Moduli spaces for right ideals of the Weyl algebra
In [J. Algebra 167, 116-141 (1994; Zbl 0824.16022)] \textit{R. Cannings} and the reviewer classified the right ideals of \({\mathcal D} (R)\) up to isomorphism, for \(R\) a Dedekind domain, finitely generated over an algebraically closed field of characteristic zero. In this approach one shows that (1) every non-zero right ideal of \({\mathcal D} (R)\) is isomorphic to one of the form \({\mathcal D} (R,M)\), where \(M\) is a fractional ideal for a subalgebra \(S\) of \(R\) with integral closure \(R\) and such that \(\text{Spec } R \to \text{Spec }S\) is injective. (2) \({\mathcal D} (R,M) \cong {\mathcal D} (R, M')\) if and only if \(M = qM'\), for some \(q \in \text{Frac }R\). This approach has a number of applications; for example, see \textit{R. C. Cannings} and the reviewer [Math. Z. 216, 179-194 (1994; Zbl 0807.13011)]. M. Kouakou has recently used the above result in his classification of the domains Morita equivalent to \(A_1 (\mathbb{C})\), up to isomorphism. In the paper under review the author uses a different method to classify the right ideals of the first Weyl algebra \(A_1 (\mathbb{C}) = {\mathcal D} (\mathbb{C} [y]) = \mathbb{C} \langle x,y\rangle /(xy- yx - 1)\). Homogenizing this relation one obtains the graded regular algebra \(H\) with generators \(X, Y, Z\) and relations: \(XY - YX = Z^2\), \(Z\) central. Now a right ideal \(P\) of \(A_1 (\mathbb{C})\) can be homogenized to give a graded reflexive right ideal \(h(P)\) of \(H\). This graded module determines a `vector bundle' \(\mathcal P\) over Artin's quantum plane, that is, in the quotient category \(\text{Coh-}\mathbb{P}_q^2 : =\) finitely graded \(H\)-modules / finite-dimensional graded \(H\)-modules. Finally, the analogue of Beilinson's equivalence of derived categories is \(D^b(\text{Coh- }\mathbb{P}_q^2) @>\cong>> D^b(\text{mod-}B)\), for a certain finite- dimensional algebra \(B\). Standard theory now dictates that a suitable twist of \(\mathcal P\) will be determined by a unique \(B\)-module; in fact, even by a representation of the Kronecker quiver. It follows that each isomorphism class of right ideals of \(A_1 (\mathbb{C})\) determines two discrete invariants \((m,n)\) and that the set of isomorphism classes of right ideals with invariants \((m,n)\) is in bijection with the space of orbits of \(\text{GL}_m \times \text{GL}_n\) on \(M_{m,n} (\mathbb{C}) \times M_{m,n} (\mathbb{C}) \times M_{m,n} (\mathbb{C})\). The author illustrates his results by considering the right ideals \(x^{n+1}A_1(\mathbb{C}) + (xy + n) A_1(\mathbb{C})\) and making a connection with some moduli spaces of vector bundles on \(\mathbb{P}^2\).
scientific article; zbMATH DE number 763904