Graded thread modules over the positive part of the Witt (Virasoro) algebra

Abstract: We study

m a t h b b Z

-graded thread

W^{+}

-modules

V=oplus_i V_i, ; dim{V_i}=1, -infty le k< i < Nle +infty, ; dim{V_i}=0, ; { m ; otherwise},

over the positive part

W^{+}

of the Witt (Virasoro) algebra

W

. There is well-known example of infinite-dimensional (

k = - i n f t y, N = i n f t y

) two-parametric family

V_{l a m b d a, m u}

of

W^{+}

-modules induced by the twisted

W

-action on tensor densities

P (x) x^{m u} (d x)^{- l a m b d a}, m u, l a m b d a i n m a t h b b K, P (x) i n m a t h b b K [t]

. Another family

of

W^{+}

-modules is defined by the action of two multiplicative generators

e_{1}, e_{2}

of

W^{+}

as

e_{1} f_{i} = a l p h a f_{i + 1}

and

for

i, j i n m a t h b b Z

and

are two arbitrary constants (

e_{i} f_{j} = 0, i g e 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_{l a m b d a, m u}

and

were found.