## LIFTINGS OF GRADED QUASI-HOPF ALGEBRAS WITH RADICAL OF PRIME CODIMENSION (2004)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Etingof04liftingsof,

author = {Pavel Etingof and Shlomo Gelaki},

title = {LIFTINGS OF GRADED QUASI-HOPF ALGEBRAS WITH RADICAL OF PRIME CODIMENSION},

year = {2004}

}

### OpenURL

### Abstract

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. In [EG1],[EG2] we completely describe the set RG(p). Namely, we show that for p> 2, RG(p) consists of the quasi-Hopf algebras A(q) constructed in [G]

### Citations

38 |
Finite quantum groups and Cartan matrices
- Andruskiewitsch, Schneider
- 2000
(Show Context)
Citation Context ... RG(p). Namely, we show that for p > 2, RG(p) consists of the quasi-Hopf algebras A(q) constructed in [G] for each primitive root of unity q of order p 2 , the Andruskiewitsch-Schneider Hopf algebras =-=[AS1]-=-, and semisimple quasi-Hopf algebras H+(p) and H−(p) of dimension p; on the other hand, RG(2) consists of the Nichols Hopf algebras of dimension 2 n , n ≥ 1, and the special quasi-Hopf algebras H(2), ... |

30 |
Cohomology of quantum groups at roots of unity
- Ginzburg, Kumar
- 1993
(Show Context)
Citation Context ... first show that H is twist equivalent to a Hopf algebra. By Proposition 2.3, for this it is sufficient to show that H 3 (H ∗ 0 , C) = 0. If H0 is the Taft algebra then H∗ 0 ∼ = H0 and one finds (see =-=[GK]-=-), that H• (H∗ 0 , C) is the polynomial algebra on an element of degree 2. In particular, H3 (H0, C) = 0, as required. On the other hand, if H0 is the book algebra of dimension 64 then H∗ 0 is generat... |

21 |
Quasi-Hopf algebras, (Russian) Algebra i Analiz 1
- Drinfeld
- 1989
(Show Context)
Citation Context ...T for its warm hospitality. Both authors were supported by BSF grant No. 2002040. 2. Preliminaries All constructions in this paper are done over the field of complex numbers C. We refer the reader to =-=[D]-=- for the definition of a quasi-Hopf algebra and a twist of a quasi-Hopf algebra. 2.1. The quasi-Hopf algebras A(q). Theorem 2.1. [G] Let n ≥ 2 be an integer. There exist n3−dimensional quasi-Hopf alge... |

16 | Integral theory for quasi-Hopf algebras - Hausser, Nill - 1999 |

10 | Finite quantum groups over abelian groups of prime exponent
- Andruskiewitsch, Schneider
(Show Context)
Citation Context ...gebra of dimension 2 n , while for p > 2 it is the dual to a finite dimensional pointed Hopf algebra with group of grouplike elements Zp. Such Hopf algebras are completely classified for p ̸= 5, 7 in =-=[AS2]-=- (see Remark 1.10(v), [M]). Note also that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebr... |

6 |
Finite-dimensional quasi-Hopf algebras with radical of codimension 2
- Etingof, Gelaki
(Show Context)
Citation Context ...OMO GELAKI 1. Introduction Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. In =-=[EG1]-=-,[EG2] we completely describe the set RG(p). Namely, we show that for p > 2, RG(p) consists of the quasi-Hopf algebras A(q) constructed in [G] for each primitive root of unity q of order p 2 , the And... |

5 |
An analogue of Radford’s S4 formula for finite tensor categories
- Etingof, Nikshych, et al.
- 2004
(Show Context)
Citation Context ...lgebra automorphisms of A; it is a linear algebraic group. Let S0 be the antipode of A0. Then S2n 0 = Ad(a). Hence the class [S2n] of S2n is unipotent in G. On the other hand, from the main result of =-=[ENO]-=- (see also [HN]) we know that on the category of representations of A there exists an isomorphism of tensor functors θV : V → χ ⊗V ∗∗∗∗ ⊗ χ−1 , where χ is a 1−dimensional representation of A. In our c... |

3 |
Basic quasi-Hopf algebras of dimension n3
- Gelaki
(Show Context)
Citation Context ...pf algebras over C, whose radical has codimension p. In [EG1],[EG2] we completely describe the set RG(p). Namely, we show that for p > 2, RG(p) consists of the quasi-Hopf algebras A(q) constructed in =-=[G]-=- for each primitive root of unity q of order p 2 , the Andruskiewitsch-Schneider Hopf algebras [AS1], and semisimple quasi-Hopf algebras H+(p) and H−(p) of dimension p; on the other hand, RG(2) consis... |

3 |
Finite quantum groups and pointed Hopf algebras. Quantum groups and Lie theory
- Musson
- 1999
(Show Context)
Citation Context ...ile for p > 2 it is the dual to a finite dimensional pointed Hopf algebra with group of grouplike elements Zp. Such Hopf algebras are completely classified for p ̸= 5, 7 in [AS2] (see Remark 1.10(v), =-=[M]-=-). Note also that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this p... |

1 | On radically graded finite dimensional quasi-Hopf algebras, submitted - Etingof, Gelaki |