Algebraic theory of the KP equations.

*(English)*Zbl 0837.35132
Penner, Robert (ed.) et al., Perspectives in mathematical physics. Proceedings of the conference on interface between mathematics and physics, held in Taiwan in summer 1992 and the special session on topics in geometry and physics, held in Los Angeles, CA, USA in winter of 1992. Boston, MA: International Press. Conf. Proc. Lect. Notes Math. Phys. 3, 151-217 (1994).

The author considers the quantum version of the Gaudin model. It contains some explanations on the work by B. Feigin, E. Frenkel and N. Reshetikhin [Commun. Math. Phys. 166, No. 1, 27-62 (1994; Zbl 0812.35103)]. The classic version of the Gaudin model is also considered in terms of loop algebras, R-matrices, Lax operators and factorization problems. He exhibits a theorem about Casimirs of the symmetric algebra of a Lie algebra \({\mathfrak g}\). A quantum version of this theorem also holds by replacing the symmetric algebra by the universal enveloping algebra of \({\mathfrak g}\). The quantum model considered in this report allows to include into the Lie algebraic picture the generalized Bethe ansatz. He also previews exciting developments in quantum integrability.

For the entire collection see [Zbl 0816.00027].

For the entire collection see [Zbl 0816.00027].

Reviewer: L.Piovan (Bahia Blanca)

##### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

17B81 | Applications of Lie (super)algebras to physics, etc. |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

##### Keywords:

Gaudin model; loop algebras; R-matrices; Lax operators; symmetric algebra; universal enveloping algebra; generalized Bethe ansatz
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\textit{M. Mulase}, in: Perspectives in mathematical physics. Proceedings of the conference on interface between mathematics and physics, held in Taiwan in summer 1992 and the special session on topics in geometry and physics, held in Los Angeles, CA, USA in winter of 1992. Boston, MA: International Press. 151--217 (1994; Zbl 0837.35132)