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This is a continuation of various questions about Chevalley groups over rings, cf these two questions (and a rather bad question of mine here). Consider a semisimple Lie algebra $\mathfrak g$ over $\mathbb C$ and let $G$ be the associated simply-connected algebraic group over $\mathbb C$. Let $A$ be a $\mathbb C$-algebra. We may construct the map algebra $ \mathfrak g_A := \mathfrak g \otimes_{\mathbb C} A $, which is a Lie algebra with bracket given by $ [X \otimes f, Y \otimes g] = [X, Y]\otimes fg $ for all $X, Y \in \mathfrak g$ and $f, g \in A$. We then have the enveloping algebra $ U(\mathfrak g_A) $ of $ \mathfrak g_A $. As this is a cocommutative Hopf algebra over $\mathbb C$, it should be the hyperalgebra of an associated $ \mathbb C $-group scheme (which will in general not be of finite type over $\mathbb C$, of course). I am wondering what this group scheme is. Note that the naive guess of $ G_A = G \times_{ \textrm{Spec} \mathbb C } \textrm{Spec} A $ is not correct; indeed, this is not even a $\mathbb C$-group scheme functor (it is only an $A$-group scheme functor).

I have the following guess: there should be a $\mathbb C$-group scheme representing the functor $B \mapsto \textrm{Hom}_{\mathbb C-\textrm{schemes}}( \textrm{Spec} \big(A \otimes_{\mathbb C} B), \, G \big)$ for all $\mathbb C$-algebras $B$. In the case that $A$ is a finite-dimensional $\mathbb C$-algebra, this functor is represented by the Weil restriction of $G_A$. However, in general, I don't know if (1) this functor is representable by a $\mathbb C$-group scheme, and (2) if it is, I don't know if its hyperalgebra is actually $U(\mathfrak g_A)$. To refine the question at the end of the first paragraph above, I am wondering if (1) and (2) are actually true, in which case this does answer my question.

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    $\begingroup$ A (co-commutatie) Hopf algebra $R$ over $\mathbf{C}$ is equipped with an "identity section" $R \rightarrow \mathbf{C}$ as $\mathbf{C}$-algebras and a co-multiplication $R \rightarrow R \otimes_{\mathbf{C}} R$ as $\mathbf{C}$-algebras. For the Hopf algebra $U(\mathfrak{g}_A)$ over $A$, what such data do you have in mind? Also, your proposed functor fails to be representable for $A=\mathbf{C}[t]$ and $G$ any linear algebraic $\mathbf{C}$-group with positive dimension. (Weil restriction through an algebra map not of finite dimension arises for "affine Grassmannians", which are not representable.) $\endgroup$
    – user76758
    Commented Dec 26, 2013 at 12:37
  • $\begingroup$ 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? $\endgroup$
    – Chuck Hague
    Commented Dec 26, 2013 at 16:07
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    $\begingroup$ Dear Chuck: Your $\epsilon$ defines a homomorphism from $U:=U(\mathfrak{g}_A)$ into $A$, not into $\mathbf{C}$; what does $\epsilon$ do to the natural image of $A$ in $U$? Likewise, your $\Delta$ is a map from the $A$-algebra $U$ into $U\otimes_A U$, not into $U \otimes_{\mathbf{C}}U$; if you think $\Delta$ is a map into the latter then where do $aX$ and $a$ go under this map for $a \in A$? For representability when $A=\mathbf{C}[t]$, if $G=\mathbf{G}_a$ then this amounts to a "universal polynomial" in $t$; considering nilpotent coefficients rules it out (EGA IV$_3$, 8.14.2 makes it rigorous). $\endgroup$
    – user76758
    Commented Dec 26, 2013 at 16:41
  • $\begingroup$ 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. $\endgroup$
    – Chuck Hague
    Commented Dec 26, 2013 at 17:40
  • $\begingroup$ Oh, OK, so you're taking the Lie algebra $\mathfrak{g}_A$ over $A$ and viewing it instead just as a Lie algebra over $\mathbf{C}$ for the formation of the universal enveloping algebra (so it isn't an $A$-algebra, etc.). Then your question makes sense, though for $\mathbf{C}$-finite $A$ do you know if the Weil restriction ${\rm{R}}_{A/\mathbf{C}}(G_A)$ has coordinate ring equal to that enveloping algebra? (I would be surprised if this is true, but maybe there is a simple trick.) $\endgroup$
    – user76758
    Commented Dec 26, 2013 at 18:00

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