Past exam of the Lie algebras course of the University of Oxford / 2011 / 2

By the classification of finite-dimensional sl2 representations over $\mathbb{C}$, every finite-dimensional module is a direct sum of simple modules $L_m$, indexed by integers $m\ge 0$. The module $L_m$ has dimension $m+1$ and a basis $v_0,\dots ,v_m$ withwith out-of-range vectors interpreted as zero. In particular $v_0$ is a highest weight vector of weight $m$. The weights are $m,m-2,\dots ,-m$, each with multiplicity one. The multiplicities of the $L_m$ in a finite module determine that module up to isomorphism.

In the simple module $L_m$, weight $0$ occurs once if $m$ is even and never if $m$ is odd. Weight $1$ occurs once if $m$ is odd and never if $m$ is even. Thus each simple summand contributes exactly one to $\dim V_0+\dim V_1$.

Complete reducibility of semisimple Lie algebra representations writes $V={⨁}_m{L}_m^{\oplus a_m}$. Taking the two weight spaces commutes with this finite direct sum, soThe right side counts irreducible summands with multiplicity, independently of a chosen decomposition.

Every $2\times 2$ matrix has the unique decompositionBoth summands define invariant subspaces for the Adjoint representation of a Lie algebra. Scalar matrices commute with everything, so $\mathbb{C}I$ is $L_0$. The traceless matrices form ${\mathfrak{sl}}_2$ with basis $e,h,f$ of weights $2,0,-2$. Their brackets satisfy $[e,f]=h$, $[h,e]=2e$, $[h,f]=-2f$; the raising and lowering operations connect all three weight lines. This is the simple module $L_2$. ThereforeAs a check, its zero-weight space has dimension two and its one-weight space is zero, agreeing with part (ii).

On a monomial $X^a{Y}^b$, direct differentiation givesFor example $(EF-FE)(X^a{Y}^b)=[a(b+1)-b(a+1)]X^a{Y}^b=(a-b)X^a{Y}^b$. The same calculation with $H$ givesThese are exactly the brackets of $h,e,f$, so the linear map $\rho$ respects every bracket and is a Lie algebra representation.

For $m\ge 0$, let $W_m$ be the space of homogeneous polynomials of total degree $m$. Each operator preserves total degree. Every polynomial is a finite sum of homogeneous polynomials, soIn $W_m$, the monomials $v_j=X^{m-j}{Y}^j$ have distinct weights $m-2j$ and exactly the actions described in part (i). To see simplicity directly, a nonzero invariant subspace is invariant under $H$ and hence contains one of its weight components: polynomial interpolation in $H$ projects onto each of its distinct eigenspaces. Starting with a nonzero multiple of $v_j$, repeated application of $E$ reaches $v_0$, and repeated application of $F$ then reaches all $v_j$. All coefficients used are nonzero in characteristic zero. Therefore $W_m$ is simple and isomorphic to $L_m$.

ThusThis is an algebraic direct sum, not an infinite product or completion. Its simple submodules are precisely the $W_m$: they are pairwise nonisomorphic and each occurs once, so projection onto homogeneous degrees and Schur's lemma exclude diagonal copies between distinct degrees. Complete reducibility here follows from the explicit decomposition, rather than applying a finite-dimensional theorem directly to the infinite-dimensional $W$.