In [28, 38], the coordinate ring of the cusp y2 = x3 is seen to be a quantum homogeneous space. Using this as a starting example, the coordinate ring of the nodal cubic y2 = x2 +x3 was shown to be a quantum homogeneous space in [40]. This thesis focuses on finding singular plane curves which are quantum homogeneous spaces. We begin by discussing the background theory of Hopf algebras, algebraic groups and the set up for Bergman’s Diamond Lemma [9]. Next, we recall the theory of quantum homogeneous spaces in the commutative (classical) and noncommutative (nonclassical) settings. Examples and theorems on these spaces are stated. Then main theorem in this thesis is that decomposable plane curves (curves of the form f(y) = g(x)) of degree less than or equal to five are quantum homogeneous spaces. In order to prove this, we construct two new families of Hopf algebras, A(x; a; g) and A(g; f). Then we use Bergman’s Diamond lemma to prove that A(g; f) is faithfully flat over the coordinate ring of f(y) = g(x). These new Hopf algebras that we have discovered have nice properties when deg(g); deg(f) B 3. The properties include being noetherian domains, finite Gelfand-Kirillov dimensions, AS-regular and finite modules over their centres. We derive these properties from the isomorphism between A(x; a; g) and well studied algebras, the localised quantum plane and down-up algebras [7] when deg(g) = 2; 3.
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