 Solution

Use the generators $e,f,h$ of the sl2 Lie algebra with $[h,e]=2e$, $[h,f]=-2f$, and $[e,f]=h$. For a finite-dimensional Lie algebra representation, $h$ is diagonalizable with integral weights. Its formal character of a weight module isHere $q$ is a formal variable; equivalently the expression is ${\mathrm{tr}}_V(q^h)$, or the formal sum ${\sum }_m(\dim V_m)e^m$ with $e^m=q^m$. This records every weight multiplicity, rather than only the dimension.

The classification of finite-dimensional sl2 representations says that $L_d$, the irreducible of highest weight $d\ge 0$, has the weight string $d,d-2,\dots ,-d$, each with multiplicity one. HenceThe quotient denotes the displayed Laurent polynomial and has the removable value $d+1$ at $q=1$. Formal characters are additive on direct sums, so these formulas determine the character of every finite-dimensional module.