For Lie groups (or algebraic groups over fields) there is a strong relation between the cohomology of the group and the cohomology of its Lie algebra. Some MO-question where this is discussed can be found here or here. I am wondering what happens to this relation for groups like $SL_n(k[T])$, with $k$ a field.

For $SL_n(k[T])$, we know of "homotopy invariance": there is an isomorphism of homology groups $H_\bullet(SL_n(k),\mathbb{Z})\cong H_\bullet(SL_n(k[T]),\mathbb{Z})$. This is due to Knudson (ASENS 30 (1997), 385-416), in the case of $n=2$ it follows from Nagao's amalgam decomposition $SL_2(k[T])\cong SL_2(k)\ast_{B_2(k)}B_2(k[T])$.

On the other hand, this is not true for the Lie algebra; the homology of $\mathfrak{sl}_n\otimes k[T]$ is a lot larger than the homology of $\mathfrak{sl}_n$. This follows from results of Garland-Lepowsky, see also this MO-question. In the special case $n=2$, one can see more directly that $\mathfrak{sl}_2\ast_{\mathfrak{b}_2}(\mathfrak{b}_2\otimes k[T])\to \mathfrak{sl}_2\otimes k[T]$ is not an isomorphism.

My questions concern the relation - if any - between these results:

  1. Is there a relation between these results? Maybe the Lie algebra $\mathfrak{sl}_2\otimes k[T]$ is just the wrong thing to look at in this context? Or is it simply the case that the close relation between homology of groups and Lie algebras breaks down in this case?
  2. The group homology result is proved by looking at the action of $SL_n(k[T])$ on the associated Bruhat-Tits building. Soulé computed a fundamental domain, and Knudson worked out the resulting spectral sequence. Can the computation of the Lie algebra homology for $\mathfrak{sl}_n\otimes k[T]$ be understood from this perspective? (Maybe the Lie algebra acts on suitable functions on the building...)
  3. More generally, is there an "additive analogue of geometric group theory" which would allow to study structure and cohomology of infinite-dimensional Lie algebras by actions on some more or less geometric objects?

Maybe a bonus question: is anything known about the Lie algebra cohomology of $\mathfrak{sl}_2\otimes k[T_1,T_2]$?

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    $\begingroup$ I think that the Lie algebra (co)homology of $\mathfrak{sl}_n\otimes A$ for an arbitrary $\mathbf{Q}$-algebra $A$ is described in Loday's book "cyclic homology". $\endgroup$ – YCor Aug 2 '14 at 8:54
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    $\begingroup$ Thanks, I will have to look it up (I don't have a copy on me right now). All I remembered from Loday's book was that there were formulas for the stable case $\mathfrak{sl}_\infty\otimes A$ and the stabilization from $\mathfrak{sl}_{n-1}$ to $\mathfrak{sl}_n$. In these cases, the relation is the one between algebraic K-theory and cyclic homology (viewed as additive version of K-theory). Part of the motivation behind the question is wanting to understand how much of the stable relation is true on the unstable level. $\endgroup$ – Matthias Wendt Aug 2 '14 at 15:24
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    $\begingroup$ Ctd: Wasn't there also a conjecture in Loday's book on the relation between stabilization and the Hodge-type decomposition for homology of $\mathfrak{sl}_\infty\otimes A$? $\endgroup$ – Matthias Wendt Aug 2 '14 at 15:25
  • $\begingroup$ Actually I don't remember well. I'm only familiar with the case of homology of degree 2, in which case $H_2(\mathfrak{sl}_n(A))$ is known to be isomorphic to $HC_1(A)$ for all $n\ge 2$. $\endgroup$ – YCor Aug 3 '14 at 0:00

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