Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 6 / 5

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.

Choose a Cartan subalgebra, simple roots ${\alpha }_1,\dots ,{\alpha }_r$, and root vectors $e_i,f_i$ normalized by $[e_i,f_i]=h_i={\alpha }_i^{\vee }$. LetHere the coroots lie in the Cartan subalgebra, and ${\alpha }_i({\rho }^{\vee })=1$. This last identity is the Weyl vector identity applied to the dual root system. For distinct simple roots, ${\alpha }_i-{\alpha }_j$ is not a root, hence $[e_i,f_j]=0$. ConsequentlyThese are exactly the sl2 Lie algebra relations. For nonzero $\mathfrak{g}$ the resulting homomorphism from ${\mathfrak{sl}}_2$ is nonzero and therefore injective, since ${\mathfrak{sl}}_2$ is simple. Its image is a Principal sl2 subalgebra. Equivalently its raising element is a principal nilpotent element, a nilpotent element with Lie algebra centralizer of dimension equal to the rank. The construction works for a semisimple Lie algebra with several simple factors by taking the triple in every factor, giving one diagonally embedded ${\mathfrak{sl}}_2$. This is the construction of a principal sl2 triple.

For a representation $V$, restrict to this subalgebra and take its formal character. If $V_{\mu }$ is the weight space for the original Cartan subalgebra, the resulting q-character of a highest-weight representation, or the analogous q-character for any finite-dimensional representation, isIt is independent of conjugating the chosen principal triple. In particular a root $\beta ={\sum }_i{k}_i{\alpha }_i$ has grading $\beta (h)=2{\sum }_i{k}_i$. Some conventions use $q^{h/2}$ instead; that convention divides all exponents here by two. The normalization above agrees with the ordinary sl2 Lie algebra character in the preceding part.

Under the Principal sl2 subalgebra, a root space of root $\beta$ has $h$-eigenvalue $\beta (2{\rho }^{\vee })=2\mathrm{ht}(\beta )$, where the height is the sum of its simple-root coefficients. The six positive roots of a root system of the G2 root system have heightsEach positive and negative root space has dimension one. The two-dimensional Cartan subalgebra contributes two copies of the zero weight. Therefore the Principal q-character of the G2 adjoint representation isEquivalently,The classification of finite-dimensional sl2 representations therefore gives the actual restricted-module decomposition $\mathfrak{g}\cong L_{10}\oplus L_2$, of dimensions $11+3=14$. In the convention using $q^{h/2}$, the same result is $q^5+q^4+q^3+q^2+2q+2+2q^{-1}+q^{-2}+q^{-3}+q^{-4}+q^{-5}$. Both forms encode the same two weight strings.