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Complex Lie Group

A complex Lie group is a group G that is also a complex manifold for which the maps G×G->G, (g,h)|->gh, and G->G, g|->g^(-1), are holomorphic maps. Every complex Lie group of complex dimension d is therefore a real Lie group of real dimension 2d.

The tangent space at the identity element is naturally a complex vector space. Its Lie bracket makes it a Lie algebra over C. Examples include the additive group C^n, the multiplicative group C^×, the general linear group GL_n(C), and the special linear group SL_n(C).

See also

Complex Manifold, General Linear Group, Holomorphic Map, Lie Algebra, Lie Group, Special Linear Group

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References

Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkhäuser, 1996.

Cite this as:

Weisstein, Eric W. "Complex Lie Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexLieGroup.html

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