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Being infinite objects, locally finite Lie algebras are very complicated in comparison with those of finite dimension. Say, semisimple locally finite Lie algebras cannot be represented as a limit of semisimple finite dimensional, which leads to the study of new asymptotic phenomena. The generation of new ideas is impossible without cooperation and exchange of ideas. These ideas are essentially "in the air" because the other classes of locally finite "algebras" (including groups) have been around for quite a long while. The comparison with the theories of locally finite groups and associative algebras seems especially fruitful. The latter theory has close connections with C*-algebras, and we believe that the wealth of the methods of this latter theory can be applied to the study of locally finite Lie algebras. The classification result by Baranov - Zhilinski for diagonal embeddings of simple finite-dimensional Lie algebras provides just one example of the successful application of the associative methods to the theory of locally finite Lie algebras. But much more research is needed to get a better understanding of more general locally finite Lie algebras.
The theory of locally finite groups is also a very well developed subject with a number of excellent contributions by many mathematicians -- Brian Hartley, Otto Kegel, Donald Passman, and Alexander Zalesskii to cite just a few. The book by Kegel - Wehrfritz summarizes an early stage of development of this theory. A number of conferences have been held on this topic in the past, including a large conference in Turkey in 1996. The theory of locally finite groups is very rich with methods, but so far there have been very few attempts to apply them to locally finite Lie algebras.
Finally, it should be mentioned that in the last decade a number of important papers have been published on locally finite Lie algebras, devoted to various aspects, such as root systems, representation theory, structure theory, etc. Some of these results will be reported by the participants of the workshop.
By holding a workshop on the topic, there is a very good chance of giving a boost to the subject. The goal of the meeting would be to bring together people from all those different areas mentioned above, to compare the results and the approaches, and to devise new strategies for studying locally finite Lie algebras.
It should be mentioned in conclusion that Canada is a country with excellent traditions in Lie theory, developed by a number of good researchers both in mathematics and physics. So it is only very appropriate to hold the c onference suggested on the Canadian soil.
Final Report (in PDF format)
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