Graded Thread Modules Over the Positive Part of the Witt (Virasoro) Algebra

Abstract

We study ${\mathbb Z}$-graded thread $W^+$-modules $V=\oplus _i V_i, \, \dim {V_i}=1, -\infty \le k< i < N\le +\infty , \, \dim {V_i}=0, \, {\mathrm{otherwise}},$ over the positive part $W^+$ of the Witt (Virasoro) algebra W. There is well-known example of infinite-dimensional ($k=-\infty , N=\infty$) two-parametric family $V_{\lambda , \mu }$ of $W^+$-modules induced by the twisted W-action on tensor densities $P(x)x^{\mu }(dx)^{-\lambda }, \mu , \lambda \in {\mathbb K}, P(x) \in {\mathbb K}[t]$. Another family $C_{\alpha , \beta }$ of $W^+$-modules is defined by the action of two multiplicative generators $e_1, e_2$ of $W^+$ as $e_1f_i=\alpha f_{i+1}$ and $e_2f_j=\beta f_{j+2}$ for $i,j \in {\mathbb Z}$ and $\alpha , \beta$ are two arbitrary constants ($e_if_j=0, i \ge 3$).

We classify $(n+1)$-dimensional graded thread $W^+$-modules of three important types for sufficiently large n. New examples of graded thread $W^+$-modules different from finite-dimensional quotients of $V_{\lambda , \mu }$ and $C_{\alpha , \beta }$ are found.

Supported by the grant N 16-51-55017 of Russian Foundation for Basic Research (RFBR)