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 ClebschGordan 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!
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$\begingroup$ In "they are no longer irreducible" what do you refer to as "they"? $\endgroup$– YCorAug 5, 2021 at 21:47

$\begingroup$ @YCor Hello! I am referring to the $V_n$. $\endgroup$– Catherine LiAug 5, 2021 at 21:57

2$\begingroup$ 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. $\endgroup$– Joshua GrochowAug 6, 2021 at 1:53

3$\begingroup$ 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 ClebschGordan. See the book of Jantzen (Representations of Algebraic Groups: Second Edition)? $\endgroup$– Wilberd van der KallenAug 6, 2021 at 6:50
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