Bounded length 3 representations of the Virasoro Lie algebra. (Q2729308)

As the title suggests, this interesting paper deals with representations of the (infinite-dimensional, ``two-sided'') Virasoro algebra.NEWLINENEWLINEThe representations considered are bounded, i.e. decomposable as a direct sum of finite-dimensional eigenspaces with respect to \(e_0\)-action, and dimensions of eigenspaces are uniformly bounded. Such representations of length 1 (i.e. irreducible) and length 2 were classified by \textit{O.~Mathieu} [Invent. Math. 107, 225--234 (1992; Zbl 0779.17025)] and \textit{C.~Martin} and \textit{A.~Piard} [Commun. Math. Phys. 137, 109--132 (1991; Zbl 0728.17015) and Commun. Math. Phys. 150, 465--493 (1992; Zbl 0774.17036)].NEWLINENEWLINEHere the author makes a next step by classifying indecomposable generic bounded representations of length 3 (``generic'' means that \(e_0^2 + e_0 - e_{-1}e_1\), the Casimir operator of the zero term in the standard grading, acts on composition series with pairwise distinct eigenvalues; the non-generic case is promised to be considered in a future paper). He also provides a new classification in the length 2 case.NEWLINENEWLINEThe main computational tool is a certain interesting (though straightforward) cohomological interpretation of module extensions in terms of cup-products. The author treats it as folklore (without providing a reference), while, in reviewer's opinion, it is not.NEWLINENEWLINEAlthough the most of the proofs are done by laborous computations, the author often outlines possible ways for a more ``conceptual'' alternative proofs.NEWLINENEWLINEBoth length 2 and length 3 representations occur to be exactly restrictions of the corresponding representations of the distinguished ``one-sided'' subalgebra \(\langle e_{-1}, e_0, e_1, e_2, \dots \rangle\). The author admits that he has no conceptual explanation of this fact.NEWLINENEWLINEThe paper concludes with observations and conjectures about the cases of higher length.