Basic quasi-Hopf algebras of dimension n^3

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dc.creatorGelaki, Shlomo
dc.date2004-02-10
dc.date2004-09-24
dc.date.accessioned2026-07-07T05:05:19Z
dc.date.available2026-07-07T05:05:19Z
dc.descriptionIt is shown in math.QA/0310253 that a finite dimensional quasi-Hopf algebra over the complex numbers with radical of codimension 2 is twist equivalent to a Nichols Hopf algebra, or to a lifting of one of four special quasi-Hopf algebras of dimensions 2, 8, 8, and 32. The purpose of this paper is to construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n^3, n>2, parametrized by primitive roots of unity q of order n^2, with radical of codimension n, which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in math.QA/0310253. These quasi-Hopf algebras are not twist equivalent to a Hopf algebra, and may be regarded as quasi-Hopf analogs of Taft Hopf algebras. By math.QA/0301027, our construction is equivalent to the construction of new finite tensor categories whose simple objects form a cyclic group of order n, and which are not tensor equivalent to a representation category of a Hopf algebra. In a later publication we plan to use our construction to classify finite tensor categories whose simple objects form a cyclic group of prime order n. We also prove that if H is a finite dimensional radically graded quasi-Hopf algebra with H[0]=(C[\Z/n\Z],Φ), where n is prime and Φis a nontrivial associator, such that H[1] is a free left module over H[0] of rank 1 (it is always free) then H is isomorphic to A(q).
dc.description8 pages, latex; minor corrections. The paper will appear in Journal of Pure and Applied Algebra
dc.identifierhttps://arxiv.org/abs/math/0402159
dc.identifierhttp://arxiv.org/abs/math/0402159
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70122
dc.subjectQuantum Algebra
dc.subject16W30
dc.titleBasic quasi-Hopf algebras of dimension n^3
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