Some Representations of Nongraded Lie Algebras of Generalized Witt Type

In a paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them, are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented. 1. INTRODUCTION The four well-known series of infinite dimensional simple Lie algebras of Cartan type have played important roles in the structure theory of Lie algebras. Generalizations of the simple Lie algebras of Witt type have been obtained by Kawamoto [Kaw], Dokovic and Zhao [DZ1,DZ2], Xu [X1] and Zhao [Z2]. Passman [P] studied the Lie algebras AD = A⊗D of generalized Witt type constructed from the pair of a commutative associative algebra A with an identity element and its commutative derivation subalgebra D over a field IF of arbitrary characteristic. Xu [X1] studied some of these simple Lie algebras of generalized Witt type and other generalized Cartan types Lie algebras, based on the pairs of the tensor algebra of the group algebra of an additive subgroup of IF n with the polynomial algebra in several variables and the subalgebra of commuting locally finite derivations. Su, Xu and Zhang [SXZ], Su and Xu [SX] determined the structure spaces of the generalized simple Lie algebras of Witt type and of special type constructed in [X1]. Su and Zhao [SZ2] determined the second cohomology group and gave some representations of some Lie algebras of generalized Witt type.