## 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 ...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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110 |
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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 ...(e ′ ⊗U m) = ee ′ ⊗U m and B-coaction extends map e ⊗ m ↦→ ∑ (e (0) ⊗U v) ⊗k e (1). The adjunctions are given by M ↦→ (E ⊗U M) coB , m ↦→ 1 ⊗ m, E ⊗U McoB → M, e ⊗U m ↦→ em. H.-J. Schneider’s theorem =-=[75, 60]-=- says that these two adjoint functors are equivalences iff E is a faithfully flat Hopf Galois extension of U. We’ll now sketch interplay between the functors playing role in the Schneider’s theorem an... |

104 |
Calculus of Fractions and Homotopy Theory
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- 1967
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Citation Context ...ation at thick subcategories is a common framework in noncommutative algebraic geometry [71, 68]. Starting from a pair (A, T ) where A is Abelian and T is thick, one forms a (Serre) quotient category =-=[4, 19, 20, 34, 64, 68]-=-. As objects one takes the objects of the original category, but in addition to the original morphisms one adds to the class of morphisms the formal inverses of those morphisms f for which both Kerf a... |

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Citation Context ...constructed as algebras of regular functions on the underlying group variety. Hence, taking the viewpoint of noncommutative geometry [8] we view Hopf algebras as noncommutative affine group varieties =-=[17, 50, 62]-=-. Groups are useful as they describe the notion of symmetries: they act on spaces. A B-variety is an algebraic variety E with a regular action ν : E × B → E of an algebraic group B. Hopf algebra O(B) ... |

100 |
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Citation Context ...constructed as algebras of regular functions on the underlying group variety. Hence, taking the viewpoint of noncommutative geometry [8] we view Hopf algebras as noncommutative affine group varieties =-=[17, 50, 62]-=-. Groups are useful as they describe the notion of symmetries: they act on spaces. A B-variety is an algebraic variety E with a regular action ν : E × B → E of an algebraic group B. Hopf algebra O(B) ... |

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84 |
Quantum Groups, and Their Primitive Ideals
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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 ... 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 ... |

51 |
Cohomologie non abélienne. Grundlehren
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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... |

51 |
and Ieke Moerdijk, Sheaves in geometry and logic
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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... |

50 |
Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct
- Gelfand, S
- 1992
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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 ... |

50 |
Noncommutative algebraic geometry and representations of quantized algebras
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(Show Context)
Citation Context ...hat C ↦→ HomC(C, G) is a faithful functor), for details cf. [34, 64, 83, 90, 81]. Such categories are a natural place to study noncommutative algebraic geometry beyond the affine and projective cases =-=[71, 81, 59]-=-. A thick subcategory of an Abelian category A is a replete (= full and closed under isomorphisms) subcategory T of A which is closed under extensions, subobjects and quotients. In other words, an obj... |

49 |
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(Show Context)
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)... |

48 |
Multiparametric quantum deformation of the general linear supergroup
- Manin
- 1989
(Show Context)
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... |

44 |
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(Show Context)
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
- Hajac
- 1996
(Show Context)
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... |

41 |
Noncommutative smooth spaces
- Kontsevich, Rosenberg
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(Show Context)
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... |

39 | Minor identities for quasi-determinants and quantum determinants
- Krob, Leclerc
- 1995
(Show Context)
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 ... |

38 | algebras and cohomology
- Beck
(Show Context)
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;... |

37 | Abstract Homotopy and Simple Homotopy Theory - Kamps, Porter - 1997 |

32 |
Quantum deformation of flag schemes and Grassmann schemes. I. A q-deformation of the shape-algebra for GL(n
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- 1991
(Show Context)
Citation Context ...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)graded Fq-modules. In classical limit q = 1, t... |

31 |
ie l, Introduction to Algebraic Geometry and Algebraic Groups
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- 1980
(Show Context)
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... |

31 |
Noncommutative schemes
- Rosenberg
(Show Context)
Citation Context .... Torsion theories correspond to idempotent preradicals. Hereditary torsion theories correspond to Gabriel localizations, which in turn correspond to idempotent radicals. 5. Covers via localizations. =-=[72, 69, 34, 80]-=- For a moment, we take the most general view [69, 20, 4] that a localization is a functor Q ∗ : C → C ′ , which is universal [20] with respect to inverting some class Σ of morphisms in C. A functor Q ... |

26 |
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(Show Context)
Citation Context ...ree matrix bialgebra of rank n2 . Every bialgebra quotient of that bialgebra is a matrix bialgebra. 13.2 Free Hopf algebras. We are going to sketch the construction of free Hopf algebras due Takeuchi =-=[87]-=-. Let C be a coalgebra over k. Let Ci = C for i even nonnegative integer, and Ci = C cop i (the cooposite coalgebra of C) for odd positive integer. Then define V to be the external direct sum (coprodu... |

25 |
Noncommutative descent and non-abelian cohomology, K-Theory 12
- Nuss
- 1997
(Show Context)
Citation Context ...(e ′ ⊗U m) = ee ′ ⊗U m and B-coaction extends map e ⊗ m ↦→ ∑ (e (0) ⊗U v) ⊗k e (1). The adjunctions are given by M ↦→ (E ⊗U M) coB , m ↦→ 1 ⊗ m, E ⊗U McoB → M, e ⊗U m ↦→ em. H.-J. Schneider’s theorem =-=[75, 60]-=- says that these two adjoint functors are equivalences iff E is a faithfully flat Hopf Galois extension of U. We’ll now sketch interplay between the functors playing role in the Schneider’s theorem an... |

22 |
et al., Revêtements Étale et Groupe Fondamental, Séminaire de Géométrie Algébrique SGA
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(Show Context)
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 |
Localization for quantum groups
- Lunts, Rosenberg
(Show Context)
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... |

21 |
The classifying topos of a continuous groupoid
- Moerdijk
- 1988
(Show Context)
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... |

20 | Matric bialgebras and quantum groups - Takeuchi - 1990 |

19 | Stacks and the homotopy theory of simplicial sheaves, in Equivariant stable homotopy theory and related areas
- Jardine
- 2000
(Show Context)
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... |

13 |
Oystaeyen, “Algebraic Geometry for Associative Algebras
- Van
- 2000
(Show Context)
Citation Context ...eft Ore in S −1 R, hence, by symmetry, iff iT(S) is left Ore in T −1 R. If this is true, what is8 ZORAN ˇ SKODA rare in noncommutative case, we say that S and T are mutually compatible left Ore sets =-=[34, 89]-=- (not to confuse with the compatibility with coaction which is a central topic in this paper). Hence, if S and T are compatible, module T −1 S −1 R has a natural ring structure (characterized also by ... |

11 |
Semiquantum Geometry. Algebraic geometry, 5
- RESHETIKHIN, VORONOV, et al.
- 1996
(Show Context)
Citation Context ...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 that case the centers of our localized algebras Wσ are also big, what enables detailed comparison of the tw... |

10 |
Théorie de Hodge III,” Publ
- Deligne
- 1974
(Show Context)
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 |
equations in noncommutative
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- 1931
(Show Context)
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... |