Quantum Homogeneous Space And The Twisted Calabi - Yau Algebra

Quantum homogeneous spaces are a class of right coideal subalgebras of Hopf algebras, which can be viewed as quantum deformation of homogeneous spaces in the theory of Lie groups. Because of a shortage of quantum quotient groups (or Hopf subalgebras), quantum homogeneous spaces, which are a larger class objects, attract the attention of mathematicians. In the past decades, they have been studied widely in the fields of mathematical physics, non-commutative geometry, and so on. In this dissertation, we concentrate on the homological properties of quantum homogeneous spaces and the classification of the quantum homogeneous spaces for some concrete quantum groups.Specifically, the homological properties of quantum homogeneous spaces, including the various homological dimensions, Auslander condition, dualizing complexes, are studied. Some general results are deduced, especially the Van den Bergh duality for an AS-Gorenstein quantum homogeneous space is established. Some conclusions on Hopf algebras, proved in [BZ08], are generalized to quantum homogeneous spaces.After studying the rigid dualizing complexes of quantum homogeneous spaces, we recall the notion of twisted Calabi-Yau algebras. A twisted Calabi-Yau alge-bra is Calabi-Yau in the sense of Ginzburg [Gin06] if and only if its Nakayama automorphism is inner. So how to compute the Nakayama automorphism of a twisted Calabi-Yau algebra becomes a crucial question.Inspired by [KT81],[Sch92], we define the normal basis property for quan-tum homogeneous spaces. Xormal basis property plays an important role in proving of the AS-Gorenstein property and determining the homological inte-grals and Nakayama automorphisms.We pay attention to quantized enveloping algebras Uq((?)) and find out that a quantum homogeneous space can be recognized by iterated Ore extension if it is contained in the quantum Borel part. It is firstly proved that an Ore extension preserves the twisted Calabi-Yau property, and the relation between their Nakayama automorphisms is also described. It follows that the quantum homogeneous spaces in the quantum Borel part are twisted Calabi-Yau and their Nakayama automorphisms can be computed. In general case, by using a filtration introduced in [Let02], all quantum homogeneous spaces of Uq(g) are proved to be AS-regular and twisted Calabi-Yau.In the final part, we classify the quantum homogeneous spaces of Ug(sl(2, C)) and study the Podles quantum spheres which are a class of quantum homoge-neous spaces of Oq(SL(2,C)). The standard Podles quantum sphere is AS-regular, as is showed in [Kra12]. We deal with the non-standard case and prove that Podles quantum spheres are all Auslander-regular, Cohen-Macaulay and AS-regular.