Section III.4.5

Irreducible modules over a Lie algebra 51(2)

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Problem III.4.5.1

Define on VnV_{n} the structure of an su(2)\mathfrak {su}(2)-module, using formulas (2), returning to the basis elements

k1=12(e−1+e2),k2=i2(e−1−e1),k3=−i2e0 \mathbf{k}_{\mathbf{1}}=\frac{1}{2}\left(e_{-1}+e_{2}\right), \quad \mathbf{k}_{2}=\frac{i}{2}\left(e_{-1}-e_{1}\right), \quad \mathbf{k}_{3}=-\frac{i}{2} e_{0}

and associating with them the linear operators K1,K2,K3K_{1}, K_{2}, K_{3}.

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Problem III.4.5.2

Let L=⟨e−1,e0,e1⟩L= \left\langle e_{-1}, e_{0}, e_{1}\right\rangle be a simple Lie algebra with multiplication table (1) over an algebraically closed field FF of characteristic p>2p>2. Consider the square matrices of order pp

E−1=∥0γ10…0000γ2…00………………000…γp−20000…0γp−1γ000…00∥,E1=∥00…00β10…00001…000………………00…10000…010∥,E0=[E−1,E1]=diag⁡(λ,λ+2,…,λ+2(p−2),λ+2(p−1)). \begin{aligned} E_{-1}= & \left\Vert \begin{smallmatrix} 0 & \gamma _{1} & 0 & \ldots & 0 & 0 \\ 0 & 0 & \gamma _{2} & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & \gamma _{p-2} & 0 \\ 0 & 0 & 0 & \ldots & 0 & \gamma _{p-1} \\ \gamma _{0} & 0 & 0 & \ldots & 0 & 0 \end{smallmatrix}\right\Vert , E_{1}=\left\Vert \begin{smallmatrix} 0 & 0 & \ldots & 0 & 0 & \beta \\ 1 & 0 & \ldots & 0 & 0 & 0 \\ 0 & 1 & \ldots & 0 & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & 1 & 0 & 0 \\ 0 & 0 & \ldots & 0 & 1 & 0 \end{smallmatrix}\right\Vert , \\ & E_{0}=\left[E_{-1}, E_{1}\right]=\operatorname {diag}(\lambda , \lambda +2, \ldots , \lambda +2(p-2), \lambda +2(p-1)). \end{aligned}

Verify that the correspondence ei↦Ei,i=−1,0,1e_{i} \mapsto E_{i}, \quad i=-1,0,1, establishes an irreducible matrix representation of the Lie algebra LL, depending on three parameters (λ,β,γ0)\left(\lambda , \beta , \gamma_{0}\right). Here

γk=βγ0+kλ+k(k−1),k=1,2,…,p−1 \gamma _{k}=\beta \gamma _{0}+k \lambda +k(k-1), \quad k=1,2, \ldots , p-1

(This entirely new situation is typical for representations over a field of finite characteristic.)

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