An infinite-dimensional representation refers to a way of representing elements of a Lie group or Lie algebra as linear transformations on an infinite-dimensional vector space. This type of representation is significant as it allows for the study of more complex structures and phenomena that cannot be captured by finite-dimensional representations. Infinite-dimensional representations often arise in various areas of mathematics and physics, such as quantum mechanics, where states are represented by functions in an infinite-dimensional Hilbert space.
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