CoReLG

Computing with real Lie algebras: real forms, Cartan decompositions, and Cartan subalgebras. We describe algorithms for performing various tasks related to real simple Lie algebras. These algorithms form the basis of our software package CoReLG, written in the language of the computer algebra system GAP4. First, we describe how to efficiently construct real simple Lie algebras up to isomorphism. Second, we consider a real semisimple Lie algebra 𝔤. We provide an algorithm for constructing a maximally (non-)compact Cartan subalgebra of 𝔤; this is based on the theory of Cayley transforms. We also describe the construction of a Cartan decomposition 𝔤=𝔨⊕𝔭. Using these results, we provide an algorithm to construct all Cartan subalgebras of 𝔤 up to conjugacy; this is a constructive version of a classification theorem due to Sugiura.


References in zbMATH (referenced in 12 articles )

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  1. Bocheński, Maciej; Jastrzȩbski, Piotr; Tralle, Aleksy: Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a computational approach (2022)
  2. Ali, Sajid; Azad, Hassan; Biswas, Indranil; de Graaf, Willem A.: A constructive method for decomposing real representations (2021)
  3. Dietrich, Heiko; de Graaf, Willem A.: Computing the real Weyl group (2021)
  4. Bächle, Andreas; Sambale, Benjamin: Orders generated by character values (2020)
  5. Bocheński, Maciej; Jastrzȩbski, Piotr; Tralle, Aleksy: Nonexistence of standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups (2020)
  6. Letzter, Gail: Cartan subalgebras for quantum symmetric pair coideals (2019)
  7. Ali, Sajid; Azad, Hassan; Biswas, Indranil; Ghanam, Ryad; Mustafa, M. T.: Embedding algorithms and applications to differential equations (2018)
  8. Levasseur, T.; Stafford, J. T.: Higher symmetries of powers of the Laplacian and rings of differential operators (2017)
  9. Dietrich, Heiko; Faccin, Paolo; de Graaf, Willem A.: Regular subalgebras and nilpotent orbits of real graded Lie algebras (2015)
  10. Faccin, Paolo; de Graaf, Willem A.: Constructing semisimple subalgebras of real semisimple Lie algebras (2015)
  11. Dietrich, Heiko; Faccin, Paolo; de Graaf, Willem A.: A GAP package for computing with real semisimple Lie algebras (2014)
  12. Dietrich, Heiko; Faccin, Paolo; de Graaf, Willem A.: Computing with real Lie algebras: real forms, Cartan decompositions, and Cartan subalgebras (2013)