From the introduction: Lie conformal superalgebras encode the singular part of the operator product expansion of chiral fields in two-dimensional quantum field theory [[K1] V. G. Kac, Vertex algebras for beginners. Providence, RI: AMS (1996; Zbl 0861.17017), 2nd ed. (1998; Zbl 0924.17023)]. On the other hand, they are closely connected to the notion of a formal distribution Lie superalgebra \((\mathfrak g, \mathcal F)\), i.e. a Lie superalgebra \(\mathfrak g\) spanned by the coefficients of a family \(\mathcal F\) of mutually local formal distributions. Namely, to a Lie conformal superalgebra \(R\) one can associate a formal distribution Lie superalgebra \((\text{Lie}\,R,R)\) which establishes an equivalence between the category of Lie conformal superalgebras and the category of equivalence classes of formal distribution Lie superalgebras obtained as quotients of \(\text{Lie}\,R\) by irregular ideals [see K1].
The classification of finite simple Lie conformal superalgebras was completed in [[DK] D. Fattori and V. G. Kac, J. Algebra 258, No. 1, 23–59 (2002; Zbl 1050.17023)]. The proof relies on the methods developed in [DK] for the classification of finite simple and semisimple Lie conformal algebras, and the classification of simple linearly compact Lie superalgebras [[K4] V. G. Kac, Adv. Math. 26, 8–96 (1977; Zbl 0366.17012)].
The main result of the present paper is the classification of finite semisimple Lie conformal superalgebras (Theorem 5.1). Unlike in [DK], we do not use the connection to formal distribution algebras in the proof of this theorem (which would require us to take care of numerous technical difficulties). We work instead entirely in the category of Lie conformal superalgebras. Our key result is the determination of finite differentiably simple Lie conformal superalgebras (Theorem 2.1). The proof of this result uses heavily the ideas of R. E. Block (1969) and S.-J. Cheng (1995).
Given a finite Lie conformal superalgebra \(R\), denote by \(\text{Rad}\,R\) the sum of all solvable ideals of \(R\). Since the rank of a finite solvable Lie conformal (super)algebra is greater than the rank of its derived subalgebra [DK], we conclude that \(\text{Rad}\,R\) is the maximal solvable ideal, hence \(R/\text{Rad}\,R\) is a finite semisimple Lie conformal superalgebra. Thus in some sense the study of general finite Lie conformal superalgebras reduces to that of semisimple and solvable superalgebras.
In the Lie conformal algebra case there is a conformal analog of Lie’s theorem, stating that any non-trivial finite irreducible module over a finite solvable Lie conformal algebra is free of rank 1 [DK]. However, a similar result in the “super” case is certainly false. We hope that we can develop a theory of finite irreducible modules over solvable Lie conformal superalgebras similar to that in the Lie superalgebra case [K4]. We make several observations to that end in the last section of this paper.
The paper is organized as follows: we define the main objects of our study and provide some general statements in Section 1. In Section 2 we establish the structure of differentiably simple Lie conformal superalgebras. Our proof is, in fact, quite general and the result is valid for non-Lie finite conformal superalgebras as well. We list finite simple Lie conformal superalgebras in Section 3 and describe their conformal derivations. Then in Section 4 we describe conformal derivations in the differentiably simple case and thus complete the classification of finite differentiably simple Lie conformal superalgebras. This allows us to describe the structure of finite semisimple Lie conformal superalgebras in Section 5. In Section 6 we classify simple physical Virasoro pairs and, as a consequence, obtain a classification of physical Lie conformal superalgebras which generalizes that of [K4]. Finally, in Section 7 we initiate the study of representations of finite solvable Lie conformal superalgebras.
For the entire collection see [Zbl 1081.17001].