Final Report Summary - CORELG (Computation with real Lie Groups)
One of the main topics of research in the mathematics of the 20'st century has been the classification of the complex simple Lie algebras. On the one hand this area has a lot of surprising applications in such fields as physics, geometry and group theory. On the other hand, the theory is built on the interplay of linear algebra and combinatorics. This second aspect has made it very suitable for investigation by computer. Indeed, since the 60's many computer programs and packages have been developed for working with complex semisimple Lie algebras.
Although the classification of the simple real Lie algebras has also been of paramount importance in fields like differential geometry and physics, there has not been a comparable endeavour to produce computer algebra systems able to deal with these objects. An exception is the Atlas project, based in the US, which aims at developing programs to study the unitary dual of a real Lie group.
The overall aim of the CORELG (Computation with Real Lie Groups) project is to develop algorithms and computer algebra packages for working with real Lie algebras, and their adjoint real Lie groups. The difference to the Atlas project is that we aim to work with Lie algebras given by a multiplication table. On the one hand this makes it possible to investigate the real Lie algebras in a much more detailed manner, on the other hand this also presents a lot of specific problems that have to be overcome.
One of the main goals of the CORELG project has been to investigate orbits of real theta-groups by computational means. These orbits naturally are divided into three classes: nilpotent, semisimple and mixed. After an initial phase of studying the theory, work was started on the nilpotent orbits. Moreover, in view of the complexity of the subject, it was decided to first work on a special case: the nilpotent orbits of a real simple Lie algebra under the action of its adjoint group. This special case has already been studied quite extensively in the literature, and for many single real simple Lie algebras, lists of the nilpotent orbits had been produced. However, no uniform methods for obtaining such lists were known. In our work we have developed such a method, based on the Kostant-Sekiguchi correspondence that has already been studied extensively in the literature. Using our methods, along with extensive computations, we put together a database containing the nilpotent orbits for each real simple Lie algebra of rank not exceeding 8. Also we devised algorithms for constructing an isomorphism between real simple Lie algebras. This makes it possible to translate the data in our database to real simple Lie algebras constructed in a different way (as algebra of matrices, for example). This work has led to the first research publication connected to the project (Heiko Dietrich and Willem de Graaf, A computational approach to the Kostant-Sekiguchi correspondence, Pacific Journal of Mathematics, to appear).
In order to get further progress on nilpotent orbits, it turned out to be necessary to create a computational infrastructure for real Lie algebras. This work has been undertaken with the Ph.D. student from Trento, Paolo Faccin. A first step was the development of efficient methods to create the simple real Lie algebras in the computer algebra system GAP4. This involved an algorithm for computing the multiplication table of such a Lie algebra. The second step concerned a method to find a Cartan decomposition of a real simple Lie algebra given only by a multiplication table. This decomposition is a theoretical device that completely identifies the Lie algebra, and on which many more theoretical constructions are built. So it is of the highest importance for our project to have an algorithm for computing it. The third and last step was to develop algorithms for constructing all Cartan subalgebras of a given simple real Lie algebra (up to conjugacy). Our methods for this build on the work of Sugiura from the 50's. This work has led to a second research paper connected to the project (Heiko Dietrich, Paolo Faccin, and Willem de Graaf, Computing with real Lie algebras: Real forms, Cartan decompositions, and Cartan subalgebras, submitted).
It has been an integral part of the project to implement the algorithms in the computational algebra system GAP4. This is an open source, freely available system. Moreover, it gives the user the possibility to create packages that interact seamlessly with the core system. Therefore it is ideally suited for our purposes. We have created a package called CORELG (the same acronym as the Marie Curie project) - again in collaboration with the Ph.D. student Paolo Faccin. This package contains implementations of all our algorithms, and most notably, the database that we created of all nilpotent orbits of all simple real Lie algebras of ranks not exceeding 8.
Due to the fact that the project terminated ahead of schedule we have not been able to finish one of the main aims of the project, namely, to create computational methods for listing the nilpotent orbits of a real theta-group. However, we have done considerable work on this problem. Here the main idea is to generalise the theory of Vinberg on so-called carrier algebras to the case of real Lie algebras. Using our previously developed and implemented algorithm for finding all Cartan subalgebras of a real Lie algebra, we get a short list of Cartan subalgebras, and with respect to each of them we have to find carrier algebras. Then we have to decide whether two given carrier algebras, corresponding to the same Cartan subalgebra, are conjugate. Work on this is still in progress, and we expect it to lead to the third research publication connected to the project.
Although the classification of the simple real Lie algebras has also been of paramount importance in fields like differential geometry and physics, there has not been a comparable endeavour to produce computer algebra systems able to deal with these objects. An exception is the Atlas project, based in the US, which aims at developing programs to study the unitary dual of a real Lie group.
The overall aim of the CORELG (Computation with Real Lie Groups) project is to develop algorithms and computer algebra packages for working with real Lie algebras, and their adjoint real Lie groups. The difference to the Atlas project is that we aim to work with Lie algebras given by a multiplication table. On the one hand this makes it possible to investigate the real Lie algebras in a much more detailed manner, on the other hand this also presents a lot of specific problems that have to be overcome.
One of the main goals of the CORELG project has been to investigate orbits of real theta-groups by computational means. These orbits naturally are divided into three classes: nilpotent, semisimple and mixed. After an initial phase of studying the theory, work was started on the nilpotent orbits. Moreover, in view of the complexity of the subject, it was decided to first work on a special case: the nilpotent orbits of a real simple Lie algebra under the action of its adjoint group. This special case has already been studied quite extensively in the literature, and for many single real simple Lie algebras, lists of the nilpotent orbits had been produced. However, no uniform methods for obtaining such lists were known. In our work we have developed such a method, based on the Kostant-Sekiguchi correspondence that has already been studied extensively in the literature. Using our methods, along with extensive computations, we put together a database containing the nilpotent orbits for each real simple Lie algebra of rank not exceeding 8. Also we devised algorithms for constructing an isomorphism between real simple Lie algebras. This makes it possible to translate the data in our database to real simple Lie algebras constructed in a different way (as algebra of matrices, for example). This work has led to the first research publication connected to the project (Heiko Dietrich and Willem de Graaf, A computational approach to the Kostant-Sekiguchi correspondence, Pacific Journal of Mathematics, to appear).
In order to get further progress on nilpotent orbits, it turned out to be necessary to create a computational infrastructure for real Lie algebras. This work has been undertaken with the Ph.D. student from Trento, Paolo Faccin. A first step was the development of efficient methods to create the simple real Lie algebras in the computer algebra system GAP4. This involved an algorithm for computing the multiplication table of such a Lie algebra. The second step concerned a method to find a Cartan decomposition of a real simple Lie algebra given only by a multiplication table. This decomposition is a theoretical device that completely identifies the Lie algebra, and on which many more theoretical constructions are built. So it is of the highest importance for our project to have an algorithm for computing it. The third and last step was to develop algorithms for constructing all Cartan subalgebras of a given simple real Lie algebra (up to conjugacy). Our methods for this build on the work of Sugiura from the 50's. This work has led to a second research paper connected to the project (Heiko Dietrich, Paolo Faccin, and Willem de Graaf, Computing with real Lie algebras: Real forms, Cartan decompositions, and Cartan subalgebras, submitted).
It has been an integral part of the project to implement the algorithms in the computational algebra system GAP4. This is an open source, freely available system. Moreover, it gives the user the possibility to create packages that interact seamlessly with the core system. Therefore it is ideally suited for our purposes. We have created a package called CORELG (the same acronym as the Marie Curie project) - again in collaboration with the Ph.D. student Paolo Faccin. This package contains implementations of all our algorithms, and most notably, the database that we created of all nilpotent orbits of all simple real Lie algebras of ranks not exceeding 8.
Due to the fact that the project terminated ahead of schedule we have not been able to finish one of the main aims of the project, namely, to create computational methods for listing the nilpotent orbits of a real theta-group. However, we have done considerable work on this problem. Here the main idea is to generalise the theory of Vinberg on so-called carrier algebras to the case of real Lie algebras. Using our previously developed and implemented algorithm for finding all Cartan subalgebras of a real Lie algebra, we get a short list of Cartan subalgebras, and with respect to each of them we have to find carrier algebras. Then we have to decide whether two given carrier algebras, corresponding to the same Cartan subalgebra, are conjugate. Work on this is still in progress, and we expect it to lead to the third research publication connected to the project.