# linear complex structure

A on a real vector space $V$, with $\mathrm{dim}(V)=m$, is a linear automorphism $J\in\mathrm{Aut}(V)$ such that $J^{2}=J\circ J=-\mathrm{id}_{V}$. With a complex structure $J$ we can consider $V$ as a complex vector space with the product $\mathbb{C}\times V\rightarrow V$ given by

 $(x+iy)\mathbf{v}=x\mathbf{v}+yJ(\mathbf{v}),\ \ \forall x,y\in\mathbb{R},\ % \mathbf{v}\in V.$

This implies that the dimension $m$ of $V$ must be even.

A common example is $V=\mathbb{R}^{2n}$ with the standard basis $\mathbf{e}_{1},...,\mathbf{e}_{n},\mathbf{f}_{1},...,\mathbf{f}_{n}$, for which we can obtain a complex structure $J_{0}\in\mathrm{Aut}(\mathbb{R}^{2n})$ represented by the matrix

 $\left(\begin{array}[]{cc}\mathbf{0}&\mathbf{I}_{n}\\ -\mathbf{I}_{n}&\mathbf{0}\end{array}\right).$

Here $\mathbf{I}_{n}\in\mathrm{M}_{n}(\mathbb{R})$ is the identity $n\times n$ matrix and $\mathbf{0}\in\mathrm{M}_{n}(\mathbb{R})$ is the zero $n\times n$ matrix.

Title linear complex structure LinearComplexStructure 2013-03-22 16:16:48 2013-03-22 16:16:48 Mazzu (14365) Mazzu (14365) 12 Mazzu (14365) Definition msc 15-00 ComplexificationOfVectorSpace linear complex structure