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Let $V$ be a finite dimensional vector space and $\left<~,\right>$ be a symmetric bilinear form on $V$. Then I would like to extend this to a bilinear form $\left< ~,\right>$ on $V_\mathbb{C}=V\otimes_{\mathbb{R}}\mathbb{C}$ i.e., we need to define what is $\left< v\otimes \lambda ,u\otimes \mu\right>$ for $v,u\in V$ and $\lambda,\mu\in \mathbb{C}$.

It is natural to define $\left<v\otimes \lambda ,u\otimes \mu\right>=\lambda\bar{ \mu}\left<v ,u\right>$. But this is not symmetric. I want to define some symmetric bilinear form.

I am reading Complex Geometry: An Introduction by Daniel Huybrechts.

The book defines an almost complex structure on a real vector space to be an endomorphism $I:V\rightarrow V$ such that $I^2=-id$. Then it says almost complex structure is compatible with a positive definite symmetric bilinear form $\left<~,\right>$ on $V$ if $\left<Iu,Iv\right>=\left<u,v\right>$ for all $u,v\in V$.

For $V_{\mathbb{C}}=V\otimes\mathbb{C}$ we have extension $I$ of $I:V\rightarrow V$ defined as $I(v\otimes \lambda)=Iv\otimes \lambda$ and it turns out that this extension is an almost complex structure on $V_{\mathbb{C}}$. I fail to extend $\left<~,\right>$ to $\mathbb{C}$. Forget about compatibility. I could not even define a symmetric bilinear form.

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  • $\begingroup$ If you really want it symmetric... then why not just forget about the complex conjugate? $\endgroup$ – b00n heT Jan 26 '17 at 16:36
  • $\begingroup$ @b00nheT : I can always do that but i am afraid that is not natural.. If that is this simple i would be more than happy but i do not want to regret after say 10 days assuming this is so simple and then get stuck because of this.. $\endgroup$ – user87543 Jan 26 '17 at 16:38
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    $\begingroup$ I do understand. But keep in mind that in the complex world "bilinear symmetry" is usually naturally replaced by sesquilinearity $\endgroup$ – b00n heT Jan 26 '17 at 16:40
  • $\begingroup$ @b00nheT : I do believe that symmetry is naturally replaced by sesquilinearity in complex space. I want to know some first hand experience of some one who actually studied this material. So, are you saying it is natural to consider extension that is not symmetric but sesquilinear? $\endgroup$ – user87543 Jan 26 '17 at 16:44
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    $\begingroup$ From my experience: yes, it is natural. But (regrettably) my experience in this field is quite limited. $\endgroup$ – b00n heT Jan 26 '17 at 17:29

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