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Representations of Lie Groups and Lie Algebras

In: A History of Algebra

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  • Bartel Leenert van der Waerden

    (Universität Zürich, Mathematisches Institut)

Abstract

According to Sophus Lie, every local representation of a Lie group G by linear transformations is generated by a representation of the Lie algebra L G of G. The matrices of this representation are linear combinations $$ A{\rm{ }} = {\rm{ }}{A_1}{x_1} + {\rm{ }}... + {A_n}{x_n} $$ , and a neighbourhood of the unity element of G is represented by the matrices $$ {\rm{exp}}A = {\rm{ exp}}({A_1}{x_1} + {\rm{ }}... + {A_n}{x_n}) $$ , the x i varying in a neighbourhood of the origin in ℝ n or ℂn. It is true that Lie does not use the modern expressions “exp” and “neighbourhood”, but his statements are equivalent to what I have just said.

Suggested Citation

  • Bartel Leenert van der Waerden, 1985. "Representations of Lie Groups and Lie Algebras," Springer Books, in: A History of Algebra, chapter 0, pages 252-263, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-51599-6_14
    DOI: 10.1007/978-3-642-51599-6_14
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