Research topics in finite groups and vertex algebras

We suggest a few projects for studying vertex algebras with emphasis on finite group viewpoints.Comment: dedicated to Geoffrey Maso

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over the Twisted Schr\"{o}dinger-Virasoro Lie algebra

It is shown that the support of an irreducible weight module over the Schr\"{o}dinger-Virasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schr\"{o}dinger-Virasoro Lie algebra with a nontrivial finite-dimensional weight space, is a Harish-Chandra module.Comment: 6 page

Verma modules over a Block Lie algebra

Let B be the Lie algebra with basis {L_{i,j},C|i,j\in Z} and relations [L_{i,j},L_{k,l}]=((j+1)k-i(l+1))L_{i+k,j+l}+i\delta_{i,-k}\delta_{j+l,-2}C, [C,L_{i,j}]=0. It is proved that an irreducible highest weight B-module is quasifinite if and only if it is a proper quotient of a Verma module. For an additive subgroup G of the base field F, there corresponds to a Lie algebra B(G) of Block type. Given a totalorder \succ on G and a weight \Lambda, a Verma B(G)-module M(\Lambda,\succ) is defined. The irreducibility of M(\Lambda,\succ) is completely determined.Comment: 7 pages. The previous version was posted by a mistak

Classification of irreducible weight modules with a finite dimensional weight space over twisted Heisenberg-Virasoro algebra

We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenberg-Virasoro algebra, having a nontrivial finite dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate seriesComment: LaTeX, 5 page

A characterization of the moonshine vertex operator algebra by means of Virasoro frames

In this article, we show that a framed vertex operator algebra V satisfying the conditions: (1) V is holomorphic (i.e., V is the only irreducible V-module); (2) V is of rank 24; and (3) V_1=0; is isomorphic to the moonshine vertex operator algebra constructed by Frenkel-Lepowsky-Meurman.Comment: 10 pages, no figur

N-derivations for finitely generated graded Lie algebras

$N$-derivation is the natural generalization of derivation and triple derivation. Let ${\cal L}$ be a finitely generated Lie algebra graded by a finite dimensional Cartan subalgebra. In this paper, a sufficient condition for Lie $N$-derivation algebra of ${\cal L}$ coinciding with Lie derivation algebra of ${\cal L}$ is given. As applications, any $N$-derivation of Schr\"{o}dinger-Virasoro algebra, generalized Witt algebras, Kac-Moody algebras and their Borel subalgebras, is a derivation.Comment: 9 page

Leibniz Central Extension on the Twisted Schr\"{o}dinger-Virasoro Algebra

In this paper we present all the Leibniz 2-cocycles of the twisted Schr\"{o}dinger-Virasoro algebra, which determine its second Leibniz cohomology group.Comment: 9 page

Explicit Formulae for Cocycles of Holomorphic Vector Fields with values in lambda-Densities

We give explicit formulae for the generators of H^2(Hol(\Sigma_r,{\cal F}_{\lambda}(\Sigma_r)) in terms of affine and projective connections. This is done using the cocycles of V. Ovsienko and C. Roger for the case of the circle and globalizing them to an open Riemann surface \Sigma_r.Comment: 10 pages, submitted in J. Lie Theory, replaced version contains minor change

Derivations and automorphisms of a Lie algebra of Block type

Let \BB be the Lie algebra of Block type with basis \{L_{\a,i}|\,\a,i\in\Z, i\geq0\} and relations [L_{\a,i},L_{\b,j}]=\left((\a-1)(j+1)-(\b-1)(i+1)\right)L_{\a+\b,i+j}. In the present paper, the derivation algebra and the automorphism group of \BB are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of \BB is trivial.Comment: LaTeX, 10 page

The Index Theorem for Homogeneous Differential Operators on Supermanifolds

In mid 60s Bott proved that (1) the index theorem for homogeneous, G-invariant, elliptic differential operators acting in the spaces of sections of induced representations of G over G/H reduces to the Weyl character formula and (2) the index of an equivariant elliptic operator does not depend on the operator, but on the representations. Here the same theorem is formulated for the unitary supergroup G=U(p|q). For atypical representations the character formula does not reduce to that for the Lie group underlying the supergroup G and this contradicts a statement of Rempel and Schmitt on index on supermanifolds (Pseudodifferential operators and the index theorem on supermanifolds. Seminar Analysis, 1981/82, 92--131, Akad. Wiss. DDR, Berlin, 1982; id., Pseudodifferential operators and the index theorem on supermanifolds. Math. Nachr. 111 (1983), 153--175).Comment: 4 pages, Late