Twisted graded algebras [J.J. Zhang]

Question

I am studying the article by James J. Zhang Twisted Graded Algebras And Equivalences of Graded Categories, I have consulted to know more about the topics they address such as twisted algebras, torsion systems and I have not found much, does anyone know where I can do it ? Or could you recommend me what to read? Thank you.

Answer 1

Let $A=\bigoplus_{i \in \mathbb{Z}} A_i$ be a $\mathbb{Z}$-graded algebra and $\sigma$ an automorphism of $A$ that respects the grading (so $\sigma(A_i) \subset A_i$). The "Zhang twist" of $A$, commonly denoted $A^\sigma$, is the same as $A$ as a vector space but with a new multiplication $\star$ defined by $a\star b = a\sigma^n(b)$ for $a \in A_n$ and $b \in A_m$. Here the multiplication on the right takes place in the original algebra $A$. What I've defined is actually a right twist. One could similarly define a left twist.

This definition goes back at least to Artin, Tate, and Van den Bergh ("Modules over regular algebras of dimension 3"), but what Zhang studied was much more general (a "twisting system"). I won't define this here, but it's a fairly straight forward definition one can get from Zhang's paper. Perhaps one of the most useful features of such these twists is that they preserve the graded module category of the algebra.

These twists show up all over the place in noncommutative (projective) algebraic geometry, in part for the reason I mentioned above. In this direction, one might also consider reading Mori's work, "Noncommutative projective schemes and point schemes", as well as a paper of Mori and Smith related to superpotential algebras, "m-Koszul Artin-Schelter regular algebras". Or one could just go to Google Scholar or MathSciNet and check out the (numerous) papers that cite Zhang's work.

A recent paper where this shows up is "Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation" by the mega-group of Huang, Nguyen, Ure, Vashaw, Veerapen, and Wang. They study twists of Hopf algebras and the related notion of 2-cocycle twists.