An Invitation to Noncommutative Algebra

Walton, Chelsea

doi:10.1007/978-3-030-19486-4_23

Cited by 4 publications

(2 citation statements)

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“…This framework works especially well when the algebras under investigation are commutative. However, for noncommutative algebras, a broader notion of symmetry is needed, and actions and coactions of bialgebras and of Hopf algebras provide a suitable framework [Dri87] [Kas95, Part I]; see also [Wal19,Section 4]. One prototypical example is when A is a (noncommutative) deformation of a commutative polynomial ring R over a base field k, and the bialgebra (or Hopf algebra) L is a corresponding deformation of the coordinate ring of an algebraic group that coacts on R. Typically the bialgebra L and the algebra A are connected graded or, more generally, augmented over k. A second motivation for this work is to establish a framework for quantum symmetry for algebras with a base algebra which is larger than the base field k. Such algebras include path algebras of quivers [ASS06, Chapter II], smash product algebras [Mon93, Chapter 4], and skew Calabi-Yau algebras [RRZ14].…”

Section: Introductionmentioning

confidence: 99%

Algebraic structures in comodule categories over weak bialgebras

Walton,

Wicks,

Won

2019

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For a bialgebra L coacting on a k-algebra A, a classical result states that A is a right L-comodule algebra if and only if A is an algebra in the monoidal category M L of right L-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras H. The category M H admits a monoidal structure by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in M H are not canonically k-algebras. Nevertheless, we prove that there is an isomorphism between the category of right H-comodule algebras and the category of algebras in M H . We also recall and introduce the formulaic notion of H coacting on a k-coalgebra and on a Frobenius k-algebra, respectively, and prove analogous category isomorphism results. Our work is inspired by the physical applications of Frobenius algebras in tensor categories and by symmetries of algebras with a base algebra larger than the ground field (e.g. path algebras). We produce examples of the latter by constructing a monoidal functor from a certain corepresentation category of a bialgebra L to the corepresentation category of a weak bialgebra built from L (a "quantum transformation groupoid"), thereby creating weak quantum symmetries from ordinary quantum symmetries.

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Section: Introductionmentioning

confidence: 99%

Algebraic structures in comodule categories over weak bialgebras

Walton,

Wicks,

Won

2019

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“…More generally, actions of Hopf algebras on k-algebras are one way to formalize the mathematical notion of quantum symmetry. One may see the following (highly incomplete) list of recent works as entry points to the extensive literature on the topic [CEW16, CKWZ16, EW16, EW17, CKWZ18, CKWZ19, Cli19, BHZ19, Neg19, CG20, EKW, DNN20, LNY20, CY20], and the survey articles [Kir16,Wal19]. In this paper we study actions of several families of Hopf algebras:…”

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confidence: 99%

Hopf actions of some quantum groups on path algebras

Kinser¹,

Oswald²

2020

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Our first collection of results parametrize (filtered) actions of a quantum Borel Uq(b) ⊂ Uq(sl2) on the path algebra of an arbitrary (finite) quiver. When q is a root of unity, we give necessary and sufficient conditions for these actions to factor through corresponding finite-dimensional quotients, generalized Taft algebras T (r, n) and small quantum groups uq(sl2).In the second part of the paper, we shift to the language of tensor categories. Here we consider a quiver path algebra equipped with an action of a Hopf algebra H to be a tensor algebra in the tensor category of representations H. Such a tensor algebra is generated by an algebra and bimodule in this tensor category. Our second collection of results describe the corresponding bimodule categories via an equivalence with categories of representations of certain explicitly described quivers with relations.

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