Talk:Linear complex structure

This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Mathematics Low‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Low This article has been rated as Low-priority on the project's priority scale.

It seems to me that there ought to be something in here, say, in the section on preserving other structures, that given a real inner product g on V, we can induce a Hermitean inner product on (V, J) via the rule h(v, w) = g(v, w) + ig(Jv, w). This then satisfies h(v, v) = g(v, v), h(iv, w) = ih(v, w), and h(v, w) = h(w, v)* Rwilsker 17:24, 24 July 2007 (UTC)Reply

currently there's something, right? Commentor (talk) 05:03, 23 March 2008 (UTC)Reply

Quote: "In general, if a vector space U admits a decompositon U = S ⊕ T then the exterior powers of U can be decomposed as follows: ${\displaystyle \Lambda ^{r}U=\bigoplus _{p+q=r}(\Lambda ^{p}S)\otimes (\Lambda ^{q}T).}$" Can somebody comment on this? Why is it true? Let ${\displaystyle U=R^{3},S=R^{2},T=R^{1},r=2}$. Then the space of antisymmetric 2-forms on U is 3-dimensional, yet the expression on the right-hand side gives only one dimension. Should one raise the right hand side to the appropriate binomial coefficient? Commentor (talk) 05:03, 23 March 2008 (UTC)Reply

The first sentence is overly intimate with its context, and requires some introductory framing. This is most likely another sentence in front of it. For reading, I dropped the "disambiguation needed" template from "complex" -- by linking "Generalized complex" out of my own search for algebraic legend and rules. Perhaps the author or the peerage can add a preface. JohnPritchard (talk) 20:29, 7 June 2013 (UTC)Reply

More formally, a linear complex structure on a real vector space is an algebra representation of the complex numbers C, thought of as an associative algebra over the real numbers. This algebra is realized concretely as ${\displaystyle \mathbf {C} =\mathbf {R} [x]/(x^{2}+1),}$ which corresponds to ${\displaystyle i^{2}=-1.}$

I understand the first sentence. Not so the second. Please explain (in the article, not here). YohanN7 (talk) 07:46, 14 May 2014 (UTC)Reply

Quotient of the real algebra of polynomial by the ideal ${\displaystyle \mathbf {R} [x]\cdot (x^{2}+1)\cdot \mathbf {R} [x]}$ and I guess, one has to identify i with x so that ${\displaystyle x^{2}+1\sim 0\ \Leftrightarrow \ x^{2}\sim -1}$ Noix07 (talk) 20:33, 16 March 2015 (UTC)Reply

Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure.

Isn't ${\displaystyle J:V\rightarrow V,v\rightarrow iv}$ a complex structure independent of any basis? Noix07 (talk) 20:33, 16 March 2015 (UTC)Reply