Segre product of Artin-Schelter regular algebras of dimension 2 and embeddings in quantum \(\mathbb{P}^ 3\)'s
Let \(A,B\) be two quantum planes; they are quadratic AS-regular algebras of global dimension 2 [see \textit{M. Artin} and \textit{W. Schelter}, Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)]. In this paper all the embeddings of the Segre product \(A \circ B = \oplus_{i \geq 0} A_i \otimes B_i\) on quantum \(\mathbb{P}^3\)'s are listed. Here ``quantum \(\mathbb{P}^3\)'' means a quadratic AS-regular algebra \(C\) of global dimension 4, and ``embedding'' means a graded, degree zero, surjective algebra map \(C \to A \circ B\) such that \(C_1 \simeq A_1 \otimes B_1\) [see \textit{M. Artin} and \textit{J. J. Zhang}, Adv. Math. 109, 228-287 (1994; Zbl 0833.14002)]. Schelter's computer program Affine was used to prove that the list contains all such embeddings. See also [\textit{M. Vancliff} and the author, Embedding a quantum non-singular quadric in a quantum \(\mathbb{P}^3\), Preprint (1995)].
quadratic Artin-Schelter regular algebras
scientific article; zbMATH DE number 869789