## Pointed and copointed Hopf algebras as cocycle deformations,” arxiv:0709.0120

Citations: | 8 - 2 self |

### BibTeX

@MISC{Grunenfelder_pointedand,

author = {L. Grunenfelder and M. Mastnak},

title = {Pointed and copointed Hopf algebras as cocycle deformations,” arxiv:0709.0120},

year = {}

}

### OpenURL

### Abstract

Abstract. We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of a result of Masuoka about Morita-Takeuchi equivalence and of Schauenburg about Hopf Galois extensions. The “infinitesimal ” part of the deforming cocycle and of the deformation determine the deformed multiplication and can be described explicitly in terms of Hochschild cohomology. Applications to, and results for copointed Hopf algebras are also considered. Finite dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero, particularly when the group of points is abelian, have been studied quite extensively with various methods in [AS, BDG, Gr1, Mu]. The most far reaching results as yet in this area have been obtained in [AS], where a large

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Citation Context ...ings’ can be viewed as cocycle deformations with convolution invertible coalgebra cocycles, which again can be discussed in a formal setting. In both cases the deformations are formal in the sense of =-=[GS]-=-, the infinitesimal parts of these deformations determine the deformed multiplication and comultiplication, respectively, and are determined by the G-invariant part of the Hochschild cohomology of B(V... |

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10 | Finite quantum groups over abelian groups of prime exponent
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Citation Context ... ≤ i ̸= j ≤ θ) define root vectors xα ∈ A(V ) for α ∈ Φ + by the same iterated braided commutators of the elements x1, x2, . . .,xθ as in Lusztig’s case but with respect to the general braiding. (See =-=[AS2]-=-, and the inductive definition of root vectors in [Ri] or also [CP, Section 8.1 and Appendix].) Let K(D) be the subalgebra of R(D) generated by { xN ∣ ∣α α ∈ Φ +}. Theorem 3.2. [AS, Theorem 2.6] Let D... |

10 | Lifting of Nichols algebras of type B2
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Citation Context ...s of the connection between ‘liftings’ and cocycle deformations are considered. After we had posted our paper A. Masuoka informed us that his Theorem 2 of [Ma] was missing a condition, as observed in =-=[BDR]-=-. For the verification of this additional condition, needed in our 3.5, we refer to the second version of [Ma2], where it now appears as an appendix. 1. Regularity and coregularity An algebra A is (Vo... |

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Citation Context ...a(g r − 1), y r = b(g r − 1) and comultiplication ∆(g) = g ⊗ g, ∆x = x ⊗ 1 + g ⊗ x, ∆(y) = y ⊗ 1 + g ⊗ y. If c ̸= 0 then G ′ = 〈 g r , g 2〉 = G and hence G(H ∗ ) ∼ = ̂ G/G ′ = {ε}. 1. The examples in =-=[BDG]-=- are of that form. If p is an odd prime number and q is a primitive p-th root of unity, then the Hopf algebra defined by generators g, x, y, relations g p2 = 1, gx = qxg, gy = q −1 yq, x p = a(g p − 1... |

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Citation Context ...e find a description of these lifted Hopf algebras, which is suitable for the application of a result of Masuoka about Morita-Takeuchi equivalence [Ma] and of Schauenburg about Hopf Galois extensions =-=[Sch]-=-, to prove that all liftings of a given H(V ) in this class are cocycle deformations of each other. As a result we see here that in the class of finite dimensional pointed Hopf algebras classified by ... |

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Citation Context ... deformations. Similar arguments work for liftings [DCY], but now the linear maps µr, δr and fr are of degree −r. □POINTED AND COPOINTED HOPF ALGEBRAS AS COCYCLE DEFORMATIONS 35 Lemma 5.3 (cf. [GS], =-=[MW]-=-). If ζ : A ⊗ A → k is a Hochschild cocycle of degree -n then the linear map µ = (ζ ⊗ m − m ⊗ ζ)∆A⊗A : A ⊗ A → A is of degree −n and satisfies the cocycle condition m(µ ⊗ 1) + µ(m ⊗ 1) = m(1 ⊗ µ) + µ(... |

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Citation Context ...re crossed H-modules, where adj(h ⊗ h ′ ) = h1h ′ S(h2) and coadj(h) = h1S(h3) ⊗ h2. 2.4. The pushout construction for bi-cross products. Recall Masuoka’s pushout construction for Hopf algebras [Ma], =-=[Gr1]-=-. If A is a Hopf algebra then Alg(A, k) s a group under convolution which acts on A by conjugation as Hopf algebra automorphisms. Lemma 2.2. For every Hopf algebra A the group Alg(A, k) acts on A by c... |

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Citation Context ...e deformations of each other. The special case of quantum linear spaces has been studied by Masuoka [Ma], and that of a crossed kG-module corresponding to a finite number of copies of type An by Didt =-=[Di]-=-. Theorem 4.3. Let G be a finite abelian group, V a crossed kG-module of special finite Cartan type, B(V ) its Nichols algebra with bosonization A = B(V )#kG. Then: (1) All liftings of A are monoidall... |

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Citation Context ...k[[t]]/(tl+1 ) → k[[t]]/(tl induces a restriction map resl: Defl+1(A) → Defl(A) and Def(A) = lim ← Defl(A) is the set of isomorphism classes of formal deformations (∞-deformations) of A. Theorem 5.1. =-=[Gr2]-=- The restriction map resl : Defl+1(A) → Defl(A) fits into an exact sequence of pointed sets H 2 (A, A) −−−−→ Defl+1(A) for l ≥ 0. In particular: resl −−−−→ Defl(A) obsl −−−−→ H 3 (A, A) • H 2 (A, A) ∼... |

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Citation Context ...co-filtered bialgebra structure G = (A, M, ∆) such that grr(G) ∼ = A. Let Lift(A) and co-Lift(A) be the sets of equivalence classes of liftings and of co-liftings of A, respectively. Theorem 5.2 (cf. =-=[DCY]-=-). There are bijections Lift(A) ∼ = Def(A) ∼ = co-Lift(A). Proof. We deal with the co-lifting part of the theorem. Let G = (A, M, ∆) be a co-lifting of the graded bialgebra H, so that grr G = H. The m... |

1 | Abelian and non-abelian second cohomo;ogy of quantized enveloping algebras, arXiv:math.QA/0708.1982 v1 - Masuoka - 2007 |

1 | The Coradical filtration of Uq(g) at roots of unity - Mueller - 2000 |

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Citation Context ...s over all cofinite ideals of A. If A is regular and I is a cofinite ideal then A/I is semisimple and (A/I) ∗ is cosemisimple, so that Cor(A◦ ) = A◦ . Assertion (d) also appears as Proposition 3.2 in =-=[Cu]-=-. □ Lemma 1.4. If A is a Von Neumann regular subring of the ring B then A ∩ RadB = 0. Proof. If a ∈ A ∩ Rad B then a = axa has a solution in A, say x = a ′ , since A is Von Neumann regular, and 1 − a ... |