scholarly journals Quantum homogeneous spaces and quasi-Hopf algebras

2000 ◽  
pp. 105-121 ◽  
Author(s):  
Benjamin Enriquez ◽  
Yvette Kosmann-Schwarzbach
Keyword(s):  
Hopf Algebras ◽  
Homogeneous Spaces ◽  
Quantum Homogeneous Spaces

scholarly journals Quantum homogeneous spaces of connected Hopf algebras

2016 ◽  
Vol 454 ◽  
pp. 400-432 ◽  
Author(s):  
Ken Brown ◽  
Paul Gilmartin
Keyword(s):  
Hopf Algebras ◽  
Homogeneous Spaces ◽  
Quantum Homogeneous Spaces

scholarly journals Plane Curves Which are Quantum Homogeneous Spaces

2021 ◽  
Author(s):  
Ken Brown ◽  
Angela Ankomaah Tabiri
Keyword(s):  
Hopf Algebras ◽  
Final Section ◽  
Homogeneous Spaces ◽  
Plane Curve ◽  
Plane Curves ◽  
Closed Field ◽  
Quantum Homogeneous Space ◽  
Central Elements ◽  
Pointed Hopf Algebras ◽  
Future Work

AbstractLet $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

scholarly journals Quantum homogeneous spaces as quantum quotient spaces

1996 ◽  
Vol 37 (5) ◽  
pp. 2388-2399 ◽  
Author(s):  
Tomasz Brzeziński
Keyword(s):  
Homogeneous Spaces ◽  
Quantum Homogeneous Spaces ◽  
Quotient Spaces

scholarly journals Subgroups of quantum groups

2018 ◽  
Vol 70 (2) ◽  
pp. 509-533
Author(s):  
Georgia Christodoulou
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Homogeneous Spaces ◽  
General Definition ◽  
Monoidal Categories ◽  
Categorical Approach ◽  
Quantum Homogeneous Spaces

Abstract We investigate the notion of a subgroup of a quantum group. We suggest a general definition, which takes into account the work that has been done for quantum homogeneous spaces. We further restrict our attention to reductive subgroups, where some faithful flatness conditions apply. Furthermore, we proceed with a categorical approach to the problem of finding quantum subgroups. We translate all existing results into the language of module and monoidal categories and give another characterization of the notion of a quantum subgroup.

Principal homogeneous spaces for arbitrary Hopf algebras

1990 ◽  
Vol 72 (1-2) ◽  
pp. 167-195 ◽  
Author(s):  
Hans-Jürgen Schneider
Keyword(s):  
Hopf Algebras ◽  
Homogeneous Spaces

ON THE QUANTUM RUIJSENAARS MODEL, SOME QUANTUM HOMOGENEOUS SPACES AND HECKE ALGEBRAS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 224-239
Author(s):  
N. IORGOV
Keyword(s):  
Homogeneous Space ◽  
Explicit Form ◽  
Homogeneous Spaces ◽  
Hecke Algebras ◽  
The Other ◽  
Difference Operators ◽  
Quantum Homogeneous Spaces ◽  
Quantum Analogue ◽  
Affine Hecke Algebras

The aim of the article is to derive in the explicit form the radial components of Casimir elements of Uq( gl n) corresponding to a quantum analogue of the homogeneous space GL (n)/ SO (n). They coincide with the Macdonald–Ruijsenaars difference operators (MRDOs), if one starts from a special set of Casimir elements from the center of Uq( gl n). The derivation is essentially based on Cherednik's approach to MRDOs by means of affine Hecke algebras. From the other side, MRDOs coincide with commuting Hamiltonians of quantum trigonometric n-particle Ruijsenaars model.

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