singular part of the OPE encodes the commutation relations of fields, which leads to the notion of a Lie conformal algebra [K1-2].In the past few years a structure theory [DK], representation theory [CK, CKW] and cohomology theory [BKV] of finite Lie conformal algebras has been developed. The associative conformal algebra Cend N and the corresponding general Lie conformal algebra gc N are the most important examples of simple conformal algebras which are not finite (see Sect. 2.10 in [K1]). One of the most urgent open problems of the theory of conformal algebras is the classification of infinite subalgebras of Cend N and of gc N which act irreducibly on C[∂] N . (For a classification of such finite algebras, in the associative case see Theorem 5.2 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 Mat N C that acts irreducibly on C N is the whole algebra Mat N C. This is certainly not true for subalgebras of Cend N (which is the "conformal" analogue of Mat N C). There is a family of infinite subalgebras Cend N,P of Cend N , where P (x) ∈ Mat N C[x], det P (x) = 0, that still act irreducibly on C[∂] N . One of the conjectures of [K2] states that there are no other infinite irreducible subalgebras of Cend N .One of the results of the present paper is the classification of all subalgebras of Cend 1 and determination of the ones that act irreducibly on C[∂] (Theorem 2.2). This result proves the above-mentioned conjecture in the case N = 1. For generalNow, using Remark 1.1, we obtain a natural homomorphism of conformal associative algebras from Conf(Diff N C × ) to Cend N , which turns out to be an isomorphism (see [DK] and Proposition 2.10 in [K1]).

Abstract. We construct all finite irreducible modules over Lie conformal superalgebras of type K

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