Projective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities.

*(English)*Zbl 0958.16014In the paper under review the author continues the study of deformations of Kleinian singularities begun in an earlier joint paper with M. P. Holland. The main tool used in that paper was a new class of algebras called deformed preprojective algebras. In the present paper, the author provides a conceptual approach to the study of deformed preprojective algebras by constructing a wider class of algebras.

This is done as follows. Let \(A\) be an associative algebra over a field \(k\) and \(A^e\) the enveloping algebra of \(A\). Let \(\Delta\in{^*\text{Der}}(A,A^e)\) be the derivation defined by \(\Delta(a):=a\otimes 1-1\otimes a\). The space \(^*\text{Der}(A,A^e)\) is naturally an \(A^e\)-module, and therefore the tensor algebra \(T_A{^*\text{Der}}(A,A^e)\) can be formed. Now, for any \(a\in A\), define \(\Pi^a(A):=T_A{^*\text{Der}}(A,A^e)/(\Delta-a)\). It turns out, that the algebra \(\Pi^a(A)\) depends only on the class of \(a\) in \(H_0(A)=A/[A,A]\). The latter comes equipped with the trace map \(^*\text{tr}\colon K_0(A)\to A/[A,A]\), which sends a projective \(P\) to the trace of any idempotent in \(M_n(A)\) whose image is isomorphic to \(P\). This map extends to a linear map \(k\otimes_\mathbb{Z} K_0(A)\to A/[A,A]\), also denoted \(^*\text{tr}\). Now, if \(\lambda\in k\otimes_\mathbb{Z} K_0(A)\), define \[ \Pi^\lambda(A):=\Pi^{a_{\lambda}}(A)=T_A{^*\text{Der}}(A,A^e)/(\Delta-a_\lambda), \] where \(a_\lambda\) is any lift of the trace of \(\lambda\) to \(A\). The algebra \(\Pi^\lambda(A)\) exhibits a number of remarkable properties.

1. If \(A\) is finite-dimensional and hereditary, then \(\Pi^0(A)\) is a preprojective algebra, as defined by Baer, Geigle, and Lenzing.

2. If \(A\) is the path algebra \(kQ\) of a quiver \(Q\) and \(\lambda\in k\otimes_\mathbb{Z} K_0(kQ)\), then \(\Pi^\lambda(A)\) is a deformed preprojective algebra, as defined by Crawley-Boevey and Holland. In particular, if \(Q\) is the quiver with one vertex and one loop and if \(\nu\in k\), then \(\Pi^\nu(k[X])\cong k\langle X,Y\mid XY-YX=\nu\rangle=:C_\nu\).

3. If \(A\) is the coordinate ring of a smooth affine curve over \(k\), then \(\Pi^0(A)\) is the coordinate ring of the cotangent bundle of \(\text{Spec }A\) and \(\Pi^1(A)\) is the ring of differential operators for \(A\).

After a number of other properties of \(\Pi^\lambda(A)\) are established, the author applies the developed technique to the study of deformations of the path algebras of the extended Dynkin quivers \(Q\) over an algebraically closed \(k\). Under these assumptions, \(\Pi^\lambda(kQ)\) is a prime Noetherian ring of Gelfand-Kirillov dimension 2 and there exists an injective map \(\theta_\lambda\colon\Pi^\lambda(kQ)\to M_N(C_\nu)\). Here \(v\) is the linear combination of the components of the minimal positive imaginary root for \(Q\) with coefficients being the components of \(\lambda\) and \(C_\nu\) is the algebra defined in the second example above. As an application, a family of deformations \({\mathcal O}^\lambda\) of a Kleinian singularity, introduced and studied in an earlier paper of Crawley-Boevey and Holland, is embedded in \(C_\nu\). This embedding induces an isomorphism on quotient division rings.

This is done as follows. Let \(A\) be an associative algebra over a field \(k\) and \(A^e\) the enveloping algebra of \(A\). Let \(\Delta\in{^*\text{Der}}(A,A^e)\) be the derivation defined by \(\Delta(a):=a\otimes 1-1\otimes a\). The space \(^*\text{Der}(A,A^e)\) is naturally an \(A^e\)-module, and therefore the tensor algebra \(T_A{^*\text{Der}}(A,A^e)\) can be formed. Now, for any \(a\in A\), define \(\Pi^a(A):=T_A{^*\text{Der}}(A,A^e)/(\Delta-a)\). It turns out, that the algebra \(\Pi^a(A)\) depends only on the class of \(a\) in \(H_0(A)=A/[A,A]\). The latter comes equipped with the trace map \(^*\text{tr}\colon K_0(A)\to A/[A,A]\), which sends a projective \(P\) to the trace of any idempotent in \(M_n(A)\) whose image is isomorphic to \(P\). This map extends to a linear map \(k\otimes_\mathbb{Z} K_0(A)\to A/[A,A]\), also denoted \(^*\text{tr}\). Now, if \(\lambda\in k\otimes_\mathbb{Z} K_0(A)\), define \[ \Pi^\lambda(A):=\Pi^{a_{\lambda}}(A)=T_A{^*\text{Der}}(A,A^e)/(\Delta-a_\lambda), \] where \(a_\lambda\) is any lift of the trace of \(\lambda\) to \(A\). The algebra \(\Pi^\lambda(A)\) exhibits a number of remarkable properties.

1. If \(A\) is finite-dimensional and hereditary, then \(\Pi^0(A)\) is a preprojective algebra, as defined by Baer, Geigle, and Lenzing.

2. If \(A\) is the path algebra \(kQ\) of a quiver \(Q\) and \(\lambda\in k\otimes_\mathbb{Z} K_0(kQ)\), then \(\Pi^\lambda(A)\) is a deformed preprojective algebra, as defined by Crawley-Boevey and Holland. In particular, if \(Q\) is the quiver with one vertex and one loop and if \(\nu\in k\), then \(\Pi^\nu(k[X])\cong k\langle X,Y\mid XY-YX=\nu\rangle=:C_\nu\).

3. If \(A\) is the coordinate ring of a smooth affine curve over \(k\), then \(\Pi^0(A)\) is the coordinate ring of the cotangent bundle of \(\text{Spec }A\) and \(\Pi^1(A)\) is the ring of differential operators for \(A\).

After a number of other properties of \(\Pi^\lambda(A)\) are established, the author applies the developed technique to the study of deformations of the path algebras of the extended Dynkin quivers \(Q\) over an algebraically closed \(k\). Under these assumptions, \(\Pi^\lambda(kQ)\) is a prime Noetherian ring of Gelfand-Kirillov dimension 2 and there exists an injective map \(\theta_\lambda\colon\Pi^\lambda(kQ)\to M_N(C_\nu)\). Here \(v\) is the linear combination of the components of the minimal positive imaginary root for \(Q\) with coefficients being the components of \(\lambda\) and \(C_\nu\) is the algebra defined in the second example above. As an application, a family of deformations \({\mathcal O}^\lambda\) of a Kleinian singularity, introduced and studied in an earlier paper of Crawley-Boevey and Holland, is embedded in \(C_\nu\). This embedding induces an isomorphism on quotient division rings.

Reviewer: Alex Martsinkovsky (Boston)

##### MSC:

16G20 | Representations of quivers and partially ordered sets |

16S32 | Rings of differential operators (associative algebraic aspects) |

16S80 | Deformations of associative rings |

14B07 | Deformations of singularities |