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The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...
More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .
Moreover, a complex linear map g : V C → W C is the complexification of a real linear map if and only if it commutes with conjugation. As an example consider a linear transformation from R n to R m thought of as an m × n matrix .
Basic examples are (), the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group ). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group. The Lie algebra of a ...
In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra. A left ideal of a K -algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace.
Lie algebra dim/C; C n: group operation is addition N 0 0 abelian C n: n: C ×: nonzero complex numbers with multiplication N 0 Z: abelian C: 1 GL(n,C) general linear group: invertible n×n complex matrices: N 0 Z: For n=1: isomorphic to C ×: M(n,C) n 2: SL(n,C) special linear group: complex matrices with determinant. 1 N 0 0 for n=1 this is a ...
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