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Constructing Lie algebras of first order differential operators

โœ Scribed by Jan Draisma

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
305 KB
Volume
36
Category
Article
ISSN
0747-7171

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โœฆ Synopsis

We extend Guillemin and Sternberg's Realization Theorem for transitive Lie algebras of formal vector fields to certain Lie algebras of formal first order differential operators, and show that Blattner's proof of the Realization Theorem allows for a computer implementation that automatically reproduces many realizations derived in the existing literature, and that can also be used to compute new realizations. Applications include the explicit construction of quasi-exactly solvable Hamiltonians, and of finite-dimensional irreducible modules over semisimple Lie algebras.

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โœ W.A. de Graaf ๐Ÿ“‚ Article ๐Ÿ“… 2001 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 183 KB

We describe an algorithm for constructing irreducible representations of split semisimple Lie algebras in characteristic 0. The algorithm calculates a Gr obner basis of a certain left ideal in a universal enveloping algebra. It is shown that this algorithm runs in polynomial time if the Lie algebra