李代数和李群的结构和表示理论，以及自然类似物和概论是非常活跃的研究领域。它们包括有限维和无限维李代数和李群的理论，以及顶点算子代数理论和量子群的理论。它们与古典和现代的许多其他领域有着重要的联系，包括代数群，有限群，几何，谐波分析，微分方程，拓扑，数论，组合论以及弦论和共形场论，在理论物理学中都是如此。
The structure and representation theories of Lie algebras and of Lie groups, and natural analogues and generalizations, are very active areas of research. They include the theory of finite-dimensional and infinite-dimensional Lie algebras and Lie groups as well as vertex operator algebra theory and the theory of quantum groups. They have important connections with many other fields, both classical and modern, including algebraic groups, finite groups, geometry, harmonic analysis, differential equations, topology, number theory, combinatorics, and string theory and conformal field theory in theoretical physics.