BGG Category For The Witt Lie Algebra

The Witt Lie algebra W is the derivation algebra of the polynomial algebra A=C[x],i.e.W=DerCA,also denoted as the rank is the Witt algebra of 1.In this paper,we mainly study the structure and properties of BGG category O of Witt Lie algebra W.Specifically,first of all,we give the extensions of all Verma modules in category O by means of the extensions of W-modules Fλ,which defined by B.L.Feigin and D.B.Fuchs,and the relation between Fλ and Verma modules Δ(λ)of category O.Via computing extensions between all simple modules in the category O,we give the block decomposition of O.and show that the representation type of each block of O is wild using the Ext-quiver.Each block of category O has infinitely many simple objects.This result is very different from that of O for complex semisimple Lie algebras.To find a connection between category O and the module category over some associative algebra,we define a subalgebra H1={uE U(b)|u(d-1-1)(?)(d-1-1)U(W)} of U(b),where the algebra H1 is isomorphic to an induced right U(W)-module Q’1 endomorphism algebra H’1=EndW(Q’1).In addition,We give an exact functor from category O to the categoryΩ of finite dimensional modules over H1.We also construct new simple W-modules from Weyl modules and modules over the Borel subalgebra b of W,meanwhile,the conditions for determining the isomorphism of the simple module are given.