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Wednesday, November 30, 2005

**Abstract:** One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which ${\mathbb{C}}[\partial]$ is replaced by the universal enveloping algebra $U({\mathfrak{d}})$ of a finite-dimensional Lie algebra ${\mathfrak{d}}$. I will review the classification of finite simple Lie pseudoalgebras, and I will discuss their relationship to solutions of the classical Yang--Baxter equation. I will also describe the irreducible representations of the Lie pseudoalgebra $W({\mathfrak{d}})$, which is closely related to the Lie--Cartan algebra $W_N$ of vector fields, where $N=\dim{\mathfrak{d}}$. (Based on joint work with A.~D'Andrea and V.~G.~Kac.)