| Complex Structure On A Real Vector Space |
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Information AboutComplex Structure On A Real Vector Space |
| CATEGORIES ABOUT LINEAR COMPLEX STRUCTURE | |
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J such that J Here ''J''2 means ''J'' Composed with itself and id''V'' is the Identity Map on ''V''. That is, the effect of applying ''J'' twice is the same as multiplication by −1. This is reminiscent of multiplication by ''i'' . A complex structure allows one to give ''V'' the structure of a Complex Vector Space . Complex scalar multiplication can be defined by :(''x'' + ''i y'')''v'' = ''xv'' + ''yJ''(''v'') for all real numbers ''x'',''y'' and all vectors ''v'' in ''V''. One can check that this does, in fact, give ''V'' the structure of a complex vector space which we denote (''V'', ''J''). 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 2''n''. That is, ''V'' admits a complex structure only if it 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 the extend by linearity to all of ''V''). If is a basis for the complex vector space (''V'', ''J'') then 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'') Iff ''A'' commutes with ''J'', i.e. AJ Likewise, a real Subspace ''U'' of ''V'' is a complex subspace of (''V'', ''J'') iff ''J'' preserves ''U'', i.e. JU RELATION TO COMPLEXIFICATIONS If ''J'' is a complex structure on ''V'', we may extend ''J'' by linearity to the Complexification of ''V'', : Since C is Algebraically Closed , ''J'' is guaranteed to have Eigenvalue s which satisfy λ2 = −1, namely λ = ±''i''. Thus we may write ''V''C = ''V''+ ⊕ ''V''−, where ''V''+ and ''V''− are the eigenspaces of +''i'' and −''i'', respectively. Complex conjugation provides a conjugate-linear isomorphism over C between ''V''+ and ''V''−, and thus they have the same complex dimension. Thus if ''n'' is the complex dimension of ''V''+, then 2''n'' is the complex dimension of ''V''C, and so 2''n'' is also the real dimension of V. Here ''V''+ is the subspace of the complexification of ''V'' that we defined above, while ''V''− is the complex space on which ''J'' acts as multiplication by −''i''. Note that there is a complex linear isomorphism between (''V'',''J'') and ''V''+, so these vector spaces can be considered the same. COMPATIBILITY WITH OTHER STRUCTURES If ''B'' is a Bilinear Form on ''V'' then we say that ''J'' preserves ''B'' if B for all ''u'',''v'' in ''V''. An equivalent characterization is that ''J'' is Skew-adjoint with respect to ''B'': B If ''g'' is an Inner Product on ''V'' then ''J'' preserves ''g'' iff ''J'' is an Orthogonal Transformation . Likewise, ''J'' preserves a Nondegenerate , Skew-symmetric form ω iff ''J'' is a Symplectic Transformation . For symplectic forms ω there is usually an added restriction for compatibility between ''J'' and ω, namely :ω(''u'', ''Ju'') > 0 for all ''u'' in ''V''. If this condition is satisfied then ''J'' is said to tame ω. EXTENSION TO RELATED VECTOR SPACES As discussed above, a linear complex structure on a real vector space induces the decomposition : on the complexification of ''V''. This decomposition can be extended to several vector spaces built from ''V''. For example, the Dual Space of ''V'' can also be decomposed into functionals of type (1,0) and (0,1). Functionals of type (1,0) are those which vanish on ''V''-. The Tensor Algebra , Symmetric Algebra , and Exterior Algebra over ''V'' also admit decompositions. The exterior algebra is perhaps the most important application of this decomposition. The homogeneous spaces of ''k''-vectors decompose : and the entire exterior algebra becomes : where ''n'' is the real dimension of the real vector space (and so also the complex dimension of its complexification), and Λ(''p'',''q'') is the exterior power of ''p'' copies of ''V''(1,0) and ''q'' copies of ''V''(0,1). See Almost Complex Manifold for an application of this construction. Similar expansions hold for the tensor algebra and the symmetric algebra. SEE ALSO |
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