Highest weight modules of W_{1+infty}, Darboux transformations and the bispectral problem

Details Highest weight modules of W_{1+infty}, Darboux transformations and the bispectral problem Highest weight modules of W_{1+infty}, Darboux transformations and the bispectral problem

1996-01-18

arxiv.org/abs/q-alg/9601017v3

Mathematics

Quantum Algebra

16 pages, LaTeX2e, uses amsfonts.sty, no figures; eq. (19) was corrected, references were added. Presented at the Conference G

Scientific paper

We announce a systematic way for constructing bispectral algebras of commuting differential operators of any rank N. It enables us to obtain all previously known classes and examples of bispectral operators. Moreover, we give a representation-theoretic explanation of the results including those of Duistermaat and Gr\"unbaum. The manifold of bispectral operators of any order is preserved by an hierarchy of symmetries. We point out that our methods provide a completely algorithmic procedure for obtaining bispectral algebras. We conjecture that the class built in the present paper exhausts all bispectral scalar operators. The proofs and details appeared in our preprints hep-th/9510211, q-alg/9602010, q-alg/9602011, q-alg/9602012.

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