We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by newâ€¦ (More)

We derive a formula for the the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We useâ€¦ (More)

We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N . It combines and unifies the ideas of Duistermaatâ€“GrÃ¼nbaum and Wilson. Ourâ€¦ (More)

All factorizable Lie bialgebra structures on complex reductive Lie algebras were described by Belavin and Drinfeld. We classify the symplectic leaves of all related Poisson-Lie groups. A formula forâ€¦ (More)

The complexified Calogero-Moser spaces appeared in several different contexts in integrable systems, geometry, and representation theory. In this talk, we will describe a criterion for their realâ€¦ (More)

We define BÃ¤cklundâ€“Darboux transformations in Satoâ€™s Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe theâ€¦ (More)

This paper is the last of a series of papers devoted to the bispectral problem [3]â€“[6]. Here we examine the connection between the bispectral operators constructed in [6] and the Lie algebra W 1+âˆžâ€¦ (More)

De Concini, Kac and Procesi defined a family of subalgebras U + of a quantized universal enveloping algebra Uq(g), associated to the elements of the corresponding Weyl group W . They are deformationsâ€¦ (More)

For each r = (r1, r2, . . . , rN ) âˆˆ C N we construct a highest weight module Mr of the Lie algebra W1+âˆž. The highest weight vectors are specific tau-functions of the N -th Gelfandâ€“Dickey hierarchy.â€¦ (More)