Quantum homogeneous spaces of connected Hopf algebras

Abstract

Let H be a connected Hopf k-algebra of finite Gel'fand-Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H. They are shown to be deformations of commutative polynomial k-algebras. A number of well-known homological and other properties follow immediately from this fact. Further properties are described, examples are considered, invariants are constructed and a number of open questions are listed.

Connected (graded) Hopf algebras

“…Despite the example from Theorem 0.5(2), the evidence so far assembled, in § §1 and 2 of the present paper for (CGA) and in previous work [EG,Zh,Wa1,BO,BG2] for (CCA), supports the intuition that Hopf algebras of finite Gel'fand-Kirillov dimension satisfying either of these hypotheses share many properties with enveloping algebras of finite dimensional k-Lie algebras g, (with g graded and hence nilpotent in the case of (CGA)). We provide some further evidence in support of this philosophy, by generalising in Theorem 4.5 a well-known result on enveloping algebras of Latysev and Passman, [Lat, Pas], and also providing a version of it for connected graded algebras, Theorem 2.9:…”

“…Let gr c H denote the associated graded algebra of H with respect to the coradical filtration of H. The signature featuring in Corollary 0.6(4) below is a list of the degrees of the homogeneous generators of gr c H. It is defined and discussed in [BG2,Definition 5.3].…”

The Nodal Cubic is a Quantum Homogeneous Space

“…Using this basis, one also easily observes that like the nonstandard Podleś spheres, the algebra extension B ⊂ A is an example of a coalgebra Galois extension [3][4][5][6] rather than of a Hopf-Galois extension: the coalgebra is C := A/B + A, where (2) and we have (as a consequence of the faithful flatness of A over B)…”

“…in [2,7,11,[14][15][16][17][18][19]21] generalise affine varieties with a transitive action of an algebraic group: Definition A quantum homogeneous space is a right coideal subalgebra B of a Hopf algebra A which is faithfully flat as a left B-module.…”

Nakayama Automorphisms of Ore Extensions Over Polynomial Algebras

“…In general, Nakayama automorphisms are known to be tough to compute. We refer to [1], [2], [3], [8], [10], [11], [13], [14], [15], [16] and the references therein for the progress on this topic during the past years.…”