Talk:Linear complex structure

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)

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

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: $\Lambda ^{r}U=\bigoplus _{p+q=r}(\Lambda ^{p}S)\otimes (\Lambda ^{q}T).$" Can somebody comment on this? Why is it true? Let $U=R^{3},S=R^{2},T=R^{1},r=2$. 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)

Preface framing

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)

Undefined thingies

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 $\mathbf {C} =\mathbf {R} [x]/(x^{2}+1),$ which corresponds to $i^{2}=-1.$

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

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

Is that a mistake?

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

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