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Intertwining operators and Hirota bilinear equations. (English) Zbl 0860.17016
Theor. Math. Phys. 104, No. 1, 879-891 (1995) and Teor. Mat. Fiz. 104, No. 1, 144-157 (1995).
The authors gave an interpretation of the Hirota relations for the $$\tau$$-functions of hierarchies of integrable equations in terms of intertwining operators. The procedure of getting bilinear equations for the $$\tau$$-function is divided into two steps: to find some (commutative) algebra which bosonizes the representation and to find bilinear relations for the matrix elements of the representation. This interpretation gives the possibility of generalizing the relations to the case of finite-dimensional Lie algebras and quantized universal enveloping algebras.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 35Q58 Other completely integrable PDE (MSC2000) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 16S30 Universal enveloping algebras of Lie algebras
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