# 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^{} $\u2102\times V\to V$ 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 ${\mathbf{e}}_{1},\mathrm{\dots},{\mathbf{e}}_{n},{\mathbf{f}}_{1},\mathrm{\dots},{\mathbf{f}}_{n}$, for which we can obtain a complex structure ${J}_{0}\in \mathrm{Aut}({\mathbb{R}}^{2n})$ represented by the matrix

$$\left(\begin{array}{cc}\hfill \mathrm{\U0001d7ce}\hfill & \hfill {\mathbf{I}}_{n}\hfill \\ \hfill -{\mathbf{I}}_{n}\hfill & \hfill \mathrm{\U0001d7ce}\hfill \end{array}\right).$$ |

Here ${\mathbf{I}}_{n}\in {\mathrm{M}}_{n}(\mathbb{R})$ is the identity^{} $n\times n$ matrix and
$\mathrm{\U0001d7ce}\in {\mathrm{M}}_{n}(\mathbb{R})$ is the zero $n\times n$ matrix.

Title | linear complex structure |
---|---|

Canonical name | LinearComplexStructure |

Date of creation | 2013-03-22 16:16:48 |

Last modified on | 2013-03-22 16:16:48 |

Owner | Mazzu (14365) |

Last modified by | Mazzu (14365) |

Numerical id | 12 |

Author | Mazzu (14365) |

Entry type | Definition |

Classification | msc 15-00 |

Related topic | ComplexificationOfVectorSpace |

Defines | linear complex structure |