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Solutions to the quantum Yang-Baxter equation arising from pointed bialgebras. (English) Zbl 0806.16044

Let \(R \in \text{End }(M \otimes M)\) be a solution of the quantum Yang- Baxter equation, \(M\) a finite-dimensional vector space over a field \(k\), \(A(R)\) the associated bialgebra. Let \(A^{\text{red}}(R) = A(R)/I(R)\), \(I(R)\) the sum of all coideals of \(A(R)\) which annihilate \(M\). \(A^{\text{red}}(R)\) is a bialgebra, and \(M\) is a left quantum Yang- Baxter \(A^{\text{red}}(R)\)-module. The author characterizes those \(R\) for which \(A^{\text{red}}(R)\) is pointed, namely that \(M\) has a flag \(\{M_ i\}\) with \(R(M \otimes M_ i) \subseteq M \otimes M_ i\) for all \(i\). He uses this to determine all solutions \(R\) to the quantum Yang- Baxter equation when \(A^{\text{red}}(R)\) is pointed, \(M\) has dimension two and \(k\) has characteristic not two.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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