Simplicity of quadratic Lie conformal algebras

Abstract

In this paper, simplicity of quadratic Lie conformal algebras are investigated. From the point view of the corresponding Gel'fand-Dorfman bialgebras, some sufficient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several classes of new infinite simple Lie conformal algebras. These results will be useful for classification purposes.

On a class of infinite simple Lie conformal algebras

“…Therefore, we have α 3 (x α ′ , x β ) = 0 for any α ′ and β ∈ ∆. Similarly, by ( 16) and (17), one can get…”

“…Note that in what follows, when we write ϕ(3b), it means that 3b ∈ ∆. By the discussion in [17], CL(b, ϕ) is a class of infinite simple Lie conformal algebras.…”

“…Recently, in [17], the authors provide a method to construct infinite simple Lie conformal algebras using the relation between quadratic Lie conformal algebras and Gel'fand-Dorfman bialgebras (see [23] or [15]). Moreover, they present three classes of infinite simple Lie conformal algebras in [17] which are obtained from the corresponding Gel'fand-Dorfman bialgebras.…”

“…Recently, in [17], the authors provide a method to construct infinite simple Lie conformal algebras using the relation between quadratic Lie conformal algebras and Gel'fand-Dorfman bialgebras (see [23] or [15]). Moreover, they present three classes of infinite simple Lie conformal algebras in [17] which are obtained from the corresponding Gel'fand-Dorfman bialgebras. Therefore, for enriching the theory of infinite simple Lie conformal algebras, it is natural to investigate the structure theory and representation theory of these infinite simple Lie conformal algebras.…”

Representations of Some Associative Pseudoalgebras

“…After B. Bakalov, A. D'Andrea and V. Kac studied finite Lie H-pseudoalgebras in [BDK], various H-pseudoalgebras were introduced and studied ( [BL,Ro,SQ1,SQ2,SQ3,Wu2,Wu3,HW]). In present paper we focus on associative H-pseudoalgebras.…”