Representations of Lie algebras. (English) Zbl 1512.17008
Representation theory of Lie algebras is an active mainstream branch of mathematics which plays an increasingly important role in different areas of modern science. In particular, representations of Lie algebras are of fundamental use in geometry, mathematical physics, topology, combinatorics, number theory, knot theory, etc. This theory was a focus of the research group “Nonassociative algebras, their representations, identities and relations” at the Instituto de Matemática e Estatística, Universidade de São Paulo for more than 20 years. The ultimate goal is to develop a general framework and new methodologies to address challenging classification and structure problems using algebraic, geometric and combinatorics techniques. V. Futorny describes notable results obtained by the members of the research group on the representations of Lie algebras, with a focus on the Gelfand-Tsetlin theories, representations of affine Kac-Moody algebras, related vertex algebras and Lie algebras of vector fields.
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |
| 17B65 | Infinite-dimensional Lie (super)algebras |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Keywords:
Witt algebra; Galois order; Gelfand-Tsetlin module; vector field; affine Lie algebra; Krichever-Novikov algebraReferences:
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