Associative Conformal Algebras with Finite Faithful Representation

The notion of conformal algebra appears as an algebraical language describing singular part of operator product expansion (OPE) in conformal field theory [BPZ]. Explicit algebraical exposition of this theory leads to the notion of vertex (chiral) algebra (see, e.g., [B], [DL], [FLM]). Roughly speaking, conformal algebras correspond to vertex algebras by the same way as Lie algebras correspond to their associative enveloping algebras (see [K1] for detailed explanation). In the recent years a great advance in structure theory of associative and Lie conformal algebras of finite type has been obtained. In [DK], [CK1], [FK], [FKR], simple and semisimple Lie conformal (super)algebras of finite type have been described (as well as associative ones). The main result of [BDK] is the classification of simple and semisimple finite pseudoalgebras, which generalizes [DK]. Some features of structure theory and representation theory of conformal algebras of infinite type have also been considered in a series of works (see [K2], [BKL1], [BKL2], [DK1], [Re1], [Re2], [Z1], [Z2]). One of the most urgent problems in this field is to describe structure of conformal algebras with faithful irreducible representation of finite type (these algebras could be of infinite type themselves). In [K2] and [BKL1], the conjectures on the structure of such algebras (associative and Lie) have been stated. The papers [BKL1], [DK1], [Z2] contain confirmations of these conjectures under some additional conditions. Another problem is to classify simple and semisimple conformal algebras of linear growth (i.e., of Gel’fand–Kirillov dimension one) [Re1]. In the papers [Re1], [Re2], [Z1], [Z2] this problem has been solved for finitely generated associative conformal algebras which contain a unit ([Re1], [Re2]), or at least an idempotent ([Z1], [Z2]). The objects appearing from consideration of conformal algebras with faithful irreducible representation of finite type are similar to those examples of conformal algebras adduced in these papers. Combinatorial aspect of the theory of conformal algebras requires the notion of free conformal