Twisting Manin’s universal quantum groups and comodule algebras.(English)Zbl 1547.16026
In this paper the authors study a categorical notion that describes when two noncommutative projective spaces have the same quantum symmetries: Two connected graded algebras finitely generated in degree one \(A\) and \(B\) are calledweakly quantum-symmetrically equivalent if there is a monoidal equivalence between the comodule categories of their associated universal quantum groups \(\textrm{comod}(\textrm{aut}(A)) \cong_\otimes \textrm{comod}(\textrm{aut}(B))\) in the sense ofY. I. Manin [Quantum groups and noncommutative geometry. 2nd edition. Cham: Springer; Montreal: Centre de Recherches Mathématiques (CRM) (2018;Zbl 1430.16001)], and they are calledquantum-symmetrically equivalent if there is such an equivalence sending \(A\) to \(B\) as comodule algebras. Some approaches to building deformations of an algebra preserving its quantum-symmetric equivalence class are presented, first through 2-cocycle twists of Hopf algebras and their comodules, and also throughJ. J. Zhang’s twists [Proc. Lond. Math. Soc. (3) 72, No. 2, 281–311 (1996;Zbl 0852.16005)]. Certain ring-theoretical and homological invariants of connected graded algebras are shown to be invariant under quantum-symmetric equivalence.
The authors also show that Koszul Artin-Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. Further results include a characterization of 2-cocycle twists of Koszul duals, of superpotentials, of superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin-Schelter regular algebras, and the fact that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists.
The authors also show that Koszul Artin-Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. Further results include a characterization of 2-cocycle twists of Koszul duals, of superpotentials, of superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin-Schelter regular algebras, and the fact that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists.
Reviewer: Sonia Natale (Córdoba)
MSC:
| 16T05 | Hopf algebras and their applications |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
universal quantum group;Morita-Takeuchi equivalence;2-cocycle twist;Artin-Schelter regular algebra;superpotential algebraCite
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