Lie group

In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures. They were introduced by Sophus Lie in 1870 in order to study symmetries of differential equations.

Alternative definitions

Sometimes, real Lie groups are defined as topological manifolds with continuous group operations; this definition is equivalent to our definition given above. This is an interpretation of the content of Hilbert's fifth problem (see Hilbert-Smith conjecture). The precise statement, proven by Gleason, Montgomery and Zippin in the 1950s, is as follows: If G is a topological manifold with continuous group operations, then there exists exactly one differentiable structure on G which turns it into a Lie group in our sense.

Related Topics:
Manifold - Hilbert - Hilbert-Smith conjecture - Gleason - Montgomery - Zippin

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Therefore one can also take the definition to use smooth functions. This is probably the most common approach now, in textbooks.

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An excellent and unusual introduction to Lie groups and their algebras through the nontrivial example of linear groups (i.e. those defined by continuous groups of finite dimensional matrices) is given by Prof. Wulf Rossmann (see below). This approach is nontrivial, especially given that one version of Ado's Theorem is that every finite dimensional Lie algebra is isomorphic to a matrix Lie algebra. Through standard construction, it can be shown that, for every finite dimensional matrix Lie algebra, there a linear group (matrix Lie group) with this algebra as its Lie algebra.

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