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 $e_1{f}_i=\alpha f_{i+1}$ and $e_2{f}_j=\beta f_{j+2}$ for $i,j\in \mathbb{Z}$ and $\alpha ,\beta$ are two arbitrary constants ($e_i{f}_j=0,i\ge 3$). We classify $(n+1)$-dimensional graded thread $W^{+}$-modules for $n$ sufficiently large $n$ of three important types. New examples of graded thread $W^{+}$-modules different from finite-dimensional quotients of $V_{\lambda ,\mu }$ and $C_{\alpha ,\beta }$ were found.