Weyl Type Algebras And Smash Products Of Hopf Algebras

Let k be an algebraically closed field and A be a commutative associative algebra with an identity element 1. Let D be a nonzero k-vector space of commuting -derivations of A. Kaim-ing Zhao and Yucai Su studied the associative algebra A[D] = Ak[D] of Weyl type constructed from the pair of a commutative associative algebra .4 and its commutative derivation subalgebra D over a field k of arbitrary characteristic. They proved that the Lie algebra A[D] is simple if and only if A as Lie algebra is D-simple and k1[D] acts faithfully on A,and the associative algebra A[D] is simple if and only if the associative algebra A is D-simple and k1[D] acts faithfully on A. Let U(D) be the universal enveloping algebra of the Lie algebra spanned by D. In this paper,we will give some ring-theoretical properties of the Weyl type algebra A[D] = A k[D] by proving there is an isomorphism between A[D] and the smash product A#U(D).The paper is organized as follows. In the first section,we introduce some backgrounds and recall the definitions including Weyl type algebra,smash product and Ore extension. In the second section,we will present a ring-theoretical proof for the necessary and sufficient condition for A[D] to be simple,as an associative algebra. In the third section,the condition for A[D] without zero divisor has been obtained. In the last section,we prove that an isomorphism between A[D] and the smash product A#U(D). Based on Theorem 3.3,we will prove that A[D] is an iterated Ore extension. As consequences,we give the noetherian-ness of the algebra A[D] and the bounds on the global dimension.