On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

Details On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

2007-06-12

arxiv.org/abs/0706.1651v1

Mathematics

Quantum Algebra

41 pages, LATEX

Scientific paper

10.1007/s00220-008-0554-x

We study classical twists of Lie bialgebra structures on the polynomial current algebra $\mathfrak{g}[u]$, where $\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric $r$-matrices fall into classes labelled by the vertices of the extended Dynkin diagram of $\mathfrak{g}$. We give complete classification of quasi-trigonometric $r$-matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of $\mathfrak{sl}(n)$.

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