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

  • Dmitry Millionshchikov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 273)

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.

Keywords

Witt algebra Visaroro algebra Lie algebras 

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Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia

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