Simple algebras of Weyl type. II (Q2781289)

Let \(\mathbb{F}\) be a field, \(A\) a commutative unital \(\mathbb{F}\)-algebra and \(D\) a non-zero \(\mathbb{F}\)-vector space of commuting \(\mathbb{F}\)-derivations of \(A\). Define \(A[D]\subseteq\text{End}_\mathbb{F}(A)\) to be the \(\mathbb{F}\)-algebra generated by \(A\) and \(D\).NEWLINENEWLINENEWLINEThe author proves the following results on the associative algebra \(A[D]\). (1) (Theorem 2.3) \(A[D]\) is simple if and only if \(A\) is \(D\)-simple (i.e. has no \(D\)-stable non-trivial ideals). (2) (Theorem 3.5) If \(A\) is \(D\)-simple then \(A[D]\) is a free \(A\)-module (and a basis is provided).NEWLINENEWLINENEWLINENow let \(\mathbb{F}_1=\{x\in A:D(x)=0\}\). In studying the natural Lie algebra structure on \(A[D]\), denoted by \(A[D]^L\), the author shows that (3) (Theorem 3.8) If \(A\) is \(D\)-simple then the Lie algebra \([A[D]^L,A[D]^L]/(\mathbb{F}_1\cap[A[D]^L,A[D]^L])\) is simple unless the characteristic of \(\mathbb{F}\) is 2 and \(A\) and \(D\) have explicitly described forms.NEWLINENEWLINENEWLINEThese results are through calculation. They extend earlier work of the author and \textit{Y. Su}, [Part I, Sci. China, Ser. A 44, No. 4, 419-426 (2001; Zbl 1006.16003)], the crucial idea being to consider the elements of \(A[D]\) as operators on \(A\).