# Integrals of motion and quantum groups

@article{Feigin1993IntegralsOM, title={Integrals of motion and quantum groups}, author={Boris Feigin and Edward Frenkel}, journal={Lecture Notes in Mathematics}, year={1993}, volume={1620}, pages={349-418} }

A homological construction of integrals of motion of the classical and quantum Toda field theories is given. Using this construction, we identify the integrals of motion with cohomology classes of certain complexes, which are modeled on the BGG resolutions of the associated Lie algebras and their quantum deformations. This way we prove that all classical integrals of motion can be quantized. For the Toda field theories associated to finite-dimensional Lie algebras, the algebra of integrals of…

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## References

SHOWING 1-10 OF 127 REFERENCES

### New Symmetry Principles in QFT

- eds. J. Fröhlich, e.a.,
- 1992

### Varchenko, in ICM-90

- Satellite Conference Proceedings Algebraic Geometry and Analytic Geometry,
- 1991

### Affine Kac-Moody algebras at the critical level and quantum Drinfeld-Sokolov reduction

- Mathematics
- 1991

### Progr

- Theor. Phys. Suppl. 102
- 1990

### Journal

- Greece and Rome
- 1958

This paper will discuss about one part in the history of Semarang City, which is discussing the river from the historical side in the early 20th century. The main problem in this study is how the…

### Perception analytique et globale

- Psychology
- 1992

Les aspects holistiques ou analytiques de la perception sont actuellement discutes en reference soit a la structure de l'objet percu, soit a la nature du traitement effectue. L'approche structurale…