PBW deformations of Artin-Schelter regular algebras. (Q2797020)

Suppose that \(A\) is an associative algebra over an algebraically closed field \(k\) of characteristic zero which is generated by elements \(x_1,\ldots,x_n\) subject to the set of defining relations \(R\). Its homogenization \(H(A)\) is the \(k\)-algebra generated by elements \(x_0,\ldots,x_n\) with defining relations \(x_0x_j=x_jx_0\) and \(H(f)=0\), \(f\in R\), where \(H(f) =\sum_{i=0}^d f_kx_0^{d-k}\) if \(f=\sum_{i=0}^df_i\) is a homogeneous decomposition of \(f\) with respect to the total degree in \(x_1,\ldots,x_n\). An algebra \(A\) is a PBW-deformation of \(\mathrm{gr}(A)\) if the canonical surjection \(k\langle x_1,\ldots,x_n\mid\mathrm{gr}(R)\rangle\to\mathrm{gr}(A)\) is an isomorphism.NEWLINENEWLINE Suppose that \(Q\) is a PBW deformation of an AS regular algebra and \(x_0\) is not a zero divisor in \(H(A)\). Then \(A\) is prime (PI, Noetherian) if and only if so is \(H(A)\). If \(A\) is a PBW deformation of a Noetherian AS regular algebra, then \(A\) is skew Calabi-Yau. Suppose that \(A\) is a PBW deformation of a 2-dimensional AS regular algebra which is not PI. Then a finite dimensional simple \(A\)-module has dimension 1. Suppose that \(A,B\) are PBW deformations of dimension 2 AS regular algebra. Then \(H(A\otimes B)\) is AS regular of dimension 5. If \(A\) is a PBW deformation of AS regular algebra, then skew polynomial extension \(A[\xi;\sigma,\delta]\) is again a PBW deformation of an AS regular algebra.