Non-holonomic modules over Weyl algebras and enveloping algebras
Let \(R=A_ n({\mathbb{C}})\) denote the nth Weyl algebra over the field \({\mathbb{C}}\) of complex numbers, i.e. R has generators \(x_ i\), \(y_ i\) (1\(\leq i\leq n)\) satisfying the relations \(x_ iy_ i-y_ ix_ i=1\) (1\(\leq i\leq n)\) with all other commutators being zero. For any R-module M, \textit{I. N. Bernstein} [Funkts. Anal. Prilozh. 6, No.4, 26-40 (1972; Zbl 0282.46038), or see \textit{J.-E. Björk}, Rings of differential operators (1979; Zbl 0499.13009), chapter 1] has proved GK-dim \(M\geq n= GK\)-dim R, where GK-dim M denotes the Gelfand-Kirillov dimension of M, and equality holds for certain important classes of R-modules. A simple R-module U is called holonomic provided GK-dim U\(= GK\)-dim R. (In general, if T is a Noetherian ring of finite GK-dimension, a simple T- module U is called holonomic provided GK-dim U\(= GK\)-dim(T/Ann U).) Not all simple R-modules are holomonic. The author shows that for \(2\leq i\leq n\), complex numbers \(\lambda_ 2,...,\lambda_ n\) linearly independent over the rational field \({\mathbb{Q}}\) and \(\alpha =x_ 1+y_ 1(\sum^{n}_{i=2}\lambda_ ix_ iy_ i)+\sum^{n}_{i=2}(x_ i+y_ i)\), the right ideal \(E=\alpha R\) is a maximal right ideal of R and in particular \(GK\)-dim(R/E)\(=2n-1\). The Ext groups of holonomic R-modules are infinite dimensional over \({\mathbb{C}}\) [see J.-E. Björk, op. cit. p. 20, Theorem 6.6 and p. 77, Theorem 7.15]. However, if \(n=2\) and \(\beta =x_ 1+y_ 1x_ 2y_ 2+x_ 2+y_ 2\) then it is proved that \(Ext^ 1_ R(R/\beta R,R/\beta R)\) is an infinite dimensional complex vector space. In addition for general n, with \(\alpha\) as above, both \(Ext^ 1_ R(R/\alpha R,R/x_ 1R)\) and \(Ext^ 1_ R(R/x_ 1R,R/\alpha R)\) are finite dimensional complex vector spaces. Moreover \({\mathbb{C}}[\alpha]\) is a maximal commutative subalgebra of R, and C(\(\alpha)\) a maximal commutative subfield of Q(R), the quotient division ring of R. The second half of the paper concerns the enveloping algebra U of the semi-simple Lie algebra \(S\ell_ 2\times S\ell_ 2\). By Conze's theorem there exists an embedding of a primitive factor ring of U into \(A_ 2({\mathbb{C}})\) which allows calculations to be transferred from U to \(A_ 2({\mathbb{C}})\). The author exhibits a subring S of \(A_ 2({\mathbb{C}})\) which is a homomorphic image of U and an element \(\gamma\) of \(A_ 2({\mathbb{C}})\) such that \(\gamma\) \(A_ 2({\mathbb{C}})\) is a maximal right ideal of \(A_ 2({\mathbb{C}})\), \(\gamma\) \(A_ 2({\mathbb{C}})\cap S\) is a maximal right ideal of S and S/(\(\gamma\) \(A_ 2({\mathbb{C}})\cap S)\) is a non-holonomic simple U- module. Moreover the ring S, and hence U, is not weakly ideal invariant (i.e. there exist a finitely generated right S-module M and an ideal I of S such that K-dim M\(<K\)-dim(R/I) but K-dim(M\(\otimes I)\geq K\)-dim(R/I), where K-dim M denotes the Krull dimension of M). As a result, there exist simple U-modules E and A such that E is finite dimensional but \(E\otimes A\) has infinite length.
scientific article; zbMATH DE number 3889760
scientific article