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...en generalized in [1] for finite pseudoalgebras. Some features of structure theory and representation theory of conformal algebras of infinite type have also been considered in a series of works (see =-=[8, 9, 13, 21, 23, 24, 28, 29]-=-). 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 themse...

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...J. I. LIBERATI N we can prove this conjecture only under the assumption that the subalgebra in question is unital (see Theorem 5.3). This result is closely related to a difficult theorem of A. Retakh =-=[R]-=- (but we avoid using it). Next, we describe all finite irreducible modules over CendN,P (see Corollary 3.7). This is done by using the description of left ideals of the algebras CendN,P (see Propositi...

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...ve a similar algebraic structure). Clearly, the necessary conditions must be simplicity and unitality. A finitness condition must be a requirement as well; even though, Cendn is not finite over H. In =-=[Re1]-=- we classified conformal algebras of linear growth; however, this is not a good condition for the case of general H. Given a pseudoalgebra R over a Hopf subalgebra H ′ of H, it is easy to lift it to a...

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...properties, as it was done in [21, 22, 23], see also [3, 4, 5]. Another way is to introduce a “growth function” of a conformal algebra and move into the next class with respect to the growth rate. In =-=[19]-=-, the notion of Gel’fand–Kirillov dimension (GKdim) for conformal algebras was proposed. As in the case of usual algebras, finite conformal algebras have GKdim zero, and there are no conformal algebra...

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... ⊗ A under the operations (h ⊗ a) γ (f ⊗ b) = h(γ −1 )b (1)(γ)Lγf ⊗ ab (2), γ ∈ G, f, h ∈ H, a, b ∈ A, (5) is a conformal algebra over G. Moreover, if A is associative then C satisfies (4). Following =-=[12]-=-, denote the conformal algebra C obtained by Diff(A, ∆A). Note that an arbitrary algebra A is an H-comodule algebra with respect to the coaction ∆0 A (a) = 1 ⊗ a, a ∈ A. The conformal algebra Diff(A, ...

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...′ ). Let L be a finitely generated conformal Lie superalgebra. Any simple associative envelope of L of at most linear growth defines an irreducible finite representation of L. Indeed, it was shown in =-=[27, 23]-=- that every simple finitely generated associative conformal algebra of at most linear growth can be embedded into Cend V , rankV < ∞, as an irreducible subalgebra. Conversely, let ρ be a representatio...

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...ique: e.g., an element of the form Q −1 (v)Q(v−D) is a unit of Cendn for any Q ∈ Mn(k[v]) such that detQ ∈ k \ {0}. The structure of unital associative conformal algebras was considered in details in =-=[16, 17]-=-. Unfortunately, it is not so clear how to gather a unit to a conformal algebra. Throughout this section, C is an associative conformal algebra, I is a nilpotent ideal of C. Lemma 3.1 (c.f. [26]). (i)...

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...formal algebra C is said to be V-conformal algebra if A(C) belongs to V. Associative and Lie conformal algebras, their representations, and cohomologies have been studied in a series of papers, e.g., =-=[6, 2, 37, 31, 9, 4, 41]-=-. In particular, associative conformal algebras naturally appear in the study of representations of Lie conformal algebras. Example 1. Consider one of the simplest (though important) examples of confo...