Embedding a Quantum Nonsingular Quadric in a Quantum image3 Original Research Article

A definition of regularity has been given for non-commutative graded algebras and results of Artin, Schelter, Tate, and Van den Bergh classify the regular algebras of global dimension three that are generated by degree one elements. Our purpose is to classify a certain class of quadratic regular algebras of global dimension four. LetSbe a twisted homogeneous coordinate ring of a nonsingular quadricQ subset of image3. Our interest is in algebrasRsuch that there is an embedding Proj S right arrow-hooked Proj R. In this paper, we classify all the quadratic regular algebrasRof global dimension four which have the same Hilbert series as that of the polynomial ring on four variables, and which map ontoSvia a graded degree zero homomorphism. Our approach makes use of the point modules ofRand their associated geometric data. We classify the algebrasRaccording to their “point scheme”Pand corresponding automorphism σ set membership, variant Aut(P); those algebrasRwhich are determined by (P, σ) belong to at most a five-parameter family, but those which are not determined by (P, σ) belong to at most a four-parameter family. In the first case,Pis either image3or consists ofQtogether with a lineL, while in the second caseP = Q. It is also proved that under certain sufficient conditions, the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism.