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). Zhang proved that twisting using a twisting system is an equivalence relation and that certain important algebraic properties are invariant under twisting. We call a twisting system nontrivial if it is not given by an automorphism. However, there are very few examples of nontrivial twisting systems in the literature.
In 1997, the second author and K. Van Rompay and, in 1999, B. Shelton and the second author were successful in finding one example each of a nontrivial twisting system. Their twisting systems were constructed on certain quadratic algebras
A
A
(on four generators) using two invertible linear maps
t
t
and
τ
\tau
of
A
1
A_1
that satisfy
t
2
=
t^2 =
identity and
τ
2
∈
Aut
(
A
)
\tau ^2 \in \text {Aut}(A)
. We extend their work on twisting systems to any finitely generated quadratic algebra
B
B
using analogous maps that satisfy
t
n
=
t^n =
identity and
τ
n
∈
Aut
(
B
)
\tau ^n \in \text {Aut}(B)
, for some
n
∈
N
n \in \mathbb {N}
.
We illustrate our new method for producing a nontrivial twisting system on a certain quadratic algebra that is a quantum
P
3
\mathbb {P}^3
and whose point scheme is isomorphic to a rank-2 quadric in
P
3
\mathbb {P}^3
. We prove that our algebra is not determined by the zero locus of its defining relations.