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Nov 5 |
awarded | Yearling |
Sep 30 |
awarded | Explainer |
Sep 24 |
awarded | Autobiographer |
Jul 2 |
awarded | Curious |
Feb 4 |
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The structure of symmetric powers of finite-dimensional local rings
added 151 characters in body |
Feb 4 |
asked | The structure of symmetric powers of finite-dimensional local rings |
Dec 26 |
comment |
Enveloping algebras of map algebras as hyperalgebras of algebraic groups
Ok, thanks - that's very helpful! I appreciate it. I'll check out the reference and think about how to lift the Lie algebra statement to a hyperalgebraic statement. |
Dec 26 |
awarded | Civic Duty |
Dec 26 |
comment |
Enveloping algebras of map algebras as hyperalgebras of algebraic groups
I think $U(\mathfrak g_A)$ sees more than just the underlying vector space of $A$; multiplication in $A$ is part of the Lie bracket in $\mathfrak g_A$, so commutation in $U$ sees the algebra structure of $A$. For example, for the dual numbers $A = \mathbb C[x] / \langle x^2 \rangle$, the bracket relations in $\mathfrak g_A$ give a natural isomorphism $ \mathfrak g_A \cong \mathfrak g \ltimes \mathfrak g^{ab} $, where by $\mathfrak g^{ab}$ I mean the abelian Lie algebra with underlying space $\mathfrak g$. And this is indeed the Lie algebra of the tangent bundle $G \ltimes \mathfrak g$ of $G$. |
Dec 26 |
comment |
Enveloping algebras of map algebras as hyperalgebras of algebraic groups
Actually, your comment about the Weil restriction for $\mathbb C$-finite $A$ is part of the motivation for my question - as a particular special case I've been trying to navigate the literature on jet group schemes (so this is the case where $A = \mathbb C[t]/ \langle t^n \rangle$ for some $n > 0$ and then considering the Weil restriction). It seems implicit in some of the literature that we do indeed obtain this enveloping algebra $U$ in this case - for example, it's true when we consider the dual numbers - but I'm having a hard time navigating the literature. |
Dec 26 |
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Enveloping algebras of map algebras as hyperalgebras of algebraic groups
I'm considering $\mathfrak g_A$ purely as a Lie algebra (in general infinite-dimensional) over $\mathbb C$, in which case I would expect that the augmentation $\epsilon$ should take values in $\mathbb C$, not $A$, unless I'm confused here? The element $aX$ doesn't make sense in $U$ as far as I can tell; that is, I don't expect $U$ to be an $A$-algebra. Only $X \otimes a$ makes sense, for $X \in \mathfrak g$ and $a \in A$, in which case $\Delta(X \otimes a) = (X \otimes a) \otimes 1 + 1 \otimes (X \otimes a)$. Thanks for the EGA reference - I'll look at that. |
Dec 26 |
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Enveloping algebras of map algebras as hyperalgebras of algebraic groups
I'm thinking of the Hopf algebra structure where we put $\epsilon(X) = 0$ and $\Delta X = X \otimes 1 + 1 \otimes X$ for all $X \in \mathfrak g_A \subseteq U(\mathfrak g_A)$ and then extend multiplicatively. Thanks for the info on representability - is there a good reference for reading about representability/non-representability of functors like this? |
Dec 25 |
asked | Enveloping algebras of map algebras as hyperalgebras of algebraic groups |
Nov 28 |
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Equivalence of Versions of the Affine Grassmannian
There's a discussion of the Laurent polynomial point of view in chapter 13.2 of Kumar's Kac-Moody Groups book (he gives a relation between $G(\mathbb C[t,t^{-1}])$ and the group denoted by $\mathcal G^{min}$). He relates the homogeneous spaces associated to $G(\mathbb C[t,t^{-1}])$ and $G(\mathbb C((t)))$ there, although I don't know much about the details. |
Nov 19 |
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Chevalley groups over $k[t]/t^n$
Okay, great - I'll look that up. Is there a standard reference for jet schemes of groups? |
Nov 17 |
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Chevalley groups over $k[t]/t^n$
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 |
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Chevalley groups over $k[t]/t^n$
Ah, I think I see. I'm confusing group schemes over $\mathbb Z$ with group schemes over $k$. |
Nov 17 |
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Chevalley groups over $k[t]/t^n$
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 16 |
awarded | Necromancer |
Nov 16 |
comment |
Chevalley groups over $k[t]/t^n$
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). |