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Plane curves which are quantum homogeneous spaces. (English) Zbl 1501.16030

Let \(k\) be an algebraically closed field of characteristic \(0\). In this paper, the authors consider the following question: Which a commutative affine algebra can be realized as a quantum homogeneous space? This question can be regarded as a way to consider the interaction between commutative algebra and Hopf algebras, which should be a potential direction of Hopf theory.
The main purpose of this paper is to show that more or less a curve \(\mathcal{C}\in k^2\) defined by \(f(x)=g(y)\) is a quantum homogeneous space. Here the phrase “more or less” means that the curve needs to satisfy the following hypothesis: each of \(f(x)\) and \(g(y)\) either has degree at most 5 or is a power of the variable. To show this result, the authors construct explicitly a pointed Hopf algebra, denoted by \(A(g,f)\), and prove that \(\mathcal{O}(\mathcal{C})\) is a faithfully flat right coideal subalgebra of \(A(g,f).\) I do believe that this will contribute to the development of this subject, and in any case the result itself is undoubtedly interesting.

MSC:

16T20 Ring-theoretic aspects of quantum groups
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16P90 Growth rate, Gelfand-Kirillov dimension
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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References:

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