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In mathematics, a complex structure on a real vector space V is an automorphism of V which squares to the minus identity -I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.

Complex structures have applications in representation theory as well as in complex geometry where they play an essential role in the definition of almost complex manifolds.

A complex structure on a real vector space V is a real linear transformation

- J : V → V

- J
^{2}= −id_{V}.

- (x + i y)v = xv + yJ(v)

Going in the other direction, if one starts with a complex vector space W then one can define a complex structure on the underlying real space by defining Jw = i w for all w in W.

If V_{ J} has complex dimension n then V must have real dimension 2n. That is, V admits a complex structure only if it is even-dimensional. It is not hard to see that every even-dimensional vector space admits a complex structure. One can define J on pairs e,f of basis vectors by Je = f and Jf = −e and then extend by linearity to all of V. If $(v\_1,\; ldots,\; v\_n)$ is a basis for the complex vector space V_{ J} then $(v\_1,\; J\; v\_1,\; ldots,\; v\_n,\; J\; v\_n)$ is a basis for the underlying real space V.

A real linear transformation A : V → V is a complex linear transformation of the corresponding complex space V_{ J} if and only if A commutes with J, i.e.

- AJ = JA

- JU = U

If V is any real vector space there is a canonical complex structure on the direct sum V ⊕ V given by

- $J(v,w)\; =\; (-w,v).,$

- $J\; =\; begin\{bmatrix\}0\; \&\; -1\; 1\; \&\; 0end\{bmatrix\}$

If B is a bilinear form on V then we say that J preserves B if

- B(Ju, Jv) = B(u, v)

- B(Ju, v) = −B(u, Jv)

If g is an inner product on V then J preserves g if and only if J is an orthogonal transformation. Likewise, J preserves a nondegenerate, skew-symmetric form ω if and only if J is a symplectic transformation (that is, if ω(Ju,Jv) = ω(u,v)). For symplectic forms ω there is usually an added restriction for compatibility between J and ω, namely

- ω(u, Ju) > 0

Given a symplectic form ω and a linear complex structure J, one may define an associated symmetric bilinear form g_{J} on V_{J}

- g
_{J}(u,v) = ω(u,Jv).

- $V^\{mathbb\; C\}=Votimes\_\{mathbb\{R\}\}mathbb\{C\}.$

- $overline\{votimes\; z\}\; =\; votimesbar\; z$

If J is a complex structure on V, we may extend J by linearity to V^{C}:

- $J(votimes\; z)\; =\; J(v)otimes\; z.$

Since C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ^{2} = −1, namely λ = ±i. Thus we may write

- $V^\{mathbb\; C\}=\; V^\{+\}oplus\; V^\{-\}$

- $mathcal\; P^\{pm\}\; =\; \{1over\; 2\}(1mp\; iJ).$

- $V^\{pm\}\; =\; \{votimes\; 1\; mp\; Jvotimes\; i:\; v\; in\; V\}.$

There is a natural complex linear isomorphism between V_{J} and V^{+}, so these vector spaces can be considered the same, while V^{−} may be regarded as the complex conjugate of V_{J}.

Note that if V_{J} has complex dimension n then both V^{+} and V^{−} have complex dimension n while V^{C} has complex dimension 2n.

Abstractly, if one starts with a complex vector space W and takes the complexification of the underlying real space, one obtains a space isomorphic to the direct sum of W and its conjugate:

- $W^\{mathbb\; C\}\; cong\; Woplus\; overline\{W\}.$

Let V be a real vector space with a complex structure J. The dual space V* has a natural complex structure J* given by the dual (or transpose) of J. The complexification of the dual space (V*)^{C} therefore has a natural decomposition

- $(V^*)^mathbb\{C\}\; =\; (V^*)^\{+\}oplus\; (V^*)^-$

into the ±i eigenspaces of J*. Under the natural identification of (V*)^{C} with (V^{C})* one can characterize (V*)^{+} as those complex linear functionals which vanish on V^{−}. Likewise (V*)^{−} consists of those complex linear functionals which vanish on V^{+}.

The (complex) tensor, symmetric, and exterior algebras over V^{C} also admit decompositions. The exterior algebra is perhaps the most important application of this decomposition. In general, if a vector space U admits a decompositon U = S ⊕ T then the exterior powers of U can be decomposed as follows:

- $Lambda^r\; U\; =\; bigoplus\_\{p+q=r\}(Lambda^p\; S)otimes(Lambda^q\; T).$

A complex structure J on V therefore induces a decomposition

- $Lambda^r,V^mathbb\{C\}\; =\; bigoplus\_\{p+q=r\}\; Lambda^\{p,q\},V\_J$

- $Lambda^\{p,q\},V\_J;stackrel\{mathrm\{def\}\}\{=\},\; (Lambda^p,V^+)otimes(Lambda^p,V^-).$

- $dim\_\{mathbb\; C\}Lambda^\{r\},V^\{mathbb\; C\}\; =\; \{2nchoose\; r\}qquad\; dim\_\{mathbb\; C\}Lambda^\{p,q\},V\_J\; =\; \{n\; choose\; p\}\{n\; choose\; q\}.$

The space of (p,q)-forms Λ^{p,q} V_{J}* is the space of (complex) multilinear forms on V^{C} which vanish on homogeneous elements unless p are from V^{+} and q are from V^{−}. It is also possible to regard Λ^{p,q} V_{J}* as the space of real multilinear maps from V_{J} to C which are complex linear in p terms and conjugate-linear in q terms.

See complex differential form and almost complex manifold for applications of these ideas.

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Last updated on Sunday August 10, 2008 at 04:32:17 PDT (GMT -0700)

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Last updated on Sunday August 10, 2008 at 04:32:17 PDT (GMT -0700)

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