## LOCALIZATIONS FOR CONSTRUCTION OF QUANTUM COSET SPACES (2003)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Skoda03localizationsfor,

author = {Zoran Skoda},

title = {LOCALIZATIONS FOR CONSTRUCTION OF QUANTUM COSET SPACES},

year = {2003}

}

### OpenURL

### Abstract

Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localizations on the categories of Hopf modules. Their interplay with the functor of taking coinvariants and its left adjoint is stressed out. Using localization approach, we constructed a natural class of examples of quantum coset spaces, related to the quantum flag varieties of type A of other authors. Noncommutative Gauss decomposition via quasideterminants reveals a new structure in noncommutative matrix bialgebras. In the quantum case, calculations with quantum minors yield the structure theorems. Notation. Ground field is k and we assume it is of characteristic zero. If we deal just with one k-Hopf algebra, say B, the comultiplication is ∆ : B → B ⊗ B, unit map η: k → B, counit ǫ: B → k, multiplication µ: B ⊗ B → B, and antipode (coinverse) is

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Citation Context ...uction the exactness properties of the functors and of the comonad play the decisive role; the description of objects obtained by gluing local data depends on the applicability of Barr-Beck’s theorem =-=[3, 4, 2, 40, 47, 48]-=-. In particular, this is suitable for descent-typeLOCALIZATIONS FOR COSET SPACES 9 questions, and the construction of quotients can also be understood that way. We say that a family T −1 µ R of Ore l... |

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Citation Context ... linear systems of equations in noncommutative variables. The main tool to do calculations with such matrix inverses and perform Gauss-type decompositions are quasideterminants of Gel’fand and Retakh =-=[22, 41, 23, 24, 76]-=-. Let A ∈ Mn(R) be a n×n matrix over an arbitrary noncommutative unital associative ring R. Suppose rows and columns of A are labeled. Let us choose a row label i and a column label j. By Aî ˆj we’ll ... |

49 |
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Citation Context ...ed algebras Wσ are also big, what enables detailed comparison of the two approaches. Quantum flag varieties can be defined and studied using the representation theory of quantized enveloping algebras =-=[82, 39, 42, 45, 35]-=- or by an Ansatz exploring quantum minors [86]. In these approaches, the quantum flag varieties are described by a noncommutative graded ring Fq, or by an appropriate quotient of a category of (multi)... |

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Citation Context ... δ (θ(δ > γ) − θ(α > β)) where θ(true) = 1, θ(false) = 0, and [, ] stands for the ordinary commutator. Mq(n,k) is a domain. There are other versions, including the multiparametric case MP,Q(n,k), cf. =-=[51, 85, 76]-=-. Some of the results below generalize to MP,Q(n,k). The quantum determinant D ∈ Mq(n,k) is defined by any of the formulas D = ∑ (−q) l(σ)−l(τ) t τ(1) σ∈Σ(n) σ(1) tτ(2) σ(2) ∑ · · ·tτ(n) σ(n) = (−q) σ... |

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Citation Context ...bra, quantum group, quantum principal bundle, localization, Ore set, matrix bialgebra, noncommutative Gauss decomposition. The paper is in final form and no version of it will be published elsewhere. =-=[1]-=-2 ZORAN ˇ SKODA 1. Introduction. There is an antiequivalence [32, 74] between the category of affine group schemes over k and the category of commutative Hopf algebras over k. In the framework of aff... |

41 | Strong Connections on Quantum Principal Bundles
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Citation Context ...approaches to quantum principal bundles without localization, noncommutative analogues of the differential calculus and of the connections on fibre bundles were considered in many earlier works, e.g. =-=[7, 63, 29]-=-. V. Lunts and A. Rosenberg [44, 43] properly extended the Grothendieck’s definition [27] of the rings of regular differential operators to noncommutative rings and considered extensibility of Hopf mo... |

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Citation Context ...uction the exactness properties of the functors and of the comonad play the decisive role; the description of objects obtained by gluing local data depends on the applicability of Barr-Beck’s theorem =-=[3, 4, 2, 40, 47, 48]-=-. In particular, this is suitable for descent-typeLOCALIZATIONS FOR COSET SPACES 9 questions, and the construction of quotients can also be understood that way. We say that a family T −1 µ R of Ore l... |

40 | Minor Identities for Quasi-Determinants and Quantum
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Citation Context ... linear systems of equations in noncommutative variables. The main tool to do calculations with such matrix inverses and perform Gauss-type decompositions are quasideterminants of Gel’fand and Retakh =-=[22, 41, 23, 24, 76]-=-. Let A ∈ Mn(R) be a n×n matrix over an arbitrary noncommutative unital associative ring R. Suppose rows and columns of A are labeled. Let us choose a row label i and a column label j. By Aî ˆj we’ll ... |

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Citation Context ...trivial Ore sets exist, so called schematic algebras [91, 89]. 2 nd method centers on a comonad [69] associated to given flat cover, to place it transparently into the general picture of flat descent =-=[3, 12, 28, 70]-=-. Then one associated a cosimplicial object [69] to the comonad. When applying various functors to this construction the exactness properties of the functors and of the comonad play the decisive role;... |

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Citation Context ...one of the several standard topologies for schemes. Local triviality in Zariski topology is the strongest requirement, and the local triviality in étale, fppf, fpqc topology are weaker, in that order =-=[14, 53]-=-. A principal bundle locally trivial in étale topology is often called a torsor. If the orbit space is denoted by X, one can replace B by trivial B-bundle B over X (i.e. by the product B ×X). Then B i... |

22 |
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Citation Context ...trivial Ore sets exist, so called schematic algebras [91, 89]. 2 nd method centers on a comonad [69] associated to given flat cover, to place it transparently into the general picture of flat descent =-=[3, 12, 28, 70]-=-. Then one associated a cosimplicial object [69] to the comonad. When applying various functors to this construction the exactness properties of the functors and of the comonad play the decisive role;... |

22 |
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Citation Context ...rther progress it is promising to rely on simplicial and cohomological methods; the ideas of cohomological descent [13] and the noncommutative Čech cohomologies [69, 92] provide a framework. Cf. also =-=[3, 6, 12, 25, 33, 45, 54, 55, 55]-=-. Very recently, V. Lunts suggested that it might be useful to consider more general flat covers and resolutions of would-be quotient space than the covers coming from localizations, and to use the co... |

22 |
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Citation Context ...rther progress it is promising to rely on simplicial and cohomological methods; the ideas of cohomological descent [13] and the noncommutative Čech cohomologies [69, 92] provide a framework. Cf. also =-=[3, 6, 12, 25, 33, 45, 54, 55, 55]-=-. Very recently, V. Lunts suggested that it might be useful to consider more general flat covers and resolutions of would-be quotient space than the covers coming from localizations, and to use the co... |

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Citation Context ... map, and were fk commute with structure morphisms. Composition of morphisms is defined componentwise. This construction is an analogue of a category of simplicial sheaves over a simplicial space cf. =-=[6, 13, 33, 54]-=-. Functor M ↦→ (N•, b) defined by N µν... = (Mµν...) coB and b = id defines a functor EMB → EM(U) coB . Functor M ↦→ (N•, b) defined by N µν... = Mµν... from EMB to EMB (U) is, however, an equivalence... |

10 |
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9 |
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Citation Context ...iel localizations appeared in [72](Appendix 2). Sheaf-theoretic ideas and localization were present much earlier in noncommutative geometry, namely in the study of sheaves over noncommutative spectra =-=[57, 90, 26]-=-. If q is a primitive root of unity, SLq(n, C) has a large center. This enables an alternative approach [67] to SLq(n, C) using small noncommutative sheaves over ordinary30 ZORAN ˇ SKODA SL(n, C). In... |

9 |
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Citation Context ...action of two flag quasideterminants u1u −1 2 and observe that ρσ(u1) = u1 ⊗ b and ρσ(u2) = u2 ⊗ b for the same b ∈ N B(n,k), and conclude that u1u −1 2 is a coinvariant. ✷ 14. Quantum matrix groups. =-=[62, 50, 49, 38, 31, 66]-=- Let q ∈ k, q ̸= 0. The quantum matrix bialgebra Mq(n,k) = O(Mq(n,k)) is the free matrix bialgebra N M(n,k) with basis T = (tα β ) modulo the smallest biideal I such that the following relations hold ... |

8 |
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Citation Context ...me good properties. A common generality in which this is studied are Grothendieck categories. A Grothendieck category A is an Abelian category which is cocomplete (small inductive limits always exist =-=[4, 47, 90]-=-), where filtered limits are exact, and which posses a generator (object G in A such that C ↦→ HomC(C, G) is a faithful functor), for details cf. [34, 64, 83, 90, 81]. Such categories are a natural pl... |

8 | Linear algebraic groups, 2nd edition, GTM 126 - Borel - 1991 |

8 |
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Citation Context ... without localization, noncommutative analogues of the differential calculus and of the connections on fibre bundles were considered in many earlier works, e.g. [7, 63, 29]. V. Lunts and A. Rosenberg =-=[44, 43]-=- properly extended the Grothendieck’s definition [27] of the rings of regular differential operators to noncommutative rings and considered extensibility of Hopf module algebra actions given by the re... |

6 |
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Citation Context ...opology” generated by Ore localizations would mean that the geometric projection is not continuous in the obvious sense, even in otherwise well–behaved noncommutative examples. 11. Cohn localization. =-=[9, 10]-=- Let R be a possibly noncommutative ring, and Σ a given set of square matrices of possibly different sizes with entries in R. Map f : R → S of rings is Σ-inverting if each matrix in Σ is mapped to an ... |

6 |
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Citation Context ...les over different local charts. For that purpose, a version of gauge transformations of [7] will play role. For a rather different concept of local triviality of noncommutative principal bundles see =-=[30]-=- and references therein. An early notion of locally trivial vector bundle using Gabriel localizations appeared in [72](Appendix 2). Sheaf-theoretic ideas and localization were present much earlier in ... |

5 |
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Citation Context ...uitable set of generators of a ring versus a suitable set of generators of the Ore set. One often uses induction arguments, recursively satisfying the Ore condition. 4. Ore vs. Gabriel localizations. =-=[72, 34]-=- This section could be skipped in first reading as only few remarks in the paper depend on it. The modern viewpoint on localization as touched upon here is however essential for the current research i... |

4 |
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(Show Context)
Citation Context ...ing several reductions to special cases and an induction using rather nontrivial commutation relations involving quantum minors. “Strong” denotes a usually imposed condition on deformation parameters =-=[1, 15, 76]-=-. From now on, G will denote either GLq(n,k) or SLq(n,k). Quantum Borel subgroup B = Bq(n,k) is the quotient of G by the biideal I with i < j. I is a Hopf ideal, hence B is a Hopf algebra. generated b... |

4 |
Differential operators on noncommutative
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Citation Context ... without localization, noncommutative analogues of the differential calculus and of the connections on fibre bundles were considered in many earlier works, e.g. [7, 63, 29]. V. Lunts and A. Rosenberg =-=[44, 43]-=- properly extended the Grothendieck’s definition [27] of the rings of regular differential operators to noncommutative rings and considered extensibility of Hopf module algebra actions given by the re... |

3 | Coherent states for quantum compact groups
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Citation Context ...mov coherent states can be generalized for Hopf algebras [76, 79] in the framework of this paper. Earlier a generalization We believe that in the special case of quantum groups the coherent states of =-=[36]-=- essentially coincide with our construction. Under rather general assumptions a Hopf algebraic analogue of a classical resolution of unity by coherent states has been proved by author in 1999 [79]. In... |

3 |
Quantum flag and Schubert schemes, in: Deformation Theory and Quantum Groups with Applications to
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Citation Context ...ed algebras Wσ are also big, what enables detailed comparison of the two approaches. Quantum flag varieties can be defined and studied using the representation theory of quantized enveloping algebras =-=[82, 39, 42, 45, 35]-=- or by an Ansatz exploring quantum minors [86]. In these approaches, the quantum flag varieties are described by a noncommutative graded ring Fq, or by an appropriate quotient of a category of (multi)... |

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2 |
Classifying spaces, classifying topoi, LNM 1414
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Citation Context |

1 |
Kol’ca častnyh (Russian), Algebra i logika
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Citation Context ... elementary. That is the case of inverting all elements belonging to a given subset S in the ring R of special kind, so called Ore set. This is Ore localization.4 ZORAN ˇ SKODA 3. Ore localizations. =-=[61, 18, 80]-=- A semigroup R with unit is called a monoid. A subset S of a monoid R is called multiplicative if 1 ∈ R and whenever s1, s2 ∈ S then s1s2 ∈ S. Let R be a (noncommutative) unital ring. We can also view... |

1 |
Étale topology
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Citation Context ...one of the several standard topologies for schemes. Local triviality in Zariski topology is the strongest requirement, and the local triviality in étale, fppf, fpqc topology are weaker, in that order =-=[14, 53]-=-. A principal bundle locally trivial in étale topology is often called a torsor. If the orbit space is denoted by X, one can replace B by trivial B-bundle B over X (i.e. by the product B ×X). Then B i... |