Segre Product of Artin–Schelter Regular Algebras of Dimension 2 and Embeddings in Quantum image3ʹs Original Research Article

The Segre embedding of image1×image1as a smooth quadricQin image3corresponds to the surjection of the four-dimensional polynomial ring onto the Segre productSof two copies of the homogeneous coordinate ring of image1. We study Segre products of noncommutative algebras. If in particularAandBare two copies of a quantum image1thenS=circled plusi(Aicircle times operatorkBi) is a twisted homogeneous coordinate ring of the quadricQ. The main result of this paper is the classification of all embeddings of the Segre product of two quantum planes into so-called quantum image3ʹs. These are (the Proj of) Artin–Schelter regular algebrasRof global dimension four with the Hilbert series of a commutative polynomial ring and which map ontoS. IfRis not a twist of a polynomial ring, then the point scheme ofReither is the union of the quadricQwith a line or is only the quadricQ. In the first case,Ris a central extension of a three-dimensional Artin–Schelter regular algebra and a twist of an algebra mapping onto the (commutative) homogeneous coordinate ring ofQ; in the second case, such an algebraRis the first known example of a four-dimensional Artin–Schelter regular algebra which is not determined by its point scheme.