Timeline for Reference request for generalized root systems
Current License: CC BY-SA 3.0
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| Jun 17, 2017 at 3:19 | comment | added | Nathan Reading | @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. | |
| Jun 17, 2017 at 3:18 | comment | added | Nathan Reading | @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. | |
| Jun 16, 2017 at 17:45 | comment | added | unknown | @Friedrich: my knowledge of Kac-Moody algebras is that they correspond to root systems with signature (++...++0); these have been classified and I'm aware of the classification. Do you have a more specific reference for Kac's book? especially if it handles other signatures as well | |
| Jun 16, 2017 at 12:11 | comment | added | Friedrich Knop | @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. | |
| Jun 16, 2017 at 10:55 | comment | added | nfdc23 | My point is that if you are trying to simply change the definition of a root system to rest on indefinite signature then there are no examples (other than the trivial operation of taking a direct sum of two usual root systems where one is given a positive-definite Euclidean structure and the other given a negative-definite one). For the third time: please look at the early parts of Chapter VI of Bourbaki. If your interest goes beyond this (apparent) change in the definition of "root system" to more general (finite?) reflection groups then that is not apparent in your question. | |
| Jun 16, 2017 at 7:53 | comment | added | Dirk | Do you want root systems or reflection groups? In the latter case, you might want to look at classifications of bilinear forms and quadratic forms. The corresponding reflection groups are often also known. | |
| Jun 16, 2017 at 4:27 | comment | added | unknown | but I would think at some point the signature enters the picture...for example in 4 dimensional real space all reflection groups have been classified for signature (++++)...I don't think the same has been done for (+++-) or (++--) case | |
| Jun 16, 2017 at 3:50 | comment | added | nfdc23 | 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! | |
| Jun 16, 2017 at 3:20 | comment | added | unknown | I'm thinking along the lines of root systems arising from finite real reflection groups where the reflections $\phi_w : v \to v - 2 w(v .w)/(w.w)$ are with with respect to a non euclidean inner product $(x.y)$ | |
| Jun 16, 2017 at 2:13 | comment | added | nfdc23 | In particular, there is no such thing as "generalized root system" in the sense you seem to be imagining. There is an important role for Euclidean structure in the deeper parts of the development of root systems, but they should not be mentioned in the initial definitions. The historical origins with compact connected semisimple Lie groups might suggest imposing a Euclidean structure, but this is not good to do at the start because it is not intrinsic. Connected semisimple groups over arbitrary fields yield root systems (over $\mathbf{Q}$) and these have no "natural" Euclidean structure. | |
| Jun 16, 2017 at 2:04 | comment | added | nfdc23 | "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! | |
| Jun 16, 2017 at 1:15 | history | asked | unknown | CC BY-SA 3.0 |