Twisting Systems and some Quantum P³s with Point Scheme a Rank-2 Quadric

Abstract

In 1996, J. J. Zhang introduced the concept of twisting a graded algebra by a twisting system, which generalizes the concept of twisting a graded algebra by an automorphism (the latter concept having been introduced in an article by M. Artin, J. Tate and M. Van den Bergh in 1991). Twisting using a twisting system is an equivalence relation and certain important algebraic properties of the original algebra are carried over to the twisted algebra. We call a twisting system nontrivial if it is not given by an automorphism. However, there are very few known examples of nontrivial twisting systems in the literature. M. Vancliff and K. Van Rompay in 1997, and B. Shelton and M. Vancliff in 1999, were successful in finding one example each of a nontrivial twisting system. Their twisting systems were constructed on certain quadratic algebras A (on four generators) using two invertible linear maps t and τ that satisfy t² = identity and τ² ∈ Aut(A). We extend their work on twisting systems using analogous maps that satisfy tn = identity and τn ∈ Aut(B), where n ∈ N and B is any finitely generated quadratic algebra. We illustrate our new method for producing a nontrivial twisting system on an algebra that s a quadrat quantum P³ whose point scheme is given by a rank-2 quadric in P³. Such an algebra is a new addition to the literature.