2,889 research outputs found
Regular subalgebras and nilpotent orbits of real graded Lie algebras
For a semisimple Lie algebra over the complex numbers, Dynkin (1952)
developed an algorithm to classify the regular semisimple subalgebras, up to
conjugacy by the inner automorphism group. For a graded semisimple Lie algebra
over the complex numbers, Vinberg (1979) showed that a classification of a
certain type of regular subalgebras (called carrier algebras) yields a
classification of the nilpotent orbits in a homogeneous component of that Lie
algebra. Here we consider these problems for (graded) semisimple Lie algebras
over the real numbers. First, we describe an algorithm to classify the regular
semisimple subalgebras of a real semisimple Lie algebra. This also yields an
algorithm for listing, up to conjugacy, the carrier algebras in a real graded
semisimple real algebra. We then discuss what needs to be done to obtain a
classification of the nilpotent orbits from that; such classifications have
applications in differential geometry and theoretical physics. Our algorithms
are implemented in the language of the computer algebra system GAP, using our
package CoReLG; we report on example computations
A constructive method for decomposing real representations
A constructive method for decomposing finite dimensional representations of
semisimple real Lie algebras is developed. The method is illustrated by an
example. We also discuss an implementation of the algorithm in the language of
the computer algebra system {\sf GAP}4.Comment: Final version; to appear in "Journal of Symbolic Computation
A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras
We introduce a search algorithm that utilises differential operator
realisations to find polynomial Casimir operators of Lie algebras. To
demonstrate the algorithm, we look at two classes of examples: (1) the model
filiform Lie algebras and (2) the Schr\"odinger Lie algebras. We find that an
abstract form of dimensional analysis assists us in our algorithm, and greatly
reduces the complexity of the problem.Comment: 22 page
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