On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

Abstract

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 $\mathfrak{sl}(n)$.

41 pages, LATEX

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• Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation
• New $r$-Matrices for Lie Bialgebra Structures over Polynomials
• Simple vector bundles on a nodal Weierstrass cubic and quasi-trigonometric solutions of CYBE
• One Parameter Family of Jordanian Twists
• Equivariant quantization of Poisson homogeneous spaces and Kostant's problem
• On classification of quantum groups and Belavin-Drinfeld twisted cohomologies
• Families of Frobenius seaweed Lie algebras
• Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL2
• Classification of Lie bialgebras over current algebras
• On the General Classification of Lie Bialgebra Structures over Polynomials