Central Extensions of Stephenson's Algebras

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This paper completes the classification of central extensions of three dimensional Artin-Schelter regular algebras to four dimensional Artin-Schelter regular algebras. Let A be an AS regular algebra of global dimension three and let D be an extension of A by a central graded element z, i.e., D/⟨z⟩ = A. If A is generated by elements of degree one, those algebras D which are again AS regular have been classified in Le Bruyn et al. (Le Bruyn L., Smith, S. P., Van den Bergh, M. (1996). Central extensions of three dimensional Artin-Schelter regular algebras. Math. Zeitschrift 222:171–212.) and Cassidy (Cassidy, T. (1999). Global dimension 4 extensions of Artin-Schelter regular algebras. J. Algebra 220:225–254.). If A is not generated by elements of degree one, then A falls under a classification due to Stephenson (Stephenson, D. R. (1996). Artin-Schelter regular algebras of global dimension three. J. Algebra 183(1):55–73 and Stephenson, D. R. (1997). Algebras associated to elliptic curves. Trans. Amer. Math. Soc. 349(6):2317–2340.). We classify the AS regular central extensions of Stephenson's algebras by proving that the regularity of D and z is equivalent to the regularity of z in low degree and this is equivalent to easily verifiable conditions on the defining relations for D.