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Automorphism groupoids in noncommutative projective geometry. (English) Zbl 1523.16031

It is well known that noncommutative algebras express fewer classical symmetries than their commutative counterparts. That is, the automorphism group of a quantum algebra is smaller than that of a polynomial ring. This is one manifestation of the principle of quantum rigidity. A standard approach to establishing further symmetries is to look for actions by Hopf algebras. The present paper takes a different approach by considering those noncommutative graded algebras associated to certain geometric data and relating a certain automorphism groupoid to its graded “Zhang” twists [J. J. Zhang, Proc. Lond. Math. Soc. (3) 72, No. 2, 281–311 (1996; Zbl 0852.16005)].
Let \(A\) be a connected graded \(k\)-algebra, \(k\) an algebraically closed field of characteristic zero. Following the philosophy of M. Artin and J. J. Zhang [Adv. Math. 109, No. 2, 228–287 (1994; Zbl 0833.14002)], we associate \(A\) to the triple \(\mathcal{A}=(\mathrm{QGr} A, \pi A, (1))\). Here \(\mathrm{QGr} A\) is the category of graded \(A\)-modules modulo the subcategory of torsion modules, \(\pi A\) is the image of \(A\) in \(\mathrm{QGr} A\), and \((1)\) is the shift functor. We say \(A\) is the homogeneous coordinate ring of a noncommutative \(\mathbb{P}^n\) if \(A\) is a noetherian domain, Artin-Schelter regular of dimension \(n+1\), generated in degree 1 and has Hilbert series \((1-t)^{-(n+1)}\). One can also specialize further and require that \(\mathcal{A}\) be geometric, that is, \(A\) is naturally associated to a certain triple \((E,\sigma,\mathcal{L})\) where \(E \subset \mathbb{P}^n\), \(\sigma \in \mathrm{Aut}(E)\), and \(\mathcal{L}\) is a very ample line bundle.
The authors consider the connected components of the category \(\mathcal{N}\mathscr{C}(\mathbb{P}^n)\). The objects of this category are triples \((\mathrm{QGr} A, \pi A, (1))\), where \(A\) is a geometric noncommutative \(\mathbb{P}^n\). The morphisms are pairs \((\mathcal{F},t)\) with \(\mathcal{F}:\mathrm{QGr} A \to \mathrm{QGr} B\) an equivalence of categories and \(t=\{t_m\}\) a family of isomorphisms \(t_m:\mathcal{F}(\pi A(m)) \to \pi B(m)\) for all \(m \in \mathbb{N}\).
Given a twisting system \(\tau\) for an algebra \(A\), the twisted algebra \(A^\tau\) has the same vector space basis as \(A\) but with multiplication deformed by \(\tau\). Two graded algebras are twist-equivalent if and only if they are graded Morita equivalent in a certain sense. The present paper gives a parameterization of the twists of \(\mathcal{A}\) up to isomorphism of the associated homogeneous coordinate ring. The authors provide several examples, including quantum affine spaces and Sklyanin algebras.

MSC:

16S38 Rings arising from noncommutative algebraic geometry
16D90 Module categories in associative algebras
16W50 Graded rings and modules (associative rings and algebras)
20G42 Quantum groups (quantized function algebras) and their representations
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