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On quantum groups associated to non-Noetherian regular algebras of dimension 2

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We investigate homological and ring-theoretic properties of universal quantum linear groups that coact on Artin-Schelter regular algebras A(n) of global dimension 2, especially with central homological codeterminant (or central quantum determinant). As classified by Zhang, the algebras A(n) are connected \(\mathbb {N}\)-graded algebras that are finitely generated by n indeterminants of degree 1, subject to one quadratic relation. In the case when the homological codeterminant of the coaction is trivial, we show that the quantum group of interest, defined independently by Manin and by Dubois-Violette and Launer, is Artin-Schelter regular of global dimension 3 and also skew Calabi-Yau (homologically smooth of dimension 3). For central homological codeterminant, we verify that the quantum groups are Noetherian and have finite Gelfand-Kirillov dimension precisely when the corresponding comodule algebra A(n) satisfies these properties, that is, if and only if \(n=2\). We have similar results for arbitrary homological codeterminant if we require that the quantum groups are involutory. We also establish conditions when Hopf quotients of these quantum groups, that also coact on A(n), are cocommutative.

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Acknowledgments

The authors are grateful to Julien Bichon for providing careful comments on a preliminary version of this article, and thank Alexandru Chirvasitu, Dan Rogalski, and James Zhang for helpful correspondences. The authors are also thankful for Pavel Etingof and Debashish Goswami for agreeing to allow Example 2.20 to appear in the present paper. We would also like to thank the referee for valuable comments. C. Walton was supported by the National Science Foundation grant #1550306.

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  1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Chelsea Walton & Xingting Wang

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Correspondence to Chelsea Walton.

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Walton, C., Wang, X. On quantum groups associated to non-Noetherian regular algebras of dimension 2. Math. Z. 284, 543–574 (2016). https://doi.org/10.1007/s00209-016-1666-1

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