## COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE (2002)

Citations: | 3 - 3 self |

### BibTeX

@MISC{Grunenfelder02cohomologyof,

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

title = {COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE},

year = {2002}

}

### OpenURL

### Abstract

Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations. In this paper we discuss various cohomology theories for Hopf algebras and their relation to extension theory. It is natural to think of building new algebraic objects from simpler structures, or to get information about the structure of complicated objects by

### Citations

481 | Homological Algebra - Cartan, Eilenberg - 1956 |

374 |
Quantum Groups
- Kassel
(Show Context)
Citation Context ...hort ˜σ(t ⊗n) = t1(n1) ⊗t n2 2 , (n ⊗t)(m ⊗s) = nt1(m1) ⊗t m2 2 s, and antipode S = ˜σ(S ⊗ S)σ: N ⊗ T → N ⊗ T, i.e: S(n ⊗ t) = S(t2)(S(n2)) ⊗ S(t1) S(n1) . For a proof that this is a Hopf algebra see =-=[Kas]-=-. To avoid ambiguity we will often write n ⊲⊳ t for n ⊗ t in N ⊲⊳ T. We also identify N and T with the Hopf subalgebras N ⊲⊳ k and k ⊲⊳ T, respectively, i.e: n ≡ n ⊲⊳ 1 and t ≡ 1 ⊲⊳ t. In this sense w... |

269 | On the structure of Hopf algebras - Milnor, Moore - 1965 |

267 |
Introduction to homological algebra
- Weibel
- 1994
(Show Context)
Citation Context ...ns and results from simplicial homological algebra used in the main text. The emphasis is on the cohomology of cosimplicial objects, but the considerations are similar to those in the simplicial case =-=[We]-=-. A.1. Simplicial and cosimplicial objects. Let ∆ denote the simplicial category [Mc]. If A is a category then the functor category A∆op is the category of simplicial objects while A∆ is the category ... |

100 | Graded manifolds, graded Lie theory, and prequantization, in: “Differential geometrical methods in mathematical physics - Kostant - 1977 |

50 |
Matched pairs of groups and bismash products of Hopf algebras
- Takeuchi
- 1981
(Show Context)
Citation Context ...s as in [Sch].COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE 3 Together with the five term exact sequence for a smash product of Hopf algebras H = N ⋊T [M2], generalizing that of K. Tahara =-=[Ta]-=- for a semi-direct product of groups, 1 → H 1 meas (T, Hom(N, A)) → ˜ H 2 (H, A) → H 2 (N, A) T → H 2 meas (T, Hom(N, A)) → ˜ H 3 (H, A) it is possible in principle to give a procedure to compute the ... |

44 |
Hopf Algebras and their
- Montgomery
- 1993
(Show Context)
Citation Context ...d pairs to Singer pairs. For more details we refer to [Ma3]. Definition 4.1. We say that an action µ: A ⊗ M → M is locally finite, if every orbit A(m) = {a(m)|a ∈ A} is finite dimensional. Lemma 4.2 (=-=[Mo1]-=-, Lemma 1.6.4). Let A be an algebra and C a coalgebra. (1) If M is a right C-comodule via ρ: M → M ⊗C, ρ(m) = m0 ⊗m1, then M is a left C ∗ -module via µ: C ∗ ⊗ M → M, µ(f ⊗ m) = f(m1)m0. (2) Let M be ... |

35 |
Cohomology of algebras over Hopf algebras
- Sweedler
- 1968
(Show Context)
Citation Context ...ND M. MASTNAK In view of these facts it appears natural to try to relate the cohomology of Hopf algebras to that of groups and Lie algebras. The first work in this direction was done by M.E. Sweedler =-=[Sw1]-=- and by G.I. Kac [Kac] in the late 1960’s. Sweedler introduced a cohomology theory of algebras that are modules over a Hopf algebra (now called Sweedler cohomology). He compared it to group cohomology... |

27 |
Extenions of groups to ring groups
- Kac
(Show Context)
Citation Context ...of these facts it appears natural to try to relate the cohomology of Hopf algebras to that of groups and Lie algebras. The first work in this direction was done by M.E. Sweedler [Sw1] and by G.I. Kac =-=[Kac]-=- in the late 1960’s. Sweedler introduced a cohomology theory of algebras that are modules over a Hopf algebra (now called Sweedler cohomology). He compared it to group cohomology, to Lie algebra cohom... |

11 | Families parametrized by coalgebras - Grunenfelder, Paré - 1987 |

11 | Extension theory for connected Hopf algebras - Singer - 1970 |

8 |
Calculations of Some Groups of Hopf algebra Extensions
- Masuoka
- 1997
(Show Context)
Citation Context ... the field of complex numbers and also carry the structure of a C ∗ - algebra. Such structures are now called Kac algebras. The generalization to arbitrary fields appears in recent work by A. Masuoka =-=[Ma1,2]-=-, where it is also used to show that certain groups of Hopf algebra extensions are trivial. Masuoka also obtained a version of the Kac sequence for matched pairs of Lie bialgebras [Ma3], as well as a ... |

6 |
Über die Struktur von Hopf-Algebren
- Grünenfelder
- 1969
(Show Context)
Citation Context ...r of pointed Hopf algebras, and A a trivial N ⊲⊳ T-module algebra. (1) Since chark = 0 and T and N are pointed we have T ≃ UP(T)⋊kG(T) and N ≃ UP(N)⋊kG(N) and N ⊲⊳ T ≃ U(P(T) ⊲⊳ P(N))⋊k(G(T) ⊲⊳ G(N)) =-=[Gr1,2]-=-. If H is a Hopf algebra then G(H) denotes the group of points and P(H) denotes the Lie algebra of primitives. (2) We can use the generalized Tahara sequence [M2] (see introduction) to compute H2 (T),... |

6 | Classifying finite dimensional semisimple Hopf algebras - Montgomery - 1998 |

4 |
algebra extensions and cohomology, in ”New directions in Hopf algebras
- Masuoka, Hopf
(Show Context)
Citation Context ...version of the Kac sequence for matched pairs of Lie bialgebras [Ma3], as well as a new exact sequence involving the group of quasi Hopf algebra extensions of a finite dimensional abelian Singer pair =-=[Ma4]-=-. In this paper we introduce a cohomology theory for general abelian matched pairs (T, N, µ, ν), consisting of two cocommutative Hopf algebras acting compatibly on each other with bismash product H = ... |

3 |
Composite Cotriples and Derived Functors
- Barr
- 1969
(Show Context)
Citation Context ... by GT ˜ = ( ˜ GT,δT,ǫT) , GN ˜ = ( ˜ GN, δN, ǫN) with GT ˜ = FT ˜ ˜ UT, δT(t ⊗x) = t ⊗1⊗x, ǫT(t ⊗x) = tx, and with GN ˜ = FN ˜ ˜ UN, δN(n ⊗ x) = n ⊗ 1 ⊗ x, ǫN(n ⊗ x) = nx, satisfy a distributive law =-=[Ba]-=- GT ˜ GN → ˜ GN ˜ GT ˜σ: ˜ given by ˜σ(t ⊗ n ⊗ −) = ˜σ(t ⊗ n) ⊗ − = t1(n1) ⊗ t n2 2 distributive law and ⊗ −. The equations for a GNδT ˜ · ˜σ = ˜σ ˜ GT · ˜ GT ˜σ · δT ˜ GN , δN ˜ GT · ˜σ = ˜ GN ˜σ · ˜... |

3 |
Hopf algebra extensions arising from semi-direct products
- Mastnak
(Show Context)
Citation Context ...rpretation when i = 2 and p = 1, 2; see Section 4.3. 3. The homomorphism π: H 2 (T, N, A) → H i,j (T, N, A) If T is a finite group and N is a finite T-group, then we have the following exact sequence =-=[M1]-=- H 2 (N, k • ) δT → Opext(kT, k N ) π → H 1 (T, H 2 (N, k • )). Here we define a version of homomorphism π for arbitrary smash products of cocommutative Hopf algebras. We start by introducing the Hopf... |

3 |
Extensions of Hopf Algebras and Lie
- Masuoka
(Show Context)
Citation Context ...y A. Masuoka [Ma1,2], where it is also used to show that certain groups of Hopf algebra extensions are trivial. Masuoka also obtained a version of the Kac sequence for matched pairs of Lie bialgebras =-=[Ma3]-=-, as well as a new exact sequence involving the group of quasi Hopf algebra extensions of a finite dimensional abelian Singer pair [Ma4]. In this paper we introduce a cohomology theory for general abe... |

2 | Groupes algébriques et groupes formels, in “Colloque sur la théorie des groupes algébriques” (Bruxelles - Cartier - 1962 |

2 | Cohomologies over commutative Hopf algebras - Doi - 1975 |

2 | On the cohomology of a smash product of Hopf algebras, preprint
- Mastnak
- 2002
(Show Context)
Citation Context ...st of groups and not just pointed sets as in [Sch].COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE 3 Together with the five term exact sequence for a smash product of Hopf algebras H = N ⋊T =-=[M2]-=-, generalizing that of K. Tahara [Ta] for a semi-direct product of groups, 1 → H 1 meas (T, Hom(N, A)) → ˜ H 2 (H, A) → H 2 (N, A) T → H 2 meas (T, Hom(N, A)) → ˜ H 3 (H, A) it is possible in principl... |

1 | Hopf Algebren und Coradikal - Grunenfelder - 1970 |

1 |
Extensions of Hopf Algebras and their
- Hofstetter
- 1994
(Show Context)
Citation Context ...t a natural isomorphism H ∗ (T, N, k) ∼ = H ∗ (N, T ∗ ) between the cohomology of the abelian matched pair and that of the corresponding abelian Singer pair. In particular, together with results from =-=[Ho]-=- one obtains H 1 (T, N, k) ∼ = H 1 (N, T ∗ ) ∼ = Aut(T ∗ #N) and H 2 (T, N, k) ∼ = H 2 (N, T ∗ ) ∼ = Opext(N, T ∗ ). The sequence gives information about extensions of cocommutative Hopf algebras by c... |

1 |
On Hopf Algebra Extensions and Cohomologies
- Mastnak
- 2002
(Show Context)
Citation Context .... If (T, N, µ, ν) is an abelian matched pair with µ locally finite then the quadruple (N, T ◦ , ν ′ , ρ) forms an abelian Singer pair. Remark. There is also a correspondence in the opposite direction =-=[M3]-=-. 4.2. Comparison of Singer and matched pair cohomologies. Let (T, N, µ, ν) be an abelian matched pair of Hopf algebras, with µ locally finite and (N, T ◦ , ν ′ , ρ) the Singer pair associated to it a... |

1 | Extensions of Hopf Algebras, notes by M. Graña - Masuoka - 1997 |

1 |
algebra extensions and monoidal categories, New directions
- Schauenburg, Hopf
(Show Context)
Citation Context ...(low degree) cohomology groups a Hopf algebras. Such a sequence can of course not exist for non-abelian matched pairs, at least if the sequence is to consist of groups and not just pointed sets as in =-=[Sch]-=-.COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE 3 Together with the five term exact sequence for a smash product of Hopf algebras H = N ⋊T [M2], generalizing that of K. Tahara [Ta] for a se... |