Timeline for Chevalley groups over $k[t]/t^n$
Current License: CC BY-SA 3.0
14 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Nov 20, 2013 at 12:23 | comment | added | Dave Anderson | I don't know about standard, but there is the appendix to Mustata's "Jet schemes of lci canonical singularities", and some papers of Bérczi and Szenes might have good references. All this may or may not be far from what you need. (Alan Stapledon and I also wrote a paper dealing with jet schemes of groups but that's certainly not a standard reference!) | |
| Nov 19, 2013 at 17:56 | comment | added | Chuck Hague | Okay, great - I'll look that up. Is there a standard reference for jet schemes of groups? | |
| Nov 18, 2013 at 22:07 | comment | added | Dave Anderson | @Chuck, another fact (that may help in searches) is that this "$G(R_{n+1})$" (via Weil restriction, as Marguax says) is also known as the $n$th jet scheme of $G$, sometimes written $G_n$. There is a sequence $1 \to K \to G_n \to G \to 1$, realizing $G_n$ as a unipotent extension of $G$. | |
| Nov 17, 2013 at 3:40 | comment | added | Chuck Hague | That makes sense - I was confusing $G(A)$ with $G_A$. The viewpoint that gives $G \ltimes \mathfrak g$ when $n=2$ is exactly what I'm looking for. I'm unfamiliar with Weil restriction so I will take a look at that. Thanks for your patience in answering my questions! | |
| Nov 17, 2013 at 2:53 | comment | added | Marguax | @Chuck: Every linear algebraic group is a Zariski-closed subgroup of some SL$_N$ (e.g., GL$_n$ inside SL$_{2n}$ via $n \times n$ blocks $g$ and $g^{-1}$). You are using the Weil restriction viewpoint. It is very much non-reductive (I described it for $n=2$). You write the group of points as proxy for a linear algebraic group, but these are really not the same thing; it is hard to discuss things precisely that way. For example, SL$_n(k)$ is just an abstract group; the $k$-variety SL$_n$ is an algebraic group. You can make sense of "points" of the latter in a $k$-algebra, but not for the former. | |
| Nov 17, 2013 at 1:38 | comment | added | Chuck Hague | Ah, I think I see. I'm confusing group schemes over $\mathbb Z$ with group schemes over $k$. | |
| Nov 17, 2013 at 1:34 | comment | added | Chuck Hague | Let's say that $G$ is simply-connected. Then we can embed $G$ in $SL_N(k)$ for some $N$. Now consider the $k$-variety $GL_N(R_n)$ of matrices with coefficients in $R_n$. Reduction of scalars gives a map $k[GL_N(k)] \to k[GL_n(R_n)]$ so we can consider the subvariety of $GL_N(R_n)$ defined by the pushforward of $I$ in $k[GL_N(R_n)]$; this is what I thought $G(R_n)$ was, although perhaps this is naive. Somewhere I must be confused, though, because this means in particular that if $G = SL_N(k)$ then $G(R_n)$ is not the same as $SL_N(R_n)$. | |
| Nov 17, 2013 at 1:10 | comment | added | Marguax | What does $G(R_n)$ as an "algebraic group" mean? The $R_n$-group scheme $G_{R_n}$ (coordinate ring $R_n\otimes_k k[G]$), or "$G(R_n)$ as a $k$-group" -- rigorously the smooth connected Weil restriction of scalars ${\rm{Res}}_{R_n/k}(G_{R_n})$ whose group of $k$-points is $G(R_n)$ and has a massive unipotent radical? (For $n=2$ it is $G\ltimes \mathfrak{g}$ via adjoint action.) What is your definition of "regular function" on $G(A)$? Chevalley groups $G$ are over $\mathbf{Z}$; the $A$-group scheme $G_A$ has coordinate ring $A\otimes_{\mathbf{Z}} \mathbf{Z}[G]$ and Lusztig computes that. | |
| Nov 16, 2013 at 22:51 | comment | added | Jim Humphreys | A further comment is that you deal with both a group scheme and an abstact group (which over arithmetic local rings may be just a finite group). The language such as Hopf algebras of regular functions or hyperalgbras belongs with the group scheme, whereas the structure and automorphisms of the group of points over a ring has been explored a lot. This goes back at least as far as a 1969 paper by E. Abe (one of 32 returned by MathSciNet using a search for "Chevalley group" and "local ring"). | |
| Nov 16, 2013 at 22:20 | comment | added | Chuck Hague | In this case, I suppose by "Chevalley group" I would be happy to consider the case where $G$ is a semisimple group over $k$ (and I don't care too much about isogeny, so we can take $G$ to be adjoint or simply-connected if that would make things easier). | |
| Nov 16, 2013 at 22:01 | comment | added | Jim Humphreys | Some of that large literature might well be useful here, though I'm not sure. Certainly local rings come up. In any case it's always desirable to specify exactly what you mean by "Chevalley group", since that notion has evolved somewhat since Chevalley's original construction. | |
| Nov 16, 2013 at 21:56 | history | edited | Chuck Hague | CC BY-SA 3.0 |
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| Nov 16, 2013 at 21:51 | history | asked | Chuck Hague | CC BY-SA 3.0 |