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ASSOCIATIVE CONFORMAL ALGEBRAS WITH FINITE FAITHFUL REPRESENTATION
, 2004
"... Abstract. We describe irreducible conformal subalgebras of CendN and build the structure theory of associative conformal algebras with finite faithful representation. 1. ..."
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Abstract. We describe irreducible conformal subalgebras of CendN and build the structure theory of associative conformal algebras with finite faithful representation. 1.
On the classification of subalgebras of CendN and gcN
 Journal of Algebra
, 2002
"... Abstract. The problem of classification of infinite subalgebras of CendN and of gcN that acts irreducibly on C[∂] N is discussed in this paper. Since the pioneering papers [BPZ] and [Bo], there has been a great deal of work towards understanding of the algebraic structure underlying the notion of th ..."
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Abstract. The problem of classification of infinite subalgebras of CendN and of gcN that acts irreducibly on C[∂] N is discussed in this paper. Since 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
Unital associative pseudoalgebras and their representations
"... Abstract. Pseudoalgebras, introduced in [BDK], are multidimensional analogues of conformal algebras, which provide an axiomatic description of the singular part of the operator product expansion. Our main interest in this paper is the pseudoalgebra Cendn, which is the analogue of an algebra of endo ..."
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Abstract. Pseudoalgebras, introduced in [BDK], are multidimensional analogues of conformal algebras, which provide an axiomatic description of the singular part of the operator product expansion. Our main interest in this paper is the pseudoalgebra Cendn, which is the analogue of an algebra of endomorphisms of a finite module. We study its algebraic properties. In particular, we introduce the class of unital pseudoalgebras and describe their structure and representations. Also, we classify pseudoalgebras algebraically similar to Cendn.
SIMPLE ASSOCIATIVE CONFORMAL ALGEBRAS OF LINEAR GROWTH
, 2004
"... Abstract. We describe simple finitely generated associative conformal algebras of Gel’fand–Kirillov dimension one. 1. ..."
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Abstract. We describe simple finitely generated associative conformal algebras of Gel’fand–Kirillov dimension one. 1.
On irreducible algebras of conformal endomorphisms over a linear algebraic group, Mathematics Subject Classification
"... Abstract. We study the algebra of conformal endomorphisms Cend G,G n of a finitely generated free module Mn over the coordinate Hopf algebra H of a linear algebraic group G. It is shown that a conformal subalgebra of Cendn acting irreducibly on Mn generates an essential left ideal of Cend G,G n if e ..."
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Abstract. We study the algebra of conformal endomorphisms Cend G,G n of a finitely generated free module Mn over the coordinate Hopf algebra H of a linear algebraic group G. It is shown that a conformal subalgebra of Cendn acting irreducibly on Mn generates an essential left ideal of Cend G,G n if enriched with operators of multiplication on elements of H. In particular, we describe such subalgebras for the case when G is finite. Introduction. The notion of a conformal algebra was introduced in [1] as a tool for investigation of vertex algebras [2, 3]. From the formal point of view, a conformal algebra is a linear space C over a field k (chark = 0) endowed with a linear operator T: C → C and with a family of bilinear operations ( · n ·), n ∈ Z+ (where Z+ stands for the set of nonnegative integers), satisfying the following axioms:
Universally defined representations of Lie conformal superalgebras
 Journal of Symbolic Computation
"... Abstract. We distinguish a class of irreducible finite representations of conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra L is complete ..."
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Abstract. We distinguish a class of irreducible finite representations of conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra L is completely determined by commutation relations of L and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras Wn, n≥0, with respect to a natural set of generators. We also consider the problem for superalgebras Kn. In particular, we find a universally defined representation for the Neveu–Schwartz conformal superalgebra K1 and show that the analogues of this representation for n≥2 are not universally defined. 1.
ON THE WEDDERBURN PRINCIPAL THEOREM FOR CONFORMAL ALGEBRAS
, 2005
"... Abstract. We investigate an analogue of the Wedderburn principal theorem for associative conformal algebras with finite faithful representations. It is shown that the radical splitting property for an algebra of this kind holds if the maximal semisimple factor of this algebra is unital, but does not ..."
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Abstract. We investigate an analogue of the Wedderburn principal theorem for associative conformal algebras with finite faithful representations. It is shown that the radical splitting property for an algebra of this kind holds if the maximal semisimple factor of this algebra is unital, but does not hold in general. 1.
Associative Conformal Algebras and Pseudoalgebras and Their Representations
, 2002
"... Lie conformal algebras axiomatically describe singular parts of vertex algebras. Conversely, a vertex algebra can be reconstructed from a conformal algebra and its highest weight module. The main subject of this thesis, associative conformal algebras, plays an important role in conformal representat ..."
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Lie conformal algebras axiomatically describe singular parts of vertex algebras. Conversely, a vertex algebra can be reconstructed from a conformal algebra and its highest weight module. The main subject of this thesis, associative conformal algebras, plays an important role in conformal representation theory. In particular, all pseudolinear maps of a finite module of rank n form a conformal algebra Cendn. Pseudoalgebras generalize conformal algebras and are also related to differential Lie algebras of Ritt and Hamiltonian formalism in the calculus of variations. This thesis is roughly divided into two parts. We begin by defining a particular class of associative pseudoalgebras called unital. They resemble unital algebras in "ordinary" algebra. Not every pseudoalgebra is unital; however, Cendn are. We describe how unital pseudoalgebras that satisfy a broad technical condition are completely determined by an associative algebra and a family of locally nilpotent operators acting on it. This allows us to classify representations of all semisimple unital associative pseudoalgebras. In particular, we provide an explicit description of finite modules over conformal Cendn. The second part of this thesis is devoted to classifying pseudoalgebras that are algebraically