# $\mathfrak{sl}_2(\mathbb{Z})$'s properties as a Lie algebra over a ring

I was wondering what was known about $$\mathfrak{sl}_2(\mathbb{Z})$$ as a Lie algebra; in particular, what is known about its representation theory? I know of some texts which treat Lie algebras generally over rings, but I haven't seen any that specifically discuss this example. It would be wonderful if there existed some analogue of a Clebsch-Gordan decomposition for tensor products of $$V_n$$ (by which I mean the irreps in the complex case), even if they are no longer irreducible over $$\mathbb{Z}$$. Thank you very much!

• In "they are no longer irreducible" what do you refer to as "they"?
– YCor
Aug 5, 2021 at 21:47
• @YCor Hello! I am referring to the $V_n$. Aug 5, 2021 at 21:57
• Considering complex or integer representations (eg the Vn are defined over Z)? If integer representations, what do you mean by irreducibility? Eg the multiples of 2 in Vn form a nontrivial proper submodule. Aug 6, 2021 at 1:53
• One must choose a form of the $V_n$. For instance, one may define them to be symmetric powers of the standard two dimensional representation. Then $V_n\otimes V_m$ has a "good filtration" whose layers are given by Clebsch-Gordan. See the book of Jantzen (Representations of Algebraic Groups: Second Edition)? Aug 6, 2021 at 6:50