### Abstract

An algorithm is proposed for integrating linear partial differential equations with the help of a special set of noncommuting linear differential operators - an analogue of the method of noncommutative integration of finite-dimensional Hamiltonian systems. The algorithm allows one to construct a parametric family of solutions of an equation satisfying the requirement of completeness. The case is considered when the noncommutative set of operators form a Lie algebra. An essential element of the algorithm is the representation of this algebra by linear differential operators in the space of parameters. A connection is indicated of the given method with the method of separation of variables, and also with problems of the theory of representations of Lie algebras. Let us emphasize that on the whole the proposed algorithm differs from the method of separation of variables, in which sets of commuting symmetry operators are used.

Original language | English |
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Pages (from-to) | 360-364 |

Number of pages | 5 |

Journal | Soviet Physics Journal |

Volume | 34 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 1991 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Soviet Physics Journal*,

*34*(4), 360-364. https://doi.org/10.1007/BF00898104

**Representations of Lie algebras and the problem of noncommutative integrability of linear differential equations.** / Shapovalov, A. V.; Shirokov, I. V.

Research output: Contribution to journal › Article

*Soviet Physics Journal*, vol. 34, no. 4, pp. 360-364. https://doi.org/10.1007/BF00898104

}

TY - JOUR

T1 - Representations of Lie algebras and the problem of noncommutative integrability of linear differential equations

AU - Shapovalov, A. V.

AU - Shirokov, I. V.

PY - 1991/4

Y1 - 1991/4

N2 - An algorithm is proposed for integrating linear partial differential equations with the help of a special set of noncommuting linear differential operators - an analogue of the method of noncommutative integration of finite-dimensional Hamiltonian systems. The algorithm allows one to construct a parametric family of solutions of an equation satisfying the requirement of completeness. The case is considered when the noncommutative set of operators form a Lie algebra. An essential element of the algorithm is the representation of this algebra by linear differential operators in the space of parameters. A connection is indicated of the given method with the method of separation of variables, and also with problems of the theory of representations of Lie algebras. Let us emphasize that on the whole the proposed algorithm differs from the method of separation of variables, in which sets of commuting symmetry operators are used.

AB - An algorithm is proposed for integrating linear partial differential equations with the help of a special set of noncommuting linear differential operators - an analogue of the method of noncommutative integration of finite-dimensional Hamiltonian systems. The algorithm allows one to construct a parametric family of solutions of an equation satisfying the requirement of completeness. The case is considered when the noncommutative set of operators form a Lie algebra. An essential element of the algorithm is the representation of this algebra by linear differential operators in the space of parameters. A connection is indicated of the given method with the method of separation of variables, and also with problems of the theory of representations of Lie algebras. Let us emphasize that on the whole the proposed algorithm differs from the method of separation of variables, in which sets of commuting symmetry operators are used.

UR - http://www.scopus.com/inward/record.url?scp=34249924894&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249924894&partnerID=8YFLogxK

U2 - 10.1007/BF00898104

DO - 10.1007/BF00898104

M3 - Article

AN - SCOPUS:34249924894

VL - 34

SP - 360

EP - 364

JO - Russian Physics Journal

JF - Russian Physics Journal

SN - 1064-8887

IS - 4

ER -