Where can I find information on root systems where the inner product is other than the standard (all positive) signature?

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    $\begingroup$ "Root system" is a purely algebraic concept in finite-dimensional vector spaces $V$ over any field $k$ of characteristic 0; it doesn't need inner products. See Ch. VI of Bourbaki's Lie Groups and Lie Algebras for more on this. It is proved there that $V=k\otimes_{\mathbf{Q}}V_0$ for the $\mathbf{Q}$-span $V_0$ of $\Phi$, irreducible decomposition is unique in a strong sense, and $V$ is absolutely irreducibile as a $W(\Phi)$-representation for irreducible $(V,\Phi)$. Thus, if $(V,\Phi)$ is irreducible then $V$ has a $W(\Phi)$-invariant non-degenerate quadratic form unique up to scaling! $\endgroup$
    – nfdc23
    Jun 16, 2017 at 2:04
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    $\begingroup$ Yes, I know, but please look at the early parts of Ch. VI of Bourbaki that I mentioned. What matters isn't the inner product, but rather the linear form $a^{\vee}: v \mapsto 2(v|a)/(a|a)$ satisfying $a^{\vee}(a) = 2$, in terms of which the reflection $r_a$ is given by the purely algebraic formula $v \mapsto v - a^{\vee}(v)a$. Writing things in terms of pairs $(a, a^{\vee})$ avoids any mention of Euclidean structure and makes the theory much more algebraic (and it is this that arises intrinsically from connected semisimple Lie groups and algebraic groups). Bourbaki Ch. VI explains it well! $\endgroup$
    – nfdc23
    Jun 16, 2017 at 3:50
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    $\begingroup$ @unknown: Kac considers in his book fairly general root systems. The Kac-Moody algebras which have an invariant inner product are called symmetrizable. It is indefinite in most cases. $\endgroup$ Jun 16, 2017 at 12:11
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    $\begingroup$ @unknown: Kac's book is called "Infinite-dimensional Lie algebras." He starts with a symmetrizable (generalized) Cartan matrix, which essentially tells you "angles" between "simple roots" and their relative "lengths". The point is that once you choose these lengths and angles, you have defined an inner product. The root systems you get from this are usually called "Kac-Moody root systems". (Those with signature +...+0 are "affine" root systems.) I think that only certain signatures can arise from a Cartan matrix. For example, in rank 2, only +++, ++-, and ++0 occur. $\endgroup$ Jun 17, 2017 at 3:18
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    $\begingroup$ @unknown: There is also an apparently more general approach that may or may not be what you want. Coincidentally, about 6 hours ago (before seeing your question!) I was looking at Kyoji Saito's "Extended affine root systems. I. Coxeter transformations". He starts with an arbitrary inner product and defines a root system with respect to that inner product. I don't know much about it and in particular don't know what any of it means in the various contexts where people use root systems. But it is intriguing. $\endgroup$ Jun 17, 2017 at 3:19


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