# A textbook on linear algebra where involutions on linear spaces are considered

Let us call an involution on a complex linear space $X$ an arbitrary $\mathbb R$-linear map $x\in X\mapsto x^*\in X$ that satisfies the following identities: $$x^{**}=x,\qquad (\lambda\cdot x)^*=\overline{\lambda}\cdot x^*\qquad (\lambda\in{\mathbb C},\quad x\in X).$$ This is strange, but I can't find a textbook on linear algebra where this notion is considered. Can anybody recommend something? I need a reference for some elementary facts like "$X$ is a complexification of the subspace of real elements" (i.e. elements satisfying the equality $x^*=x$), or "the dual space (of $\mathbb C$-linear functionals) also has a natural involution", and so on.

I posted this in math.stackexchange, but without success.

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I can't help with your actual question, since I never saw this in a linear algebra textbook, but I saw these mentioned in passing in some monograph on an advanced topic, as "real structures on a complex vector space". Perhaps that phrase will yield more search results – Yemon Choi Jul 7 '13 at 6:16
You don't remember details? Title, author? – Sergei Akbarov Jul 7 '13 at 6:20
I think it was somewhere in the middle of John Roe's book on the Atiyah-Singer Index Theorem - I don't even remember why it was mentioned or what role it played, to be honest, this was at least five years ago. Sorry. – Yemon Choi Jul 7 '13 at 6:24
This is indeed the notion of a real structure on a complex vector space, and does the opposite of complexification. If you replace your first equation with $x^{**} = -x$ then you get a quaternionic structure on a complex vector space. These are discussed in Adam's book 'Lie Groups' for representations, but everything he says is relevant for mere vector spaces as well. He calls both of the above objects 'structure maps'. I suppose I could put this as an answer but I've typed it here now. – Paul Reynolds Jul 7 '13 at 8:48
It should be apparent to you by now that there is no standard linear algebra textbook which does what you need, and if you find some more or less obscure counterexample it will be of little use; that someone somewhere wrote this down does not mean that giving that as a reference wil be useful for anyone! Maybe it is worth stating the properties you want, possibly —since the proofs should not be hard— omiting all details about the proofs? – Mariano Suárez-Alvarez Jul 9 '13 at 10:01

Fulton and Harris, Representation Theory: A First Course, p. 444, section 26.3, the definition of real representation.

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I do not understand. I need a source for reference, the book must contain a list of properties of this notion. – Sergei Akbarov Jul 7 '13 at 11:05
Which properties do you need? As a I wrote above, a complex vector space is the complexification of a real vector space just when it bears a conjugate linear involution operator; the real vector space is the set of fixed points of the involution. So the category of conjugate linear involutions is isomorphic to the category of real vector spaces, and the properties are just those of real vector spaces. There can't be any more or fewer properties. – Ben McKay Jul 7 '13 at 13:39
"a complex vector space is the complexification of a real vector space just when it bears a conjugate linear involution operator" -- I need this and the construction of involution on the dual space. But I think it's not nice to write this without reference. Or you mean that this is stated in the Fulton-Harris book? – Sergei Akbarov Jul 7 '13 at 13:51
This equivalence is stated in Fulton and Harris, p. 444. But it is easy to prove anyway. – Ben McKay Jul 7 '13 at 13:54
It is easy to prove of course, but it is not good to write this proof in a paper, or to use this result without reference. Ben, I don't like the formulation of this fact that Fulton and Harris give at p.444. This is not for normal mathematician, you must have some background in representation theory to recognize what I need in what they say. I believe there are texts with simpler formulations. And, by the way, I need also the constructions of real and imaginary parts, and their properties. Of course this is trivial, but I'd like to have a book on my table. – Sergei Akbarov Jul 7 '13 at 14:07

Perhaps you might be interested in Section 4.3 of Linear Algebra and Geometry by Shafarevich and Reznikov (which is my favourite Linear Algebra textbook, by the way), in which a complex structure on a vector space is introduced in a coordinate-free way starting on page 150.

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Remizov, actually... I deduce from this mistake, that you are Russian. Приятно познакомиться, сейчас погляжу. – Sergei Akbarov Jul 7 '13 at 11:29
I think this is not what I need. They introduce the operation of taking complex conjugate vector, but only in the case when $X$ is a complexification of a given real vector space $Y$, not for an arbitrary complex vector space $X$... Or I missed something? – Sergei Akbarov Jul 7 '13 at 11:59
A complex vector space is the complexification of a real vector space just when it bears a conjugate linear involution operator; the real vector space is the set of fixed points of the involution. – Ben McKay Jul 7 '13 at 13:37
No doubts, but which book to cite? – Sergei Akbarov Jul 7 '13 at 13:47

The result you want is a special case of Lemme 26, V, no 21 in Serre, Groupes Algebriques... 1959; also Borel, Linear Algebraic Groups, I, 14.1; also Milne's online Algebraic Geometry notes, 16.14.

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Excuse me, the numbers I, 14.1 in the reference to Borel, what do they mean? I can't find... – Sergei Akbarov Jul 9 '13 at 11:05